A compiler for a domain-specific language for ordinary differential equations (ODEs) of the following form:
DiffSL Language
The DSL is designed to be an easy and flexible language for specifying +DAE systems and is based on the idea that a DAE system can be specified by a set +of equations of the form:
$$ M(t) \frac{d\mathbf{u}}{dt} = F(\mathbf{u}, t) $$
As an example, the following code defines a classic DAE testcase, the Robertson -(1966) problem, which models the kinetics of an autocatalytic reaction, given -by the following set of equations:
where \( \mathbf{u}$ \) is the vector of state variables and \( t \) is the time. The DSL +allows the user to specify the state vector \( \mathbf{u} \) and the RHS function \( F \). +Optionally, the user can also define the derivative of the state vector \( d\mathbf{u}/dt \) +and the mass matrix \( M \) as a function of \( d\mathbf{u}/dt \) (note that this function should be linear!).
The user is also free to define an an arbitrary number of intermediate +scalars and vectors of the users that are required to calculate \( F \) and \( M \).
To illustrate the language, consider the following simple example of a logistic growth model:
$$ -\begin{align} -\frac{dx}{dt} &= -0.04x + 10^4 y z \ -\frac{dy}{dt} &= 0.04x - 10^4 y z - 3 \cdot 10^7 y^2 \ -0 &= x + y + z - 1 -\end{align} +\frac{dN}{dt} = r N (1 - N/K) $$
The DiffSL code for this problem is as follows:
in = [k1, k2, k3] -k1 { 0.04 } -k2 { 10000 } -k3 { 30000000 } -u_i { - x = 1, - y = 0, - z = 0, -} -dudt_i { - dxdt = 1, - dydt = 0, - dzdt = 0, -} -M_i { - dxdt, - dydt, - 0, -} -F_i { - -k1 * x + k2 * y * z, - k1 * x - k2 * y * z - k3 * y * y, - 1 - x - y - z, -} -out_i { - x, - y, - z, -} -
See the DiffSL Language section for a full description.
You will need to install the LLVM project. The easiest way to -install this is to use the package manager for your operating system. For -example, on Ubuntu you can install these with the following command:
sudo apt-get install llvm -
You can install DiffSL using cargo. You will need to indicate the llvm version you have installed using a feature flag. For example, for llvm 14:
cargo add diffsl --features llvm14-0 -
Other versions of llvm are also supported given by the features llvm4-0, llvm5-0, llvm6-0, llvm7-0, llvm8-0, llvm9-0, llvm10-0, llvm11-0, llvm12-0, llvm13-0, llvm14-0, llvm15-0, llvm16-0, llvm17-0.
llvm4-0
llvm5-0
llvm6-0
llvm7-0
llvm8-0
llvm9-0
llvm10-0
llvm11-0
llvm12-0
llvm13-0
llvm14-0
llvm15-0
llvm16-0
llvm17-0
where \( N \) is the population, \( r \) is the growth rate, and \( K \) is the carrying capacity.
To specify this model in DiffSL, we can write:
in = [r, k] u_i { N = 0.0 } F_i { r * N * (1 - N/k) } out_i { N }
Here, we define the input parameters for our model as a vector in with the growth rate r and the carrying capacity k. We then define the state vector u_i with the population N initialized to 0.0. Next, we define the RHS function F_i as the logistic growth equation. Finally, we define the output vector out_i with the population N.
in
r
k
u_i
N
0.0
F_i
out_i
Often it is useful to parameterize the system of equations using a set of input parameters. It is also useful to be able to extract certain variables from the system for further analysis. +In this section we will show how to specify inputs and outputs in the DiffSL language.
We can override the values of any scalar variables by specifying them as input +variables. To do this, we add a line at the top of the code to specify that +these are input variables:
in = [k] +k { 1.0 } +u { 0.1 } +F { k * u } +
Here we have specified a single input parameter k that is used in the RHS function F. +The value of k is set to 1.0 in the code, but this value is only a default, and can be overridden by passing in a value at solve time.
F
1.0
We can use input parameters anywhere in the code, including in the definition of other input parameters.
in = [k] +k { 1.0 } +g { 2 * k } +F { g * u } +
or in the intial conditions of the state variables:
in = [k] +k { 1.0 } +u_i { + x = k, +} +F { u } +
We can also specify the outputs of the system. These might be the state +variables themselves, or they might be other variables that are calculated from +the state variables.
Here is an example where we simply output the elements of the state vector:
u_i { + x = 1.0, + y = 2.0, + z = 3.0, +} +out_i { x, y, z } +
or we can derive additional outputs from the state variables:
u_i { + x = 1.0, + y = 2.0, + z = 3.0, +} +out { x + y + z } +
$$ +M(t) \frac{d\mathbf{u}}{dt} = F(\mathbf{u}, t) +$$
$$ +\frac{dN}{dt} = r N (1 - N/K) +$$
The DSL is designed to be an easy and flexible language for specifying -DAE systems and is based on the idea that a DAE system can be specified by a set -of equations of the form:
The primary goal of the DiffSL language is to define a system of ODEs in the following form:
$$ -M(t) \frac{d\mathbf{u}}{dt} = F(\mathbf{u}, t) +\begin{align*} +M(t) \frac{d\mathbf{u}}{dt} &= F(\mathbf{u}, t) \\ +\mathbf{u}(0) &= \mathbf{u}_0 +\end{align*} $$
where $\mathbf{u}$ is the vector of state variables and $t$ is the time. The DSL -allows the user to specify the state vector $\mathbf{u}$ and the RHS function $F$. -Optionally, the user can also define the derivative of the state vector $d\mathbf{u}/dt$ -and the mass matrix $M$ as a function of $d\mathbf{u}/dt$ (note that this function should be linear!). -The user is also free to define an an arbitrary number of intermediate -scalars and vectors of the users that are required to calculate $F$ and $M$.
The DSL allows the user to define scalars, vectors, and dense/sparse/diagonal -n-dimensional tensors. You can optionally label the elements of a vector or -tensor for later use.
For example, to define a scalar variable $k$ with value $1$, we write:
k { 1.0 } -
To define a vector variable $\mathbf{v}$ with 3 elements that are labelled, we write:
v_i { - x = 1.0, - y = 2.0, - z = 3.0, -} -
The subscript _i indicates that this is a 1D vector, and $x$, $y$, and $z$ are -defined as labels to the 3 elements of the vector. Later in the code, we could -refer to either the whole vector v_i for $\mathbf{v}$ or to the individual -elements $x$, $y$, and $z$, which are scalars.
_i
v_i
To define dense 2x3 matrix $A$ with all elements set to 1.0, we write:
A_ij { - (0:2, 0:3) = 1.0, -} -
Note the two subscript to indicate that this is a 2D tensor. The size of the -tensor is given in the brackets, and the elements are set to $1$. If we have -additional rows, we can add them as follows:
A_ij { - (0:2, 0:3) = 1.0, - (3:4, 0:3) = 2.0, -} -
We can define a sparse matrix $B$ by specifying the non-zero elements:
B_ij { - (0, 0) = 1.0, - (0, 1) = 2.0, - (1, 1) = 3.0, -} -
We can also define a diagonal identity matrix $I$ by specifying the diagonal -elements using a different range syntax:
I_ij { - (0..2, 0..2) = 1.0, -} -
We can use standard algebraic operations on variables. To refer to previously -defined variables, we use the variable name, making sure to use the correct -subscript if it is a vector or tensor.
For example, to define a scalar variable $a$ as the sum of two other scalar -variables $b$ and $c$, we write:
a { b + c } -
To define a vector variable $\mathbf{V}$ as the sum of two other vector -variables $\mathbf{u}$ and $\mathbf{w}$, we write:
v_i { u_i + w_i } -
The indexing can be used to perform translations on tensors, for example the -following will define a new tensor $C$ that is the sum of $A$ and $B^T$:
C_ij { A_ij + B_ji } -
Tensor indexing notation can also matrix-vector multiplications and any other -contraction operations. Any indices that do not appear in the output will be -summed over. For example, the following will define a new vector $v$ that is -the result of a matrix-vector multiplication:
v_i { A_ij * u_j } -
Broadcasting is also supported, for example the following will define a new -matrix $D$ that is the sum of $A$ and a vector $v$:
D_ij { A_ij + v_j } -
Here the vector $v$ is broadcast to the same shape as $A$ before the addition.
We can override the values of any scalar variables by specifying them as input -variables. To do this, we add a line at the top of the code to specify that -these are input variables:
in = [k] -k { 1.0 } -
where \( \mathbf{u} \) is the vector of state variables, \( \mathbf{u}_0 \) is the initial condition, \( F \) is the RHS function, and \( M \) is the mass matrix. +The DSL allows the user to specify the state vector \( \mathbf{u} \) and the RHS function \( F \).
Optionally, the user can also define the derivative of the state vector \( d\mathbf{u}/dt \) and the mass matrix \( M \) as a function of \( d\mathbf{u}/dt \) +(note that this function should be linear!). +The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \( F \) and \( M \).
The primary goal of the DSL is to define a set of differential equations of a -system of state variables. To define the state variables, we create a special -vector variable u_i which corresponds to the state variables $\mathbf{u}$.
To define the state variables \(\mathbf{u} \), we create a special +vector variable u_i. Note the particular name u is used to indicate that this +is the state vector.
u
The values that we use for u_i are the initial values of the state variables -at $t=0$.
u_i { x = 1, y = 0, z = 0, }
Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well.
We can optionally define the time derivatives of the state variables, -$\mathbf{\dot{u}}$ as well:
dudt_i { dxdt = 1, dydt = 0, dzdt = 0, }
Here the initial values of the time derivatives are given, these are typically +
Here the initial values of the time derivatives are given. +In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. +However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables.
Note that there is no need to define dudt if you do not define a mass matrix $M$.
dudt
Recall that the DAE system is defined by the equations:
$$ -M(t) \frac{d\mathbf{u}}{dt} = F(\mathbf{u}, t) -$$
We now define the equations $F$ and $M$ that we want to solve, using the +
We now define the right-hand-side function \( F \) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations.
For example, to define a simple system of ODEs:
$$ \begin{align*} -\frac{dx}{dt} &= y \ -\frac{dy}{dt} &= -x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= y \\ +\frac{dy}{dt} &= -x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$
We write:
u_i { - x = 1, - y = 0, -} -F_i { - y, - -x, -} +u_i { x = 1, y = 0 } +F_i { y, -x } -We can also define a mass matrix $M$ by defining a vector variable M_i. This -is optional, and if not defined, the mass matrix is assumed to be the identity +Defining the mass matrix +We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \( M \mathbf{\dot{u}} \). +This is optional, and if not defined, the mass matrix is assumed to be the identity matrix. +Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \( M \mathbf{\dot{u}} \), and not the mass matrix itself. For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal: $$ \begin{align*} -\frac{dx}{dt} &= x \ -0 &= y-x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= x \\ +0 &= y-x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: @@ -348,53 +265,18 @@ Specifying outputs -Finally, we specify the outputs of the system. These might be the state -variables themselves, or they might be other variables that are calculated from -the state variables. Here we specify that we want to output the state variables -$x$ and $y$: -out_i { - x, - y, -} - -Required variables -The DSL allows the user to specify an arbitrary number of intermediate -variables, but certain variables are required to be defined. These are: - -u_i - the state variables -F_i - the vector $F(\mathbf{u}, t)$ -out_i - the output variables - -Predefined variables -The only predefined variable is the scalar $t$ which is the current time, this -allows the equations to be written as functions of time. For example -F_i { - k1 * t + sin(t) -} - -Mathematical functions -The DSL supports the following mathematical functions: - -pow(x, y) - x raised to the power of y -sin(x) - sine of x -cos(x) - cosine of x -tan(x) - tangent of x -exp(x) - exponential of x -log(x) - natural logarithm of x -sqrt(x) - square root of x -abs(x) - absolute value of x -sigmoid(x) - sigmoid function of x -
u_i { x = 1, y = 0 } +F_i { y, -x }
We can also define a mass matrix $M$ by defining a vector variable M_i. This -is optional, and if not defined, the mass matrix is assumed to be the identity +
M_i
We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \( M \mathbf{\dot{u}} \). +This is optional, and if not defined, the mass matrix is assumed to be the identity matrix.
Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \( M \mathbf{\dot{u}} \), and not the mass matrix itself.
For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal:
$$ \begin{align*} -\frac{dx}{dt} &= x \ -0 &= y-x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= x \\ +0 &= y-x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$
Finally, we specify the outputs of the system. These might be the state -variables themselves, or they might be other variables that are calculated from -the state variables. Here we specify that we want to output the state variables -$x$ and $y$:
out_i { - x, - y, -} -
The DSL allows the user to specify an arbitrary number of intermediate -variables, but certain variables are required to be defined. These are:
The only predefined variable is the scalar $t$ which is the current time, this -allows the equations to be written as functions of time. For example
F_i { - k1 * t + sin(t) -} -
The DSL supports the following mathematical functions:
pow(x, y)
sin(x)
cos(x)
tan(x)
exp(x)
log(x)
sqrt(x)
abs(x)
sigmoid(x)
We can use standard algebraic operations on variables. To refer to previously +defined variables, we use the variable name, making sure to use the correct +subscript if it is a vector or tensor.
For example, to define a scalar variable $a$ as the sum of two other scalar +variables $b$ and $c$, we write:
b { 1.0 } +c { 2.0 } +a { b + c } +
The scalar a will therefore be equal to 3.0.
a
To define a vector variable \( \mathbf{v} \) as the sum of two other vector +variables \( \mathbf{u} \) and \( \mathbf{w} \), we write:
u_i { 1.0, 2.0 } +w_i { 3.0, 4.0 } +v_i { u_i + w_i } +
Notice that the index of the vectors within the expression must match the index of the output vector v_i. +So if we defined v_a instead of v_i, the expression would be:
v_a
v_a { u_a + w_a } +
For higher-dimensional tensors, the order of the indices in the expression relative to the output tensor is important.
For example, we can use the indices to define a translation of a matrix. Here we define a new matrix \( C \) that is the sum of \( A \) and \( B^T \)$, +where \( B^T \) is the transpose of \( B \)
C_ij { A_ij + B_ji } +
Notice that the indices of \( B^T \) are reversed in the expression compared to the output tensor \( C \), indicating that we are indexing the rows of \( B \) with j and the columns with i when we calculate the (i, j) element of \( C \).
j
i
(i, j)
Broadcasting is supported in the language, so you can perform element-wise operations on tensors of different shapes. For example, the following will define a new vector \( \mathbf{d} \) that is the sum of \( \mathbf{a} \) and a scalar \( k \):
a_i { 1.0, 2.0 } +k { 3.0 } +d_i { a_i + k } +
Here the scalar \( k \) is broadcast to the same shape as \( \mathbf{a} \) before the addition. The output vector \( \mathbf{d} \) will be \( [4.0, 5.0] \).
DiffSL uses the same broadcasting rules as NumPy, and you can read more about this in the NumPy documentation.
The tensor indexing notation can also matrix-vector multiplications and any other +contraction operations. Any indices that do not appear in the output tensor will be +summed over. For example, the following will define a new vector $v$ that is +the result of a matrix-vector multiplication:
v_i { A_ij * u_j } +
Here the j index is summed over, so the i index of the output vector v is the sum of the element-wise product of the ith row of A and the vector u.
v
A
Another way to think about this matrix-vector multiplication is by considering it as a broadcasted element-wise multiplication followed by a sum or contraction over the j index. +The A_ij * u_j expression broadcasts the vector u to the same shape as A, forming a new 2D tensor where each row is the element-wise product of the ith row of A and the vector u. +When we specify the output vector v_i, we implicitly then sum over the missing j index to form the final output vector.
A_ij * u_j
For example, lets manually break this matrix-vector multiplication into two steps using an intermediary tensor M_ij:
M_ij
M_ij { A_ij * u_j } +v_i { M_ij } +
The first step calculates the element-wise product of A and u using broadcasting into the 2D tensor M, and the second step uses a contraction to sum over the j index to form the output vector v.
M
$$ -\begin{align} -\frac{dx}{dt} &= -0.04x + 10^4 y z \ -\frac{dy}{dt} &= 0.04x - 10^4 y z - 3 \cdot 10^7 y^2 \ -0 &= x + y + z - 1 -\end{align} -$$
The DSL is designed to be an easy and flexible language for specifying DAE systems and is based on the idea that a DAE system can be specified by a set of equations of the form:
The DiffSL language only has a single type of variable, which is a n-dimensional tensor filled with double precision floating point numbers. +These tensors can be dense, sparse or diagonal, and the compiler will try to choose the representation that is most efficient, preferring diagonal and then sparse matrices over dense.
The simplest tensor is a 0th dimensional scalar. For example, to define a scalar variable \( k )\$ with value 1.0, we write:
k { 1.0 }
Here k is the label for the tensor, which we can later use to refer to it in expressions. In the curly brackets we have one or more elements of the tensor, +and here k only has a single element, which is a constant value 1.
Lets now define a 1-dimensional vector variable $\mathbf{v}$ with 3 elements:
v_i { - x = 1.0, - y = 2.0, - z = 3.0, + 1.0, + 2.0, + 3.0, }
The list of elements within a tensor are deliniated with a comma , and the trailing comma at the end of the list is optional. +Whitespace is ignored so you can also write this tensor all on the one line:
,
v_i { 1.0, 2.0, 3.0 } +
In the previous vector v_i, the subscript _i indicates that this is a 1D vector. Each subscript is a single character, +and the number of subscripts indicates the number of dimensions of the tensor. You can use any character for the subscript,
v_x { 1.0, 2.0, 3.0 } +w_a { 2.0, 3.0 } +v_i { 1.0, 2.0, 3.0, 4.0 } +
Each element of a tensor can optionally give a range of index numbers, which is used by the compiler to determine the extent of each element. +This is useful when defining higher dimensional tensors, such as matrices. For example, to define a 2x3 matrix $A$ with all elements set to 1.0, we write:
A_ij { (0:2, 0:3) = 1.0, }
0:2
0:3
Here is another example of a 4x2 matrix $B$ with rows 0 to 2 set to 1.0 and rows 3 to 4 set to 2.0:
2.0
A_ij { (0:2, 0:3) = 1.0, (3:4, 0:3) = 2.0, }
For specifying a single index, you can simply write the index number without the colon, for example to define a 3x3 identity matrix $I$:
I_ij { + (0, 0) = 1.0, + (1, 1) = 1.0, + (2, 2) = 1.0, +} +
Note that the compiler will automatically infer the size of the tensor from the ranges you provide, so you don't need to specify the size of the tensor explicitly. +Since the maximum index in the range is 2, the compiler will infer that the size of the tensor is 3x3.
Notice also that we have not defined all the elements of the matrix, only the non-zero elements. The compiler will assume that all other elements are zero.
Finally, you can also use the .. operator to specify a diagonal range of indices. For example, to define a 3x3 matrix $D$ with the diagonal elements set to 1.0:
..
D_ij { + (0..2, 0..2) = 1.0, +} +
We can automatically define a sparse matrix $B$ by simply specifying the non-zero elements:
B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 1) = 3.0, }
I_ij { - (0..2, 0..2) = 1.0, +The compiler will infer that this is a 2x2 matrix, and will automatically represent it as a sparse matrix. +We can force the compiler to use a dense representation by specifying the zeros explicitly: +B_ij { + (0, 0) = 1.0, + (0, 1) = 2.0, + (1, 0) = 0.0, + (1, 1) = 3.0, +} + +As well as specifying a sparse matrix, we can also define a diagonal matrix by specifying the diagonal elements: +D_ij { + (0, 0) = 1.0, + (1, 1) = 2.0, + (2, 2) = 3.0, } -Operations +The compiler will infer that this is a 3x3 matrix, and will automatically represent it as a diagonal matrix. +Labels +Each element of a tensor can optionally be given a name or label that can be used to refer to the element in expressions. +For example, to define a vector with two named elements: +v_i { + x = 1.0, + y = 2.0, +} + +Here we have defined a single tensor v_i with two named elements x and y. We can then refer to the individual elements in expressions, where they will act as if they were separate variables: +v_i { x = 1.0, y = 2.0 } +w_i { 2 * y, 3 * x } + +Tensor Operations We can use standard algebraic operations on variables. To refer to previously defined variables, we use the variable name, making sure to use the correct subscript if it is a vector or tensor. For example, to define a scalar variable $a$ as the sum of two other scalar variables $b$ and $c$, we write: -a { b + c } +b { 1.0 } +c { 2.0 } +a { b + c } -To define a vector variable $\mathbf{V}$ as the sum of two other vector -variables $\mathbf{u}$ and $\mathbf{w}$, we write: -v_i { u_i + w_i } +The scalar a will therefore be equal to 3.0. +To define a vector variable \( \mathbf{v} \) as the sum of two other vector +variables \( \mathbf{u} \) and \( \mathbf{w} \), we write: +u_i { 1.0, 2.0 } +w_i { 3.0, 4.0 } +v_i { u_i + w_i } -The indexing can be used to perform translations on tensors, for example the -following will define a new tensor $C$ that is the sum of $A$ and $B^T$: +Notice that the index of the vectors within the expression must match the index of the output vector v_i. +So if we defined v_a instead of v_i, the expression would be: +v_a { u_a + w_a } + +Translations +For higher-dimensional tensors, the order of the indices in the expression relative to the output tensor is important. +For example, we can use the indices to define a translation of a matrix. Here we define a new matrix \( C \) that is the sum of \( A \) and \( B^T \)$, +where \( B^T \) is the transpose of \( B \) C_ij { A_ij + B_ji } -Tensor indexing notation can also matrix-vector multiplications and any other -contraction operations. Any indices that do not appear in the output will be +Notice that the indices of \( B^T \) are reversed in the expression compared to the output tensor \( C \), indicating that we are indexing the rows of \( B \) with j and the columns with i when we calculate the (i, j) element of \( C \). +Broadcasting +Broadcasting is supported in the language, so you can perform element-wise operations on tensors of different shapes. For example, the following will define a new vector \( \mathbf{d} \) that is the sum of \( \mathbf{a} \) and a scalar \( k \): +a_i { 1.0, 2.0 } +k { 3.0 } +d_i { a_i + k } + +Here the scalar \( k \) is broadcast to the same shape as \( \mathbf{a} \) before the addition. The output vector \( \mathbf{d} \) will be \( [4.0, 5.0] \). +DiffSL uses the same broadcasting rules as NumPy, and you can read more about this in the NumPy documentation. +Contractions +The tensor indexing notation can also matrix-vector multiplications and any other +contraction operations. Any indices that do not appear in the output tensor will be summed over. For example, the following will define a new vector $v$ that is the result of a matrix-vector multiplication: v_i { A_ij * u_j } -Broadcasting is also supported, for example the following will define a new -matrix $D$ that is the sum of $A$ and a vector $v$: -D_ij { A_ij + v_j } - -Here the vector $v$ is broadcast to the same shape as $A$ before the addition. -Specifying inputs -We can override the values of any scalar variables by specifying them as input -variables. To do this, we add a line at the top of the code to specify that -these are input variables: -in = [k] -k { 1.0 } +Here the j index is summed over, so the i index of the output vector v is the sum of the element-wise product of the ith row of A and the vector u. +Another way to think about this matrix-vector multiplication is by considering it as a broadcasted element-wise multiplication followed by a sum or contraction over the j index. +The A_ij * u_j expression broadcasts the vector u to the same shape as A, forming a new 2D tensor where each row is the element-wise product of the ith row of A and the vector u. +When we specify the output vector v_i, we implicitly then sum over the missing j index to form the final output vector. +For example, lets manually break this matrix-vector multiplication into two steps using an intermediary tensor M_ij: +M_ij { A_ij * u_j } +v_i { M_ij } +The first step calculates the element-wise product of A and u using broadcasting into the 2D tensor M, and the second step uses a contraction to sum over the j index to form the output vector v. +Defining a system of ODEs +The primary goal of the DiffSL language is to define a system of ODEs in the following form: +$$ +\begin{align*} +M(t) \frac{d\mathbf{u}}{dt} &= F(\mathbf{u}, t) \\ +\mathbf{u}(0) &= \mathbf{u}_0 +\end{align*} +$$ +where \( \mathbf{u} \) is the vector of state variables, \( \mathbf{u}_0 \) is the initial condition, \( F \) is the RHS function, and \( M \) is the mass matrix. +The DSL allows the user to specify the state vector \( \mathbf{u} \) and the RHS function \( F \). +Optionally, the user can also define the derivative of the state vector \( d\mathbf{u}/dt \) and the mass matrix \( M \) as a function of \( d\mathbf{u}/dt \) +(note that this function should be linear!). +The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \( F \) and \( M \). Defining state variables -The primary goal of the DSL is to define a set of differential equations of a -system of state variables. To define the state variables, we create a special -vector variable u_i which corresponds to the state variables $\mathbf{u}$. +To define the state variables \(\mathbf{u} \), we create a special +vector variable u_i. Note the particular name u is used to indicate that this +is the state vector. The values that we use for u_i are the initial values of the state variables -at $t=0$. +at \( t=0 \), so an initial condition \( \mathbf{u|(t=0)} = [x(0), y(0), z(0)] = [1, 0, 0] \) is defined as: u_i { x = 1, y = 0, z = 0, } +Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well. We can optionally define the time derivatives of the state variables, -$\mathbf{\dot{u}}$ as well: +\( \mathbf{\dot{u}} \) as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0, } -Here the initial values of the time derivatives are given, these are typically +Here the initial values of the time derivatives are given. +In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. +However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$. Defining the ODE system equations -Recall that the DAE system is defined by the equations: -$$ -M(t) \frac{d\mathbf{u}}{dt} = F(\mathbf{u}, t) -$$ -We now define the equations $F$ and $M$ that we want to solve, using the +We now define the right-hand-side function \( F \) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. For example, to define a simple system of ODEs: $$ \begin{align*} -\frac{dx}{dt} &= y \ -\frac{dy}{dt} &= -x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= y \\ +\frac{dy}{dt} &= -x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: -u_i { - x = 1, - y = 0, -} -F_i { - y, - -x, -} +u_i { x = 1, y = 0 } +F_i { y, -x } -We can also define a mass matrix $M$ by defining a vector variable M_i. This -is optional, and if not defined, the mass matrix is assumed to be the identity +Defining the mass matrix +We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \( M \mathbf{\dot{u}} \). +This is optional, and if not defined, the mass matrix is assumed to be the identity matrix. +Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \( M \mathbf{\dot{u}} \), and not the mass matrix itself. For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal: $$ \begin{align*} -\frac{dx}{dt} &= x \ -0 &= y-x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= x \\ +0 &= y-x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: @@ -426,44 +433,54 @@
The compiler will infer that this is a 2x2 matrix, and will automatically represent it as a sparse matrix. +We can force the compiler to use a dense representation by specifying the zeros explicitly:
B_ij { + (0, 0) = 1.0, + (0, 1) = 2.0, + (1, 0) = 0.0, + (1, 1) = 3.0, +} +
As well as specifying a sparse matrix, we can also define a diagonal matrix by specifying the diagonal elements:
D_ij { + (0, 0) = 1.0, + (1, 1) = 2.0, + (2, 2) = 3.0, }
The compiler will infer that this is a 3x3 matrix, and will automatically represent it as a diagonal matrix.
Each element of a tensor can optionally be given a name or label that can be used to refer to the element in expressions.
For example, to define a vector with two named elements:
v_i { + x = 1.0, + y = 2.0, +} +
Here we have defined a single tensor v_i with two named elements x and y. We can then refer to the individual elements in expressions, where they will act as if they were separate variables:
x
y
v_i { x = 1.0, y = 2.0 } +w_i { 2 * y, 3 * x } +
We can use standard algebraic operations on variables. To refer to previously defined variables, we use the variable name, making sure to use the correct subscript if it is a vector or tensor.
For example, to define a scalar variable $a$ as the sum of two other scalar variables $b$ and $c$, we write:
a { b + c } +b { 1.0 } +c { 2.0 } +a { b + c } -To define a vector variable $\mathbf{V}$ as the sum of two other vector -variables $\mathbf{u}$ and $\mathbf{w}$, we write: -v_i { u_i + w_i } +The scalar a will therefore be equal to 3.0. +To define a vector variable \( \mathbf{v} \) as the sum of two other vector +variables \( \mathbf{u} \) and \( \mathbf{w} \), we write: +u_i { 1.0, 2.0 } +w_i { 3.0, 4.0 } +v_i { u_i + w_i } -The indexing can be used to perform translations on tensors, for example the -following will define a new tensor $C$ that is the sum of $A$ and $B^T$: +Notice that the index of the vectors within the expression must match the index of the output vector v_i. +So if we defined v_a instead of v_i, the expression would be: +v_a { u_a + w_a } + +Translations +For higher-dimensional tensors, the order of the indices in the expression relative to the output tensor is important. +For example, we can use the indices to define a translation of a matrix. Here we define a new matrix \( C \) that is the sum of \( A \) and \( B^T \)$, +where \( B^T \) is the transpose of \( B \) C_ij { A_ij + B_ji } -Tensor indexing notation can also matrix-vector multiplications and any other -contraction operations. Any indices that do not appear in the output will be +Notice that the indices of \( B^T \) are reversed in the expression compared to the output tensor \( C \), indicating that we are indexing the rows of \( B \) with j and the columns with i when we calculate the (i, j) element of \( C \). +Broadcasting +Broadcasting is supported in the language, so you can perform element-wise operations on tensors of different shapes. For example, the following will define a new vector \( \mathbf{d} \) that is the sum of \( \mathbf{a} \) and a scalar \( k \): +a_i { 1.0, 2.0 } +k { 3.0 } +d_i { a_i + k } + +Here the scalar \( k \) is broadcast to the same shape as \( \mathbf{a} \) before the addition. The output vector \( \mathbf{d} \) will be \( [4.0, 5.0] \). +DiffSL uses the same broadcasting rules as NumPy, and you can read more about this in the NumPy documentation. +Contractions +The tensor indexing notation can also matrix-vector multiplications and any other +contraction operations. Any indices that do not appear in the output tensor will be summed over. For example, the following will define a new vector $v$ that is the result of a matrix-vector multiplication: v_i { A_ij * u_j } -Broadcasting is also supported, for example the following will define a new -matrix $D$ that is the sum of $A$ and a vector $v$: -D_ij { A_ij + v_j } - -Here the vector $v$ is broadcast to the same shape as $A$ before the addition. -Specifying inputs -We can override the values of any scalar variables by specifying them as input -variables. To do this, we add a line at the top of the code to specify that -these are input variables: -in = [k] -k { 1.0 } +Here the j index is summed over, so the i index of the output vector v is the sum of the element-wise product of the ith row of A and the vector u. +Another way to think about this matrix-vector multiplication is by considering it as a broadcasted element-wise multiplication followed by a sum or contraction over the j index. +The A_ij * u_j expression broadcasts the vector u to the same shape as A, forming a new 2D tensor where each row is the element-wise product of the ith row of A and the vector u. +When we specify the output vector v_i, we implicitly then sum over the missing j index to form the final output vector. +For example, lets manually break this matrix-vector multiplication into two steps using an intermediary tensor M_ij: +M_ij { A_ij * u_j } +v_i { M_ij } +The first step calculates the element-wise product of A and u using broadcasting into the 2D tensor M, and the second step uses a contraction to sum over the j index to form the output vector v. +Defining a system of ODEs +The primary goal of the DiffSL language is to define a system of ODEs in the following form: +$$ +\begin{align*} +M(t) \frac{d\mathbf{u}}{dt} &= F(\mathbf{u}, t) \\ +\mathbf{u}(0) &= \mathbf{u}_0 +\end{align*} +$$ +where \( \mathbf{u} \) is the vector of state variables, \( \mathbf{u}_0 \) is the initial condition, \( F \) is the RHS function, and \( M \) is the mass matrix. +The DSL allows the user to specify the state vector \( \mathbf{u} \) and the RHS function \( F \). +Optionally, the user can also define the derivative of the state vector \( d\mathbf{u}/dt \) and the mass matrix \( M \) as a function of \( d\mathbf{u}/dt \) +(note that this function should be linear!). +The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \( F \) and \( M \). Defining state variables -The primary goal of the DSL is to define a set of differential equations of a -system of state variables. To define the state variables, we create a special -vector variable u_i which corresponds to the state variables $\mathbf{u}$. +To define the state variables \(\mathbf{u} \), we create a special +vector variable u_i. Note the particular name u is used to indicate that this +is the state vector. The values that we use for u_i are the initial values of the state variables -at $t=0$. +at \( t=0 \), so an initial condition \( \mathbf{u|(t=0)} = [x(0), y(0), z(0)] = [1, 0, 0] \) is defined as: u_i { x = 1, y = 0, z = 0, } +Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well. We can optionally define the time derivatives of the state variables, -$\mathbf{\dot{u}}$ as well: +\( \mathbf{\dot{u}} \) as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0, } -Here the initial values of the time derivatives are given, these are typically +Here the initial values of the time derivatives are given. +In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. +However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$. Defining the ODE system equations -Recall that the DAE system is defined by the equations: -$$ -M(t) \frac{d\mathbf{u}}{dt} = F(\mathbf{u}, t) -$$ -We now define the equations $F$ and $M$ that we want to solve, using the +We now define the right-hand-side function \( F \) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. For example, to define a simple system of ODEs: $$ \begin{align*} -\frac{dx}{dt} &= y \ -\frac{dy}{dt} &= -x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= y \\ +\frac{dy}{dt} &= -x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: -u_i { - x = 1, - y = 0, -} -F_i { - y, - -x, -} +u_i { x = 1, y = 0 } +F_i { y, -x } -We can also define a mass matrix $M$ by defining a vector variable M_i. This -is optional, and if not defined, the mass matrix is assumed to be the identity +Defining the mass matrix +We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \( M \mathbf{\dot{u}} \). +This is optional, and if not defined, the mass matrix is assumed to be the identity matrix. +Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \( M \mathbf{\dot{u}} \), and not the mass matrix itself. For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal: $$ \begin{align*} -\frac{dx}{dt} &= x \ -0 &= y-x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= x \\ +0 &= y-x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: @@ -426,44 +433,54 @@
b { 1.0 } +c { 2.0 } +a { b + c }
v_i { u_i + w_i } +The scalar a will therefore be equal to 3.0. +To define a vector variable \( \mathbf{v} \) as the sum of two other vector +variables \( \mathbf{u} \) and \( \mathbf{w} \), we write: +u_i { 1.0, 2.0 } +w_i { 3.0, 4.0 } +v_i { u_i + w_i } -The indexing can be used to perform translations on tensors, for example the -following will define a new tensor $C$ that is the sum of $A$ and $B^T$: +Notice that the index of the vectors within the expression must match the index of the output vector v_i. +So if we defined v_a instead of v_i, the expression would be: +v_a { u_a + w_a } + +Translations +For higher-dimensional tensors, the order of the indices in the expression relative to the output tensor is important. +For example, we can use the indices to define a translation of a matrix. Here we define a new matrix \( C \) that is the sum of \( A \) and \( B^T \)$, +where \( B^T \) is the transpose of \( B \) C_ij { A_ij + B_ji } -Tensor indexing notation can also matrix-vector multiplications and any other -contraction operations. Any indices that do not appear in the output will be +Notice that the indices of \( B^T \) are reversed in the expression compared to the output tensor \( C \), indicating that we are indexing the rows of \( B \) with j and the columns with i when we calculate the (i, j) element of \( C \). +Broadcasting +Broadcasting is supported in the language, so you can perform element-wise operations on tensors of different shapes. For example, the following will define a new vector \( \mathbf{d} \) that is the sum of \( \mathbf{a} \) and a scalar \( k \): +a_i { 1.0, 2.0 } +k { 3.0 } +d_i { a_i + k } + +Here the scalar \( k \) is broadcast to the same shape as \( \mathbf{a} \) before the addition. The output vector \( \mathbf{d} \) will be \( [4.0, 5.0] \). +DiffSL uses the same broadcasting rules as NumPy, and you can read more about this in the NumPy documentation. +Contractions +The tensor indexing notation can also matrix-vector multiplications and any other +contraction operations. Any indices that do not appear in the output tensor will be summed over. For example, the following will define a new vector $v$ that is the result of a matrix-vector multiplication: v_i { A_ij * u_j } -Broadcasting is also supported, for example the following will define a new -matrix $D$ that is the sum of $A$ and a vector $v$: -D_ij { A_ij + v_j } - -Here the vector $v$ is broadcast to the same shape as $A$ before the addition. -Specifying inputs -We can override the values of any scalar variables by specifying them as input -variables. To do this, we add a line at the top of the code to specify that -these are input variables: -in = [k] -k { 1.0 } +Here the j index is summed over, so the i index of the output vector v is the sum of the element-wise product of the ith row of A and the vector u. +Another way to think about this matrix-vector multiplication is by considering it as a broadcasted element-wise multiplication followed by a sum or contraction over the j index. +The A_ij * u_j expression broadcasts the vector u to the same shape as A, forming a new 2D tensor where each row is the element-wise product of the ith row of A and the vector u. +When we specify the output vector v_i, we implicitly then sum over the missing j index to form the final output vector. +For example, lets manually break this matrix-vector multiplication into two steps using an intermediary tensor M_ij: +M_ij { A_ij * u_j } +v_i { M_ij } +The first step calculates the element-wise product of A and u using broadcasting into the 2D tensor M, and the second step uses a contraction to sum over the j index to form the output vector v. +Defining a system of ODEs +The primary goal of the DiffSL language is to define a system of ODEs in the following form: +$$ +\begin{align*} +M(t) \frac{d\mathbf{u}}{dt} &= F(\mathbf{u}, t) \\ +\mathbf{u}(0) &= \mathbf{u}_0 +\end{align*} +$$ +where \( \mathbf{u} \) is the vector of state variables, \( \mathbf{u}_0 \) is the initial condition, \( F \) is the RHS function, and \( M \) is the mass matrix. +The DSL allows the user to specify the state vector \( \mathbf{u} \) and the RHS function \( F \). +Optionally, the user can also define the derivative of the state vector \( d\mathbf{u}/dt \) and the mass matrix \( M \) as a function of \( d\mathbf{u}/dt \) +(note that this function should be linear!). +The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \( F \) and \( M \). Defining state variables -The primary goal of the DSL is to define a set of differential equations of a -system of state variables. To define the state variables, we create a special -vector variable u_i which corresponds to the state variables $\mathbf{u}$. +To define the state variables \(\mathbf{u} \), we create a special +vector variable u_i. Note the particular name u is used to indicate that this +is the state vector. The values that we use for u_i are the initial values of the state variables -at $t=0$. +at \( t=0 \), so an initial condition \( \mathbf{u|(t=0)} = [x(0), y(0), z(0)] = [1, 0, 0] \) is defined as: u_i { x = 1, y = 0, z = 0, } +Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well. We can optionally define the time derivatives of the state variables, -$\mathbf{\dot{u}}$ as well: +\( \mathbf{\dot{u}} \) as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0, } -Here the initial values of the time derivatives are given, these are typically +Here the initial values of the time derivatives are given. +In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. +However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$. Defining the ODE system equations -Recall that the DAE system is defined by the equations: -$$ -M(t) \frac{d\mathbf{u}}{dt} = F(\mathbf{u}, t) -$$ -We now define the equations $F$ and $M$ that we want to solve, using the +We now define the right-hand-side function \( F \) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. For example, to define a simple system of ODEs: $$ \begin{align*} -\frac{dx}{dt} &= y \ -\frac{dy}{dt} &= -x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= y \\ +\frac{dy}{dt} &= -x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: -u_i { - x = 1, - y = 0, -} -F_i { - y, - -x, -} +u_i { x = 1, y = 0 } +F_i { y, -x } -We can also define a mass matrix $M$ by defining a vector variable M_i. This -is optional, and if not defined, the mass matrix is assumed to be the identity +Defining the mass matrix +We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \( M \mathbf{\dot{u}} \). +This is optional, and if not defined, the mass matrix is assumed to be the identity matrix. +Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \( M \mathbf{\dot{u}} \), and not the mass matrix itself. For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal: $$ \begin{align*} -\frac{dx}{dt} &= x \ -0 &= y-x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= x \\ +0 &= y-x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: @@ -426,44 +433,54 @@
u_i { 1.0, 2.0 } +w_i { 3.0, 4.0 } +v_i { u_i + w_i }
C_ij { A_ij + B_ji }
Tensor indexing notation can also matrix-vector multiplications and any other -contraction operations. Any indices that do not appear in the output will be +
The tensor indexing notation can also matrix-vector multiplications and any other +contraction operations. Any indices that do not appear in the output tensor will be summed over. For example, the following will define a new vector $v$ that is the result of a matrix-vector multiplication:
v_i { A_ij * u_j }
in = [k] -k { 1.0 } +Here the j index is summed over, so the i index of the output vector v is the sum of the element-wise product of the ith row of A and the vector u. +Another way to think about this matrix-vector multiplication is by considering it as a broadcasted element-wise multiplication followed by a sum or contraction over the j index. +The A_ij * u_j expression broadcasts the vector u to the same shape as A, forming a new 2D tensor where each row is the element-wise product of the ith row of A and the vector u. +When we specify the output vector v_i, we implicitly then sum over the missing j index to form the final output vector. +For example, lets manually break this matrix-vector multiplication into two steps using an intermediary tensor M_ij: +M_ij { A_ij * u_j } +v_i { M_ij } +The first step calculates the element-wise product of A and u using broadcasting into the 2D tensor M, and the second step uses a contraction to sum over the j index to form the output vector v. +Defining a system of ODEs +The primary goal of the DiffSL language is to define a system of ODEs in the following form: +$$ +\begin{align*} +M(t) \frac{d\mathbf{u}}{dt} &= F(\mathbf{u}, t) \\ +\mathbf{u}(0) &= \mathbf{u}_0 +\end{align*} +$$ +where \( \mathbf{u} \) is the vector of state variables, \( \mathbf{u}_0 \) is the initial condition, \( F \) is the RHS function, and \( M \) is the mass matrix. +The DSL allows the user to specify the state vector \( \mathbf{u} \) and the RHS function \( F \). +Optionally, the user can also define the derivative of the state vector \( d\mathbf{u}/dt \) and the mass matrix \( M \) as a function of \( d\mathbf{u}/dt \) +(note that this function should be linear!). +The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \( F \) and \( M \). Defining state variables -The primary goal of the DSL is to define a set of differential equations of a -system of state variables. To define the state variables, we create a special -vector variable u_i which corresponds to the state variables $\mathbf{u}$. +To define the state variables \(\mathbf{u} \), we create a special +vector variable u_i. Note the particular name u is used to indicate that this +is the state vector. The values that we use for u_i are the initial values of the state variables -at $t=0$. +at \( t=0 \), so an initial condition \( \mathbf{u|(t=0)} = [x(0), y(0), z(0)] = [1, 0, 0] \) is defined as: u_i { x = 1, y = 0, z = 0, } +Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well. We can optionally define the time derivatives of the state variables, -$\mathbf{\dot{u}}$ as well: +\( \mathbf{\dot{u}} \) as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0, } -Here the initial values of the time derivatives are given, these are typically +Here the initial values of the time derivatives are given. +In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. +However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$. Defining the ODE system equations -Recall that the DAE system is defined by the equations: -$$ -M(t) \frac{d\mathbf{u}}{dt} = F(\mathbf{u}, t) -$$ -We now define the equations $F$ and $M$ that we want to solve, using the +We now define the right-hand-side function \( F \) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. For example, to define a simple system of ODEs: $$ \begin{align*} -\frac{dx}{dt} &= y \ -\frac{dy}{dt} &= -x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= y \\ +\frac{dy}{dt} &= -x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: -u_i { - x = 1, - y = 0, -} -F_i { - y, - -x, -} +u_i { x = 1, y = 0 } +F_i { y, -x } -We can also define a mass matrix $M$ by defining a vector variable M_i. This -is optional, and if not defined, the mass matrix is assumed to be the identity +Defining the mass matrix +We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \( M \mathbf{\dot{u}} \). +This is optional, and if not defined, the mass matrix is assumed to be the identity matrix. +Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \( M \mathbf{\dot{u}} \), and not the mass matrix itself. For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal: $$ \begin{align*} -\frac{dx}{dt} &= x \ -0 &= y-x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= x \\ +0 &= y-x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: @@ -426,44 +433,54 @@
M_ij { A_ij * u_j } +v_i { M_ij }
$$ +\begin{align*} +M(t) \frac{d\mathbf{u}}{dt} &= F(\mathbf{u}, t) \\ +\mathbf{u}(0) &= \mathbf{u}_0 +\end{align*} +$$
u_i { - x = 1, - y = 0, -} -F_i { - y, - -x, -} +u_i { x = 1, y = 0 } +F_i { y, -x } -We can also define a mass matrix $M$ by defining a vector variable M_i. This -is optional, and if not defined, the mass matrix is assumed to be the identity +Defining the mass matrix +We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \( M \mathbf{\dot{u}} \). +This is optional, and if not defined, the mass matrix is assumed to be the identity matrix. +Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \( M \mathbf{\dot{u}} \), and not the mass matrix itself. For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal: $$ \begin{align*} -\frac{dx}{dt} &= x \ -0 &= y-x \ -x(0) &= 1 \ -y(0) &= 0 \ +\frac{dx}{dt} &= x \\ +0 &= y-x \\ +x(0) &= 1 \\ +y(0) &= 0 \\ \end{align*} $$ We write: @@ -426,44 +433,54 @@
Finally, we specify the outputs of the system. These might be the state +
We can also specify the outputs of the system. These might be the state variables themselves, or they might be other variables that are calculated from -the state variables. Here we specify that we want to output the state variables -$x$ and $y$:
out_i { - x, - y, +the state variables. +Here is an example where we simply output the elements of the state vector: +u_i { + x = 1.0, + y = 2.0, + z = 3.0, } +out_i { x, y, z } -Required variables -The DSL allows the user to specify an arbitrary number of intermediate -variables, but certain variables are required to be defined. These are: - -u_i - the state variables -F_i - the vector $F(\mathbf{u}, t)$ -out_i - the output variables - -Predefined variables -The only predefined variable is the scalar $t$ which is the current time, this -allows the equations to be written as functions of time. For example -F_i { - k1 * t + sin(t) +or we can derive additional outputs from the state variables: +u_i { + x = 1.0, + y = 2.0, + z = 3.0, } +out { x + y + z } -Mathematical functions -The DSL supports the following mathematical functions: - -pow(x, y) - x raised to the power of y -sin(x) - sine of x -cos(x) - cosine of x -tan(x) - tangent of x -exp(x) - exponential of x -log(x) - natural logarithm of x -sqrt(x) - square root of x -abs(x) - absolute value of x -sigmoid(x) - sigmoid function of x - diff --git a/searchindex.js b/searchindex.js index de8f5b4..6667118 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Object.assign(window.search, {"doc_urls":["../../index.html#diffsl","../../index.html#diffsl-language-features","../../index.html#dependencies","../../index.html#installing-diffsl","language.html#diffsl-language","language.html#defining-variables","language.html#operations","language.html#specifying-inputs","language.html#defining-state-variables","language.html#defining-the-ode-system-equations","language.html#specifying-outputs","language.html#required-variables","language.html#predefined-variables","language.html#mathematical-functions"],"index":{"documentStore":{"docInfo":{"0":{"body":106,"breadcrumbs":2,"title":1},"1":{"body":35,"breadcrumbs":4,"title":3},"10":{"body":22,"breadcrumbs":4,"title":2},"11":{"body":22,"breadcrumbs":4,"title":2},"12":{"body":16,"breadcrumbs":4,"title":2},"13":{"body":39,"breadcrumbs":4,"title":2},"2":{"body":21,"breadcrumbs":2,"title":1},"3":{"body":54,"breadcrumbs":3,"title":2},"4":{"body":64,"breadcrumbs":4,"title":2},"5":{"body":137,"breadcrumbs":4,"title":2},"6":{"body":110,"breadcrumbs":3,"title":1},"7":{"body":17,"breadcrumbs":4,"title":2},"8":{"body":77,"breadcrumbs":5,"title":3},"9":{"body":107,"breadcrumbs":6,"title":4}},"docs":{"0":{"body":"A compiler for a domain-specific language for ordinary differential equations (ODEs) of the following form: $$ M(t) \\frac{d\\mathbf{u}}{dt} = F(\\mathbf{u}, t) $$ As an example, the following code defines a classic DAE testcase, the Robertson (1966) problem, which models the kinetics of an autocatalytic reaction, given by the following set of equations: $$ \\begin{align} \\frac{dx}{dt} &= -0.04x + 10^4 y z \\ \\frac{dy}{dt} &= 0.04x - 10^4 y z - 3 \\cdot 10^7 y^2 \\ 0 &= x + y + z - 1 \\end{align} $$ The DiffSL code for this problem is as follows: in = [k1, k2, k3]\nk1 { 0.04 }\nk2 { 10000 }\nk3 { 30000000 }\nu_i { x = 1, y = 0, z = 0,\n}\ndudt_i { dxdt = 1, dydt = 0, dzdt = 0,\n}\nM_i { dxdt, dydt, 0,\n}\nF_i { -k1 * x + k2 * y * z, k1 * x - k2 * y * z - k3 * y * y, 1 - x - y - z,\n}\nout_i { x, y, z,\n}","breadcrumbs":"Introduction » DiffSL","id":"0","title":"DiffSL"},"1":{"body":"See the DiffSL Language section for a full description. Tensor types: Scalars (double precision floating point numbers) Vectors (1D arrays of scalars) N-dimensional tensor of scalars Sparse/dense/diagonal tensors Tensor operations: Elementwise operations Broadcasting Tensor contractions/matmul/translation etc via index notation","breadcrumbs":"Introduction » DiffSL Language Features","id":"1","title":"DiffSL Language Features"},"10":{"body":"Finally, we specify the outputs of the system. These might be the state variables themselves, or they might be other variables that are calculated from the state variables. Here we specify that we want to output the state variables $x$ and $y$: out_i { x, y,\n}","breadcrumbs":"DiffSL Language » Specifying outputs","id":"10","title":"Specifying outputs"},"11":{"body":"The DSL allows the user to specify an arbitrary number of intermediate variables, but certain variables are required to be defined. 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The user is also free to define an an arbitrary number of intermediate scalars and vectors of the users that are required to calculate $F$ and $M$.","breadcrumbs":"DiffSL Language » DiffSL Language","id":"4","title":"DiffSL Language"},"5":{"body":"The DSL allows the user to define scalars, vectors, and dense/sparse/diagonal n-dimensional tensors. You can optionally label the elements of a vector or tensor for later use. For example, to define a scalar variable $k$ with value $1$, we write: k { 1.0 } To define a vector variable $\\mathbf{v}$ with 3 elements that are labelled, we write: v_i { x = 1.0, y = 2.0, z = 3.0,\n} The subscript _i indicates that this is a 1D vector, and $x$, $y$, and $z$ are defined as labels to the 3 elements of the vector. Later in the code, we could refer to either the whole vector v_i for $\\mathbf{v}$ or to the individual elements $x$, $y$, and $z$, which are scalars. To define dense 2x3 matrix $A$ with all elements set to 1.0, we write: A_ij { (0:2, 0:3) = 1.0,\n} Note the two subscript to indicate that this is a 2D tensor. The size of the tensor is given in the brackets, and the elements are set to $1$. If we have additional rows, we can add them as follows: A_ij { (0:2, 0:3) = 1.0, (3:4, 0:3) = 2.0,\n} We can define a sparse matrix $B$ by specifying the non-zero elements: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 1) = 3.0,\n} We can also define a diagonal identity matrix $I$ by specifying the diagonal elements using a different range syntax: I_ij { (0..2, 0..2) = 1.0,\n}","breadcrumbs":"DiffSL Language » Defining variables","id":"5","title":"Defining variables"},"6":{"body":"We can use standard algebraic operations on variables. To refer to previously defined variables, we use the variable name, making sure to use the correct subscript if it is a vector or tensor. 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Language The DSL is designed to be an easy and flexible language for specifying DAE systems and is based on the idea that a DAE system can be specified by a set of equations of the form: $$ M(t) \\frac{d\\mathbf{u}}{dt} = F(\\mathbf{u}, t) $$ where \\( \\mathbf{u}$ \\) is the vector of state variables and \\( t \\) is the time. The DSL allows the user to specify the state vector \\( \\mathbf{u} \\) and the RHS function \\( F \\). Optionally, the user can also define the derivative of the state vector \\( d\\mathbf{u}/dt \\) and the mass matrix \\( M \\) as a function of \\( d\\mathbf{u}/dt \\) (note that this function should be linear!). The user is also free to define an an arbitrary number of intermediate scalars and vectors of the users that are required to calculate \\( F \\) and \\( M \\). To illustrate the language, consider the following simple example of a logistic growth model: $$ \\frac{dN}{dt} = r N (1 - N/K) $$ where \\( N \\) is the population, \\( r \\) is the growth rate, and \\( K \\) is the carrying capacity. To specify this model in DiffSL, we can write: in = [r, k] u_i { N = 0.0 } F_i { r * N * (1 - N/k) } out_i { N } Here, we define the input parameters for our model as a vector in with the growth rate r and the carrying capacity k. We then define the state vector u_i with the population N initialized to 0.0. Next, we define the RHS function F_i as the logistic growth equation. Finally, we define the output vector out_i with the population N.","breadcrumbs":"DiffSL Language » A Simple Example","id":"0","title":"A Simple Example"},"1":{"body":"The DiffSL language only has a single type of variable, which is a n-dimensional tensor filled with double precision floating point numbers. These tensors can be dense, sparse or diagonal, and the compiler will try to choose the representation that is most efficient, preferring diagonal and then sparse matrices over dense. The simplest tensor is a 0th dimensional scalar. For example, to define a scalar variable \\( k )\\$ with value 1.0, we write: k { 1.0 } Here k is the label for the tensor, which we can later use to refer to it in expressions. In the curly brackets we have one or more elements of the tensor, and here k only has a single element, which is a constant value 1. Lets now define a 1-dimensional vector variable $\\mathbf{v}$ with 3 elements: v_i { 1.0, 2.0, 3.0,\n} The list of elements within a tensor are deliniated with a comma , and the trailing comma at the end of the list is optional. Whitespace is ignored so you can also write this tensor all on the one line: v_i { 1.0, 2.0, 3.0 }","breadcrumbs":"Tensor Variables » Defining tensor variables","id":"1","title":"Defining tensor variables"},"10":{"body":"The primary goal of the DiffSL language is to define a system of ODEs in the following form: $$ \\begin{align*} M(t) \\frac{d\\mathbf{u}}{dt} &= F(\\mathbf{u}, t) \\\\ \\mathbf{u}(0) &= \\mathbf{u}_0 \\end{align*} $$ where \\( \\mathbf{u} \\) is the vector of state variables, \\( \\mathbf{u}_0 \\) is the initial condition, \\( F \\) is the RHS function, and \\( M \\) is the mass matrix. The DSL allows the user to specify the state vector \\( \\mathbf{u} \\) and the RHS function \\( F \\). Optionally, the user can also define the derivative of the state vector \\( d\\mathbf{u}/dt \\) and the mass matrix \\( M \\) as a function of \\( d\\mathbf{u}/dt \\) (note that this function should be linear!). The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \\( F \\) and \\( M \\).","breadcrumbs":"Defining ODEs » Defining a system of ODEs","id":"10","title":"Defining a system of ODEs"},"11":{"body":"To define the state variables \\(\\mathbf{u} \\), we create a special vector variable u_i. Note the particular name u is used to indicate that this is the state vector. The values that we use for u_i are the initial values of the state variables at \\( t=0 \\), so an initial condition \\( \\mathbf{u|(t=0)} = [x(0), y(0), z(0)] = [1, 0, 0] \\) is defined as: u_i { x = 1, y = 0, z = 0,\n} Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well. We can optionally define the time derivatives of the state variables, \\( \\mathbf{\\dot{u}} \\) as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0,\n} Here the initial values of the time derivatives are given. In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$.","breadcrumbs":"Defining ODEs » Defining state variables","id":"11","title":"Defining state variables"},"12":{"body":"We now define the right-hand-side function \\( F \\) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. For example, to define a simple system of ODEs: $$ \\begin{align*} \\frac{dx}{dt} &= y \\\\ \\frac{dy}{dt} &= -x \\\\ x(0) &= 1 \\\\ y(0) &= 0 \\\\ \\end{align*} $$ We write: u_i { x = 1, y = 0 }\nF_i { y, -x }","breadcrumbs":"Defining ODEs » Defining the ODE system equations","id":"12","title":"Defining the ODE system equations"},"13":{"body":"We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \\( M \\mathbf{\\dot{u}} \\). This is optional, and if not defined, the mass matrix is assumed to be the identity matrix. Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \\( M \\mathbf{\\dot{u}} \\), and not the mass matrix itself. For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal: $$ \\begin{align*} \\frac{dx}{dt} &= x \\\\ 0 &= y-x \\\\ x(0) &= 1 \\\\ y(0) &= 0 \\\\ \\end{align*} $$ We write: u_i { x = 1, y = 0,\n}\ndudt_i { dxdt = 0, dydt = 1,\n}\nM_i { dxdt, 0,\n}\nF_i { x, y-x,\n}","breadcrumbs":"Defining ODEs » Defining the mass matrix","id":"13","title":"Defining the mass matrix"},"14":{"body":"Often it is useful to parameterize the system of equations using a set of input parameters. It is also useful to be able to extract certain variables from the system for further analysis. In this section we will show how to specify inputs and outputs in the DiffSL language.","breadcrumbs":"Inputs and Outputs » Inputs & Outputs","id":"14","title":"Inputs & Outputs"},"15":{"body":"We can override the values of any scalar variables by specifying them as input variables. To do this, we add a line at the top of the code to specify that these are input variables: in = [k]\nk { 1.0 }\nu { 0.1 }\nF { k * u } Here we have specified a single input parameter k that is used in the RHS function F. The value of k is set to 1.0 in the code, but this value is only a default, and can be overridden by passing in a value at solve time. We can use input parameters anywhere in the code, including in the definition of other input parameters. in = [k]\nk { 1.0 }\ng { 2 * k }\nF { g * u } or in the intial conditions of the state variables: in = [k]\nk { 1.0 }\nu_i { x = k,\n}\nF { u }","breadcrumbs":"Inputs and Outputs » Specifying inputs","id":"15","title":"Specifying inputs"},"16":{"body":"We can also specify the outputs of the system. These might be the state variables themselves, or they might be other variables that are calculated from the state variables. Here is an example where we simply output the elements of the state vector: u_i { x = 1.0, y = 2.0, z = 3.0,\n}\nout_i { x, y, z } or we can derive additional outputs from the state variables: u_i { x = 1.0, y = 2.0, z = 3.0,\n}\nout { x + y + z }","breadcrumbs":"Inputs and Outputs » Specifying outputs","id":"16","title":"Specifying outputs"},"2":{"body":"In the previous vector v_i, the subscript _i indicates that this is a 1D vector. Each subscript is a single character, and the number of subscripts indicates the number of dimensions of the tensor. You can use any character for the subscript, v_x { 1.0, 2.0, 3.0 }\nw_a { 2.0, 3.0 }\nv_i { 1.0, 2.0, 3.0, 4.0 }","breadcrumbs":"Tensor Variables » Subscripts","id":"2","title":"Subscripts"},"3":{"body":"Each element of a tensor can optionally give a range of index numbers, which is used by the compiler to determine the extent of each element. This is useful when defining higher dimensional tensors, such as matrices. For example, to define a 2x3 matrix $A$ with all elements set to 1.0, we write: A_ij { (0:2, 0:3) = 1.0,\n} Note the two subscript to indicate that this is a 2D tensor. The size of the single element is given in the brackets, we have two ranges 0:2 and 0:3 that correspond to the two dimensions of the matrix. Here is another example of a 4x2 matrix $B$ with rows 0 to 2 set to 1.0 and rows 3 to 4 set to 2.0: A_ij { (0:2, 0:3) = 1.0, (3:4, 0:3) = 2.0,\n} For specifying a single index, you can simply write the index number without the colon, for example to define a 3x3 identity matrix $I$: I_ij { (0, 0) = 1.0, (1, 1) = 1.0, (2, 2) = 1.0,\n} Note that the compiler will automatically infer the size of the tensor from the ranges you provide, so you don't need to specify the size of the tensor explicitly. Since the maximum index in the range is 2, the compiler will infer that the size of the tensor is 3x3. Notice also that we have not defined all the elements of the matrix, only the non-zero elements. The compiler will assume that all other elements are zero. Finally, you can also use the .. operator to specify a diagonal range of indices. For example, to define a 3x3 matrix $D$ with the diagonal elements set to 1.0: D_ij { (0..2, 0..2) = 1.0,\n}","breadcrumbs":"Tensor Variables » Ranges","id":"3","title":"Ranges"},"4":{"body":"We can automatically define a sparse matrix $B$ by simply specifying the non-zero elements: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 1) = 3.0,\n} The compiler will infer that this is a 2x2 matrix, and will automatically represent it as a sparse matrix. We can force the compiler to use a dense representation by specifying the zeros explicitly: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 0) = 0.0, (1, 1) = 3.0,\n} As well as specifying a sparse matrix, we can also define a diagonal matrix by specifying the diagonal elements: D_ij { (0, 0) = 1.0, (1, 1) = 2.0, (2, 2) = 3.0,\n} The compiler will infer that this is a 3x3 matrix, and will automatically represent it as a diagonal matrix.","breadcrumbs":"Tensor Variables » Sparse and diagonal matrices","id":"4","title":"Sparse and diagonal matrices"},"5":{"body":"Each element of a tensor can optionally be given a name or label that can be used to refer to the element in expressions. For example, to define a vector with two named elements: v_i { x = 1.0, y = 2.0,\n} Here we have defined a single tensor v_i with two named elements x and y. We can then refer to the individual elements in expressions, where they will act as if they were separate variables: v_i { x = 1.0, y = 2.0 }\nw_i { 2 * y, 3 * x }","breadcrumbs":"Tensor Variables » Labels","id":"5","title":"Labels"},"6":{"body":"We can use standard algebraic operations on variables. To refer to previously defined variables, we use the variable name, making sure to use the correct subscript if it is a vector or tensor. For example, to define a scalar variable $a$ as the sum of two other scalar variables $b$ and $c$, we write: b { 1.0 }\nc { 2.0 }\na { b + c } The scalar a will therefore be equal to 3.0. To define a vector variable \\( \\mathbf{v} \\) as the sum of two other vector variables \\( \\mathbf{u} \\) and \\( \\mathbf{w} \\), we write: u_i { 1.0, 2.0 }\nw_i { 3.0, 4.0 }\nv_i { u_i + w_i } Notice that the index of the vectors within the expression must match the index of the output vector v_i. So if we defined v_a instead of v_i, the expression would be: v_a { u_a + w_a }","breadcrumbs":"Tensor Operations » Tensor Operations","id":"6","title":"Tensor Operations"},"7":{"body":"For higher-dimensional tensors, the order of the indices in the expression relative to the output tensor is important. For example, we can use the indices to define a translation of a matrix. Here we define a new matrix \\( C \\) that is the sum of \\( A \\) and \\( B^T \\)$, where \\( B^T \\) is the transpose of \\( B \\) C_ij { A_ij + B_ji } Notice that the indices of \\( B^T \\) are reversed in the expression compared to the output tensor \\( C \\), indicating that we are indexing the rows of \\( B \\) with j and the columns with i when we calculate the (i, j) element of \\( C \\).","breadcrumbs":"Tensor Operations » Translations","id":"7","title":"Translations"},"8":{"body":"Broadcasting is supported in the language, so you can perform element-wise operations on tensors of different shapes. For example, the following will define a new vector \\( \\mathbf{d} \\) that is the sum of \\( \\mathbf{a} \\) and a scalar \\( k \\): a_i { 1.0, 2.0 }\nk { 3.0 }\nd_i { a_i + k } Here the scalar \\( k \\) is broadcast to the same shape as \\( \\mathbf{a} \\) before the addition. The output vector \\( \\mathbf{d} \\) will be \\( [4.0, 5.0] \\). DiffSL uses the same broadcasting rules as NumPy, and you can read more about this in the NumPy documentation .","breadcrumbs":"Tensor Operations » Broadcasting","id":"8","title":"Broadcasting"},"9":{"body":"The tensor indexing notation can also matrix-vector multiplications and any other contraction operations. Any indices that do not appear in the output tensor will be summed over. For example, the following will define a new vector $v$ that is the result of a matrix-vector multiplication: v_i { A_ij * u_j } Here the j index is summed over, so the i index of the output vector v is the sum of the element-wise product of the ith row of A and the vector u. Another way to think about this matrix-vector multiplication is by considering it as a broadcasted element-wise multiplication followed by a sum or contraction over the j index. The A_ij * u_j expression broadcasts the vector u to the same shape as A, forming a new 2D tensor where each row is the element-wise product of the ith row of A and the vector u. When we specify the output vector v_i, we implicitly then sum over the missing j index to form the final output vector. 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Language The DSL is designed to be an easy and flexible language for specifying DAE systems and is based on the idea that a DAE system can be specified by a set of equations of the form: $$ M(t) \\frac{d\\mathbf{u}}{dt} = F(\\mathbf{u}, t) $$ where \\( \\mathbf{u}$ \\) is the vector of state variables and \\( t \\) is the time. The DSL allows the user to specify the state vector \\( \\mathbf{u} \\) and the RHS function \\( F \\). Optionally, the user can also define the derivative of the state vector \\( d\\mathbf{u}/dt \\) and the mass matrix \\( M \\) as a function of \\( d\\mathbf{u}/dt \\) (note that this function should be linear!). The user is also free to define an an arbitrary number of intermediate scalars and vectors of the users that are required to calculate \\( F \\) and \\( M \\). To illustrate the language, consider the following simple example of a logistic growth model: $$ \\frac{dN}{dt} = r N (1 - N/K) $$ where \\( N \\) is the population, \\( r \\) is the growth rate, and \\( K \\) is the carrying capacity. To specify this model in DiffSL, we can write: in = [r, k] u_i { N = 0.0 } F_i { r * N * (1 - N/k) } out_i { N } Here, we define the input parameters for our model as a vector in with the growth rate r and the carrying capacity k. We then define the state vector u_i with the population N initialized to 0.0. Next, we define the RHS function F_i as the logistic growth equation. Finally, we define the output vector out_i with the population N.","breadcrumbs":"DiffSL Language » A Simple Example","id":"0","title":"A Simple Example"},"1":{"body":"The DiffSL language only has a single type of variable, which is a n-dimensional tensor filled with double precision floating point numbers. These tensors can be dense, sparse or diagonal, and the compiler will try to choose the representation that is most efficient, preferring diagonal and then sparse matrices over dense. The simplest tensor is a 0th dimensional scalar. For example, to define a scalar variable \\( k )\\$ with value 1.0, we write: k { 1.0 } Here k is the label for the tensor, which we can later use to refer to it in expressions. In the curly brackets we have one or more elements of the tensor, and here k only has a single element, which is a constant value 1. Lets now define a 1-dimensional vector variable $\\mathbf{v}$ with 3 elements: v_i { 1.0, 2.0, 3.0,\n} The list of elements within a tensor are deliniated with a comma , and the trailing comma at the end of the list is optional. Whitespace is ignored so you can also write this tensor all on the one line: v_i { 1.0, 2.0, 3.0 }","breadcrumbs":"Tensor Variables » Defining tensor variables","id":"1","title":"Defining tensor variables"},"10":{"body":"The primary goal of the DiffSL language is to define a system of ODEs in the following form: $$ \\begin{align*} M(t) \\frac{d\\mathbf{u}}{dt} &= F(\\mathbf{u}, t) \\\\ \\mathbf{u}(0) &= \\mathbf{u}_0 \\end{align*} $$ where \\( \\mathbf{u} \\) is the vector of state variables, \\( \\mathbf{u}_0 \\) is the initial condition, \\( F \\) is the RHS function, and \\( M \\) is the mass matrix. The DSL allows the user to specify the state vector \\( \\mathbf{u} \\) and the RHS function \\( F \\). Optionally, the user can also define the derivative of the state vector \\( d\\mathbf{u}/dt \\) and the mass matrix \\( M \\) as a function of \\( d\\mathbf{u}/dt \\) (note that this function should be linear!). The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \\( F \\) and \\( M \\).","breadcrumbs":"Defining ODEs » Defining a system of ODEs","id":"10","title":"Defining a system of ODEs"},"11":{"body":"To define the state variables \\(\\mathbf{u} \\), we create a special vector variable u_i. Note the particular name u is used to indicate that this is the state vector. The values that we use for u_i are the initial values of the state variables at \\( t=0 \\), so an initial condition \\( \\mathbf{u|(t=0)} = [x(0), y(0), z(0)] = [1, 0, 0] \\) is defined as: u_i { x = 1, y = 0, z = 0,\n} Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well. We can optionally define the time derivatives of the state variables, \\( \\mathbf{\\dot{u}} \\) as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0,\n} Here the initial values of the time derivatives are given. In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$.","breadcrumbs":"Defining ODEs » Defining state variables","id":"11","title":"Defining state variables"},"12":{"body":"We now define the right-hand-side function \\( F \\) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. For example, to define a simple system of ODEs: $$ \\begin{align*} \\frac{dx}{dt} &= y \\\\ \\frac{dy}{dt} &= -x \\\\ x(0) &= 1 \\\\ y(0) &= 0 \\\\ \\end{align*} $$ We write: u_i { x = 1, y = 0 }\nF_i { y, -x }","breadcrumbs":"Defining ODEs » Defining the ODE system equations","id":"12","title":"Defining the ODE system equations"},"13":{"body":"We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \\( M \\mathbf{\\dot{u}} \\). 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In this section we will show how to specify inputs and outputs in the DiffSL language.","breadcrumbs":"Inputs and Outputs » Inputs & Outputs","id":"14","title":"Inputs & Outputs"},"15":{"body":"We can override the values of any scalar variables by specifying them as input variables. To do this, we add a line at the top of the code to specify that these are input variables: in = [k]\nk { 1.0 }\nu { 0.1 }\nF { k * u } Here we have specified a single input parameter k that is used in the RHS function F. The value of k is set to 1.0 in the code, but this value is only a default, and can be overridden by passing in a value at solve time. We can use input parameters anywhere in the code, including in the definition of other input parameters. in = [k]\nk { 1.0 }\ng { 2 * k }\nF { g * u } or in the intial conditions of the state variables: in = [k]\nk { 1.0 }\nu_i { x = k,\n}\nF { u }","breadcrumbs":"Inputs and Outputs » Specifying inputs","id":"15","title":"Specifying inputs"},"16":{"body":"We can also specify the outputs of the system. These might be the state variables themselves, or they might be other variables that are calculated from the state variables. Here is an example where we simply output the elements of the state vector: u_i { x = 1.0, y = 2.0, z = 3.0,\n}\nout_i { x, y, z } or we can derive additional outputs from the state variables: u_i { x = 1.0, y = 2.0, z = 3.0,\n}\nout { x + y + z }","breadcrumbs":"Inputs and Outputs » Specifying outputs","id":"16","title":"Specifying outputs"},"2":{"body":"In the previous vector v_i, the subscript _i indicates that this is a 1D vector. Each subscript is a single character, and the number of subscripts indicates the number of dimensions of the tensor. You can use any character for the subscript, v_x { 1.0, 2.0, 3.0 }\nw_a { 2.0, 3.0 }\nv_i { 1.0, 2.0, 3.0, 4.0 }","breadcrumbs":"Tensor Variables » Subscripts","id":"2","title":"Subscripts"},"3":{"body":"Each element of a tensor can optionally give a range of index numbers, which is used by the compiler to determine the extent of each element. This is useful when defining higher dimensional tensors, such as matrices. For example, to define a 2x3 matrix $A$ with all elements set to 1.0, we write: A_ij { (0:2, 0:3) = 1.0,\n} Note the two subscript to indicate that this is a 2D tensor. The size of the single element is given in the brackets, we have two ranges 0:2 and 0:3 that correspond to the two dimensions of the matrix. Here is another example of a 4x2 matrix $B$ with rows 0 to 2 set to 1.0 and rows 3 to 4 set to 2.0: A_ij { (0:2, 0:3) = 1.0, (3:4, 0:3) = 2.0,\n} For specifying a single index, you can simply write the index number without the colon, for example to define a 3x3 identity matrix $I$: I_ij { (0, 0) = 1.0, (1, 1) = 1.0, (2, 2) = 1.0,\n} Note that the compiler will automatically infer the size of the tensor from the ranges you provide, so you don't need to specify the size of the tensor explicitly. Since the maximum index in the range is 2, the compiler will infer that the size of the tensor is 3x3. Notice also that we have not defined all the elements of the matrix, only the non-zero elements. The compiler will assume that all other elements are zero. Finally, you can also use the .. operator to specify a diagonal range of indices. For example, to define a 3x3 matrix $D$ with the diagonal elements set to 1.0: D_ij { (0..2, 0..2) = 1.0,\n}","breadcrumbs":"Tensor Variables » Ranges","id":"3","title":"Ranges"},"4":{"body":"We can automatically define a sparse matrix $B$ by simply specifying the non-zero elements: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 1) = 3.0,\n} The compiler will infer that this is a 2x2 matrix, and will automatically represent it as a sparse matrix. We can force the compiler to use a dense representation by specifying the zeros explicitly: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 0) = 0.0, (1, 1) = 3.0,\n} As well as specifying a sparse matrix, we can also define a diagonal matrix by specifying the diagonal elements: D_ij { (0, 0) = 1.0, (1, 1) = 2.0, (2, 2) = 3.0,\n} The compiler will infer that this is a 3x3 matrix, and will automatically represent it as a diagonal matrix.","breadcrumbs":"Tensor Variables » Sparse and diagonal matrices","id":"4","title":"Sparse and diagonal matrices"},"5":{"body":"Each element of a tensor can optionally be given a name or label that can be used to refer to the element in expressions. For example, to define a vector with two named elements: v_i { x = 1.0, y = 2.0,\n} Here we have defined a single tensor v_i with two named elements x and y. We can then refer to the individual elements in expressions, where they will act as if they were separate variables: v_i { x = 1.0, y = 2.0 }\nw_i { 2 * y, 3 * x }","breadcrumbs":"Tensor Variables » Labels","id":"5","title":"Labels"},"6":{"body":"We can use standard algebraic operations on variables. To refer to previously defined variables, we use the variable name, making sure to use the correct subscript if it is a vector or tensor. For example, to define a scalar variable $a$ as the sum of two other scalar variables $b$ and $c$, we write: b { 1.0 }\nc { 2.0 }\na { b + c } The scalar a will therefore be equal to 3.0. To define a vector variable \\( \\mathbf{v} \\) as the sum of two other vector variables \\( \\mathbf{u} \\) and \\( \\mathbf{w} \\), we write: u_i { 1.0, 2.0 }\nw_i { 3.0, 4.0 }\nv_i { u_i + w_i } Notice that the index of the vectors within the expression must match the index of the output vector v_i. So if we defined v_a instead of v_i, the expression would be: v_a { u_a + w_a }","breadcrumbs":"Tensor Operations » Tensor Operations","id":"6","title":"Tensor Operations"},"7":{"body":"For higher-dimensional tensors, the order of the indices in the expression relative to the output tensor is important. For example, we can use the indices to define a translation of a matrix. Here we define a new matrix \\( C \\) that is the sum of \\( A \\) and \\( B^T \\)$, where \\( B^T \\) is the transpose of \\( B \\) C_ij { A_ij + B_ji } Notice that the indices of \\( B^T \\) are reversed in the expression compared to the output tensor \\( C \\), indicating that we are indexing the rows of \\( B \\) with j and the columns with i when we calculate the (i, j) element of \\( C \\).","breadcrumbs":"Tensor Operations » Translations","id":"7","title":"Translations"},"8":{"body":"Broadcasting is supported in the language, so you can perform element-wise operations on tensors of different shapes. For example, the following will define a new vector \\( \\mathbf{d} \\) that is the sum of \\( \\mathbf{a} \\) and a scalar \\( k \\): a_i { 1.0, 2.0 }\nk { 3.0 }\nd_i { a_i + k } Here the scalar \\( k \\) is broadcast to the same shape as \\( \\mathbf{a} \\) before the addition. The output vector \\( \\mathbf{d} \\) will be \\( [4.0, 5.0] \\). DiffSL uses the same broadcasting rules as NumPy, and you can read more about this in the NumPy documentation .","breadcrumbs":"Tensor Operations » Broadcasting","id":"8","title":"Broadcasting"},"9":{"body":"The tensor indexing notation can also matrix-vector multiplications and any other contraction operations. Any indices that do not appear in the output tensor will be summed over. 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\ No newline at end of file diff --git a/tensors.html b/tensors.html new file mode 100644 index 0000000..ebdf443 --- /dev/null +++ b/tensors.html @@ -0,0 +1,322 @@ + + + + + + Tensor Variables - DiffSL + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1. DiffSL Language2. Tensor Variables3. Tensor Operations4. Defining ODEs5. Inputs and Outputs + + + + + + + + + + + + + + + + + + + + + + + Light + Rust + Coal + Navy + Ayu + + + + + + + DiffSL + + + + + + + + + + + + + + + + + + + + + + + + + + Defining tensor variables +The DiffSL language only has a single type of variable, which is a n-dimensional tensor filled with double precision floating point numbers. +These tensors can be dense, sparse or diagonal, and the compiler will try to choose the representation that is most efficient, preferring diagonal and then sparse matrices over dense. +The simplest tensor is a 0th dimensional scalar. For example, to define a scalar variable \( k )\$ with value 1.0, we write: +k { 1.0 } + +Here k is the label for the tensor, which we can later use to refer to it in expressions. In the curly brackets we have one or more elements of the tensor, +and here k only has a single element, which is a constant value 1. +Lets now define a 1-dimensional vector variable $\mathbf{v}$ with 3 elements: +v_i { + 1.0, + 2.0, + 3.0, +} + +The list of elements within a tensor are deliniated with a comma , and the trailing comma at the end of the list is optional. +Whitespace is ignored so you can also write this tensor all on the one line: +v_i { 1.0, 2.0, 3.0 } + +Subscripts +In the previous vector v_i, the subscript _i indicates that this is a 1D vector. Each subscript is a single character, +and the number of subscripts indicates the number of dimensions of the tensor. You can use any character for the subscript, +v_x { 1.0, 2.0, 3.0 } +w_a { 2.0, 3.0 } +v_i { 1.0, 2.0, 3.0, 4.0 } + +Ranges +Each element of a tensor can optionally give a range of index numbers, which is used by the compiler to determine the extent of each element. +This is useful when defining higher dimensional tensors, such as matrices. For example, to define a 2x3 matrix $A$ with all elements set to 1.0, we write: +A_ij { + (0:2, 0:3) = 1.0, +} + +Note the two subscript to indicate that this is a 2D tensor. The size of the +single element is given in the brackets, we have two ranges 0:2 and 0:3 that correspond to the two dimensions of the matrix. +Here is another example of a 4x2 matrix $B$ with rows 0 to 2 set to 1.0 and rows 3 to 4 set to 2.0: +A_ij { + (0:2, 0:3) = 1.0, + (3:4, 0:3) = 2.0, +} + +For specifying a single index, you can simply write the index number without the colon, for example to define a 3x3 identity matrix $I$: +I_ij { + (0, 0) = 1.0, + (1, 1) = 1.0, + (2, 2) = 1.0, +} + +Note that the compiler will automatically infer the size of the tensor from the ranges you provide, so you don't need to specify the size of the tensor explicitly. +Since the maximum index in the range is 2, the compiler will infer that the size of the tensor is 3x3. +Notice also that we have not defined all the elements of the matrix, only the non-zero elements. The compiler will assume that all other elements are zero. +Finally, you can also use the .. operator to specify a diagonal range of indices. For example, to define a 3x3 matrix $D$ with the diagonal elements set to 1.0: +D_ij { + (0..2, 0..2) = 1.0, +} + +Sparse and diagonal matrices +We can automatically define a sparse matrix $B$ by simply specifying the non-zero elements: +B_ij { + (0, 0) = 1.0, + (0, 1) = 2.0, + (1, 1) = 3.0, +} + +The compiler will infer that this is a 2x2 matrix, and will automatically represent it as a sparse matrix. +We can force the compiler to use a dense representation by specifying the zeros explicitly: +B_ij { + (0, 0) = 1.0, + (0, 1) = 2.0, + (1, 0) = 0.0, + (1, 1) = 3.0, +} + +As well as specifying a sparse matrix, we can also define a diagonal matrix by specifying the diagonal elements: +D_ij { + (0, 0) = 1.0, + (1, 1) = 2.0, + (2, 2) = 3.0, +} + +The compiler will infer that this is a 3x3 matrix, and will automatically represent it as a diagonal matrix. +Labels +Each element of a tensor can optionally be given a name or label that can be used to refer to the element in expressions. +For example, to define a vector with two named elements: +v_i { + x = 1.0, + y = 2.0, +} + +Here we have defined a single tensor v_i with two named elements x and y. We can then refer to the individual elements in expressions, where they will act as if they were separate variables: +v_i { x = 1.0, y = 2.0 } +w_i { 2 * y, 3 * x } + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
u_i { + x = 1.0, + y = 2.0, + z = 3.0, } +out_i { x, y, z }
F_i { - k1 * t + sin(t) +or we can derive additional outputs from the state variables: +u_i { + x = 1.0, + y = 2.0, + z = 3.0, } +out { x + y + z } -Mathematical functions -The DSL supports the following mathematical functions: - -pow(x, y) - x raised to the power of y -sin(x) - sine of x -cos(x) - cosine of x -tan(x) - tangent of x -exp(x) - exponential of x -log(x) - natural logarithm of x -sqrt(x) - square root of x -abs(x) - absolute value of x -sigmoid(x) - sigmoid function of x - diff --git a/searchindex.js b/searchindex.js index de8f5b4..6667118 100644 --- a/searchindex.js +++ b/searchindex.js @@ -1 +1 @@ -Object.assign(window.search, {"doc_urls":["../../index.html#diffsl","../../index.html#diffsl-language-features","../../index.html#dependencies","../../index.html#installing-diffsl","language.html#diffsl-language","language.html#defining-variables","language.html#operations","language.html#specifying-inputs","language.html#defining-state-variables","language.html#defining-the-ode-system-equations","language.html#specifying-outputs","language.html#required-variables","language.html#predefined-variables","language.html#mathematical-functions"],"index":{"documentStore":{"docInfo":{"0":{"body":106,"breadcrumbs":2,"title":1},"1":{"body":35,"breadcrumbs":4,"title":3},"10":{"body":22,"breadcrumbs":4,"title":2},"11":{"body":22,"breadcrumbs":4,"title":2},"12":{"body":16,"breadcrumbs":4,"title":2},"13":{"body":39,"breadcrumbs":4,"title":2},"2":{"body":21,"breadcrumbs":2,"title":1},"3":{"body":54,"breadcrumbs":3,"title":2},"4":{"body":64,"breadcrumbs":4,"title":2},"5":{"body":137,"breadcrumbs":4,"title":2},"6":{"body":110,"breadcrumbs":3,"title":1},"7":{"body":17,"breadcrumbs":4,"title":2},"8":{"body":77,"breadcrumbs":5,"title":3},"9":{"body":107,"breadcrumbs":6,"title":4}},"docs":{"0":{"body":"A compiler for a domain-specific language for ordinary differential equations (ODEs) of the following form: $$ M(t) \\frac{d\\mathbf{u}}{dt} = F(\\mathbf{u}, t) $$ As an example, the following code defines a classic DAE testcase, the Robertson (1966) problem, which models the kinetics of an autocatalytic reaction, given by the following set of equations: $$ \\begin{align} \\frac{dx}{dt} &= -0.04x + 10^4 y z \\ \\frac{dy}{dt} &= 0.04x - 10^4 y z - 3 \\cdot 10^7 y^2 \\ 0 &= x + y + z - 1 \\end{align} $$ The DiffSL code for this problem is as follows: in = [k1, k2, k3]\nk1 { 0.04 }\nk2 { 10000 }\nk3 { 30000000 }\nu_i { x = 1, y = 0, z = 0,\n}\ndudt_i { dxdt = 1, dydt = 0, dzdt = 0,\n}\nM_i { dxdt, dydt, 0,\n}\nF_i { -k1 * x + k2 * y * z, k1 * x - k2 * y * z - k3 * y * y, 1 - x - y - z,\n}\nout_i { x, y, z,\n}","breadcrumbs":"Introduction » DiffSL","id":"0","title":"DiffSL"},"1":{"body":"See the DiffSL Language section for a full description. Tensor types: Scalars (double precision floating point numbers) Vectors (1D arrays of scalars) N-dimensional tensor of scalars Sparse/dense/diagonal tensors Tensor operations: Elementwise operations Broadcasting Tensor contractions/matmul/translation etc via index notation","breadcrumbs":"Introduction » DiffSL Language Features","id":"1","title":"DiffSL Language Features"},"10":{"body":"Finally, we specify the outputs of the system. These might be the state variables themselves, or they might be other variables that are calculated from the state variables. Here we specify that we want to output the state variables $x$ and $y$: out_i { x, y,\n}","breadcrumbs":"DiffSL Language » Specifying outputs","id":"10","title":"Specifying outputs"},"11":{"body":"The DSL allows the user to specify an arbitrary number of intermediate variables, but certain variables are required to be defined. These are: u_i - the state variables F_i - the vector $F(\\mathbf{u}, t)$ out_i - the output variables","breadcrumbs":"DiffSL Language » Required variables","id":"11","title":"Required variables"},"12":{"body":"The only predefined variable is the scalar $t$ which is the current time, this allows the equations to be written as functions of time. For example F_i { k1 * t + sin(t)\n}","breadcrumbs":"DiffSL Language » Predefined variables","id":"12","title":"Predefined variables"},"13":{"body":"The DSL supports the following mathematical functions: pow(x, y) - x raised to the power of y sin(x) - sine of x cos(x) - cosine of x tan(x) - tangent of x exp(x) - exponential of x log(x) - natural logarithm of x sqrt(x) - square root of x abs(x) - absolute value of x sigmoid(x) - sigmoid function of x","breadcrumbs":"DiffSL Language » Mathematical functions","id":"13","title":"Mathematical functions"},"2":{"body":"You will need to install the LLVM project . The easiest way to install this is to use the package manager for your operating system. For example, on Ubuntu you can install these with the following command: sudo apt-get install llvm","breadcrumbs":"Introduction » Dependencies","id":"2","title":"Dependencies"},"3":{"body":"You can install DiffSL using cargo. You will need to indicate the llvm version you have installed using a feature flag. For example, for llvm 14: cargo add diffsl --features llvm14-0 Other versions of llvm are also supported given by the features llvm4-0, llvm5-0, llvm6-0, llvm7-0, llvm8-0, llvm9-0, llvm10-0, llvm11-0, llvm12-0, llvm13-0, llvm14-0, llvm15-0, llvm16-0, llvm17-0.","breadcrumbs":"Introduction » Installing DiffSL","id":"3","title":"Installing DiffSL"},"4":{"body":"The DSL is designed to be an easy and flexible language for specifying DAE systems and is based on the idea that a DAE system can be specified by a set of equations of the form: $$ M(t) \\frac{d\\mathbf{u}}{dt} = F(\\mathbf{u}, t) $$ where $\\mathbf{u}$ is the vector of state variables and $t$ is the time. The DSL allows the user to specify the state vector $\\mathbf{u}$ and the RHS function $F$. Optionally, the user can also define the derivative of the state vector $d\\mathbf{u}/dt$ and the mass matrix $M$ as a function of $d\\mathbf{u}/dt$ (note that this function should be linear!). 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To define dense 2x3 matrix $A$ with all elements set to 1.0, we write: A_ij { (0:2, 0:3) = 1.0,\n} Note the two subscript to indicate that this is a 2D tensor. The size of the tensor is given in the brackets, and the elements are set to $1$. If we have additional rows, we can add them as follows: A_ij { (0:2, 0:3) = 1.0, (3:4, 0:3) = 2.0,\n} We can define a sparse matrix $B$ by specifying the non-zero elements: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 1) = 3.0,\n} We can also define a diagonal identity matrix $I$ by specifying the diagonal elements using a different range syntax: I_ij { (0..2, 0..2) = 1.0,\n}","breadcrumbs":"DiffSL Language » Defining variables","id":"5","title":"Defining variables"},"6":{"body":"We can use standard algebraic operations on variables. To refer to previously defined variables, we use the variable name, making sure to use the correct subscript if it is a vector or tensor. For example, to define a scalar variable $a$ as the sum of two other scalar variables $b$ and $c$, we write: a { b + c } To define a vector variable $\\mathbf{V}$ as the sum of two other vector variables $\\mathbf{u}$ and $\\mathbf{w}$, we write: v_i { u_i + w_i } The indexing can be used to perform translations on tensors, for example the following will define a new tensor $C$ that is the sum of $A$ and $B^T$: C_ij { A_ij + B_ji } Tensor indexing notation can also matrix-vector multiplications and any other contraction operations. Any indices that do not appear in the output will be summed over. 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The values that we use for u_i are the initial values of the state variables at $t=0$. u_i { x = 1, y = 0, z = 0,\n} We can optionally define the time derivatives of the state variables, $\\mathbf{\\dot{u}}$ as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0,\n} Here the initial values of the time derivatives are given, these are typically used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$.","breadcrumbs":"DiffSL Language » Defining state variables","id":"8","title":"Defining state variables"},"9":{"body":"Recall that the DAE system is defined by the equations: $$ M(t) \\frac{d\\mathbf{u}}{dt} = F(\\mathbf{u}, t) $$ We now define the equations $F$ and $M$ that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. 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Language The DSL is designed to be an easy and flexible language for specifying DAE systems and is based on the idea that a DAE system can be specified by a set of equations of the form: $$ M(t) \\frac{d\\mathbf{u}}{dt} = F(\\mathbf{u}, t) $$ where \\( \\mathbf{u}$ \\) is the vector of state variables and \\( t \\) is the time. The DSL allows the user to specify the state vector \\( \\mathbf{u} \\) and the RHS function \\( F \\). Optionally, the user can also define the derivative of the state vector \\( d\\mathbf{u}/dt \\) and the mass matrix \\( M \\) as a function of \\( d\\mathbf{u}/dt \\) (note that this function should be linear!). The user is also free to define an an arbitrary number of intermediate scalars and vectors of the users that are required to calculate \\( F \\) and \\( M \\). To illustrate the language, consider the following simple example of a logistic growth model: $$ \\frac{dN}{dt} = r N (1 - N/K) $$ where \\( N \\) is the population, \\( r \\) is the growth rate, and \\( K \\) is the carrying capacity. To specify this model in DiffSL, we can write: in = [r, k] u_i { N = 0.0 } F_i { r * N * (1 - N/k) } out_i { N } Here, we define the input parameters for our model as a vector in with the growth rate r and the carrying capacity k. We then define the state vector u_i with the population N initialized to 0.0. Next, we define the RHS function F_i as the logistic growth equation. Finally, we define the output vector out_i with the population N.","breadcrumbs":"DiffSL Language » A Simple Example","id":"0","title":"A Simple Example"},"1":{"body":"The DiffSL language only has a single type of variable, which is a n-dimensional tensor filled with double precision floating point numbers. These tensors can be dense, sparse or diagonal, and the compiler will try to choose the representation that is most efficient, preferring diagonal and then sparse matrices over dense. The simplest tensor is a 0th dimensional scalar. For example, to define a scalar variable \\( k )\\$ with value 1.0, we write: k { 1.0 } Here k is the label for the tensor, which we can later use to refer to it in expressions. In the curly brackets we have one or more elements of the tensor, and here k only has a single element, which is a constant value 1. Lets now define a 1-dimensional vector variable $\\mathbf{v}$ with 3 elements: v_i { 1.0, 2.0, 3.0,\n} The list of elements within a tensor are deliniated with a comma , and the trailing comma at the end of the list is optional. Whitespace is ignored so you can also write this tensor all on the one line: v_i { 1.0, 2.0, 3.0 }","breadcrumbs":"Tensor Variables » Defining tensor variables","id":"1","title":"Defining tensor variables"},"10":{"body":"The primary goal of the DiffSL language is to define a system of ODEs in the following form: $$ \\begin{align*} M(t) \\frac{d\\mathbf{u}}{dt} &= F(\\mathbf{u}, t) \\\\ \\mathbf{u}(0) &= \\mathbf{u}_0 \\end{align*} $$ where \\( \\mathbf{u} \\) is the vector of state variables, \\( \\mathbf{u}_0 \\) is the initial condition, \\( F \\) is the RHS function, and \\( M \\) is the mass matrix. The DSL allows the user to specify the state vector \\( \\mathbf{u} \\) and the RHS function \\( F \\). Optionally, the user can also define the derivative of the state vector \\( d\\mathbf{u}/dt \\) and the mass matrix \\( M \\) as a function of \\( d\\mathbf{u}/dt \\) (note that this function should be linear!). The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \\( F \\) and \\( M \\).","breadcrumbs":"Defining ODEs » Defining a system of ODEs","id":"10","title":"Defining a system of ODEs"},"11":{"body":"To define the state variables \\(\\mathbf{u} \\), we create a special vector variable u_i. Note the particular name u is used to indicate that this is the state vector. The values that we use for u_i are the initial values of the state variables at \\( t=0 \\), so an initial condition \\( \\mathbf{u|(t=0)} = [x(0), y(0), z(0)] = [1, 0, 0] \\) is defined as: u_i { x = 1, y = 0, z = 0,\n} Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well. We can optionally define the time derivatives of the state variables, \\( \\mathbf{\\dot{u}} \\) as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0,\n} Here the initial values of the time derivatives are given. In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$.","breadcrumbs":"Defining ODEs » Defining state variables","id":"11","title":"Defining state variables"},"12":{"body":"We now define the right-hand-side function \\( F \\) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. For example, to define a simple system of ODEs: $$ \\begin{align*} \\frac{dx}{dt} &= y \\\\ \\frac{dy}{dt} &= -x \\\\ x(0) &= 1 \\\\ y(0) &= 0 \\\\ \\end{align*} $$ We write: u_i { x = 1, y = 0 }\nF_i { y, -x }","breadcrumbs":"Defining ODEs » Defining the ODE system equations","id":"12","title":"Defining the ODE system equations"},"13":{"body":"We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \\( M \\mathbf{\\dot{u}} \\). This is optional, and if not defined, the mass matrix is assumed to be the identity matrix. Notice that we are defining a vector variable M_i, which is the LHS of the ODE equations \\( M \\mathbf{\\dot{u}} \\), and not the mass matrix itself. For example, lets define a simple DAE system using a singular mass matrix with a zero on the diagonal: $$ \\begin{align*} \\frac{dx}{dt} &= x \\\\ 0 &= y-x \\\\ x(0) &= 1 \\\\ y(0) &= 0 \\\\ \\end{align*} $$ We write: u_i { x = 1, y = 0,\n}\ndudt_i { dxdt = 0, dydt = 1,\n}\nM_i { dxdt, 0,\n}\nF_i { x, y-x,\n}","breadcrumbs":"Defining ODEs » Defining the mass matrix","id":"13","title":"Defining the mass matrix"},"14":{"body":"Often it is useful to parameterize the system of equations using a set of input parameters. It is also useful to be able to extract certain variables from the system for further analysis. In this section we will show how to specify inputs and outputs in the DiffSL language.","breadcrumbs":"Inputs and Outputs » Inputs & Outputs","id":"14","title":"Inputs & Outputs"},"15":{"body":"We can override the values of any scalar variables by specifying them as input variables. To do this, we add a line at the top of the code to specify that these are input variables: in = [k]\nk { 1.0 }\nu { 0.1 }\nF { k * u } Here we have specified a single input parameter k that is used in the RHS function F. The value of k is set to 1.0 in the code, but this value is only a default, and can be overridden by passing in a value at solve time. We can use input parameters anywhere in the code, including in the definition of other input parameters. in = [k]\nk { 1.0 }\ng { 2 * k }\nF { g * u } or in the intial conditions of the state variables: in = [k]\nk { 1.0 }\nu_i { x = k,\n}\nF { u }","breadcrumbs":"Inputs and Outputs » Specifying inputs","id":"15","title":"Specifying inputs"},"16":{"body":"We can also specify the outputs of the system. These might be the state variables themselves, or they might be other variables that are calculated from the state variables. Here is an example where we simply output the elements of the state vector: u_i { x = 1.0, y = 2.0, z = 3.0,\n}\nout_i { x, y, z } or we can derive additional outputs from the state variables: u_i { x = 1.0, y = 2.0, z = 3.0,\n}\nout { x + y + z }","breadcrumbs":"Inputs and Outputs » Specifying outputs","id":"16","title":"Specifying outputs"},"2":{"body":"In the previous vector v_i, the subscript _i indicates that this is a 1D vector. Each subscript is a single character, and the number of subscripts indicates the number of dimensions of the tensor. You can use any character for the subscript, v_x { 1.0, 2.0, 3.0 }\nw_a { 2.0, 3.0 }\nv_i { 1.0, 2.0, 3.0, 4.0 }","breadcrumbs":"Tensor Variables » Subscripts","id":"2","title":"Subscripts"},"3":{"body":"Each element of a tensor can optionally give a range of index numbers, which is used by the compiler to determine the extent of each element. This is useful when defining higher dimensional tensors, such as matrices. For example, to define a 2x3 matrix $A$ with all elements set to 1.0, we write: A_ij { (0:2, 0:3) = 1.0,\n} Note the two subscript to indicate that this is a 2D tensor. The size of the single element is given in the brackets, we have two ranges 0:2 and 0:3 that correspond to the two dimensions of the matrix. Here is another example of a 4x2 matrix $B$ with rows 0 to 2 set to 1.0 and rows 3 to 4 set to 2.0: A_ij { (0:2, 0:3) = 1.0, (3:4, 0:3) = 2.0,\n} For specifying a single index, you can simply write the index number without the colon, for example to define a 3x3 identity matrix $I$: I_ij { (0, 0) = 1.0, (1, 1) = 1.0, (2, 2) = 1.0,\n} Note that the compiler will automatically infer the size of the tensor from the ranges you provide, so you don't need to specify the size of the tensor explicitly. Since the maximum index in the range is 2, the compiler will infer that the size of the tensor is 3x3. Notice also that we have not defined all the elements of the matrix, only the non-zero elements. The compiler will assume that all other elements are zero. Finally, you can also use the .. operator to specify a diagonal range of indices. For example, to define a 3x3 matrix $D$ with the diagonal elements set to 1.0: D_ij { (0..2, 0..2) = 1.0,\n}","breadcrumbs":"Tensor Variables » Ranges","id":"3","title":"Ranges"},"4":{"body":"We can automatically define a sparse matrix $B$ by simply specifying the non-zero elements: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 1) = 3.0,\n} The compiler will infer that this is a 2x2 matrix, and will automatically represent it as a sparse matrix. We can force the compiler to use a dense representation by specifying the zeros explicitly: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 0) = 0.0, (1, 1) = 3.0,\n} As well as specifying a sparse matrix, we can also define a diagonal matrix by specifying the diagonal elements: D_ij { (0, 0) = 1.0, (1, 1) = 2.0, (2, 2) = 3.0,\n} The compiler will infer that this is a 3x3 matrix, and will automatically represent it as a diagonal matrix.","breadcrumbs":"Tensor Variables » Sparse and diagonal matrices","id":"4","title":"Sparse and diagonal matrices"},"5":{"body":"Each element of a tensor can optionally be given a name or label that can be used to refer to the element in expressions. For example, to define a vector with two named elements: v_i { x = 1.0, y = 2.0,\n} Here we have defined a single tensor v_i with two named elements x and y. We can then refer to the individual elements in expressions, where they will act as if they were separate variables: v_i { x = 1.0, y = 2.0 }\nw_i { 2 * y, 3 * x }","breadcrumbs":"Tensor Variables » Labels","id":"5","title":"Labels"},"6":{"body":"We can use standard algebraic operations on variables. To refer to previously defined variables, we use the variable name, making sure to use the correct subscript if it is a vector or tensor. For example, to define a scalar variable $a$ as the sum of two other scalar variables $b$ and $c$, we write: b { 1.0 }\nc { 2.0 }\na { b + c } The scalar a will therefore be equal to 3.0. To define a vector variable \\( \\mathbf{v} \\) as the sum of two other vector variables \\( \\mathbf{u} \\) and \\( \\mathbf{w} \\), we write: u_i { 1.0, 2.0 }\nw_i { 3.0, 4.0 }\nv_i { u_i + w_i } Notice that the index of the vectors within the expression must match the index of the output vector v_i. So if we defined v_a instead of v_i, the expression would be: v_a { u_a + w_a }","breadcrumbs":"Tensor Operations » Tensor Operations","id":"6","title":"Tensor Operations"},"7":{"body":"For higher-dimensional tensors, the order of the indices in the expression relative to the output tensor is important. For example, we can use the indices to define a translation of a matrix. Here we define a new matrix \\( C \\) that is the sum of \\( A \\) and \\( B^T \\)$, where \\( B^T \\) is the transpose of \\( B \\) C_ij { A_ij + B_ji } Notice that the indices of \\( B^T \\) are reversed in the expression compared to the output tensor \\( C \\), indicating that we are indexing the rows of \\( B \\) with j and the columns with i when we calculate the (i, j) element of \\( C \\).","breadcrumbs":"Tensor Operations » Translations","id":"7","title":"Translations"},"8":{"body":"Broadcasting is supported in the language, so you can perform element-wise operations on tensors of different shapes. For example, the following will define a new vector \\( \\mathbf{d} \\) that is the sum of \\( \\mathbf{a} \\) and a scalar \\( k \\): a_i { 1.0, 2.0 }\nk { 3.0 }\nd_i { a_i + k } Here the scalar \\( k \\) is broadcast to the same shape as \\( \\mathbf{a} \\) before the addition. The output vector \\( \\mathbf{d} \\) will be \\( [4.0, 5.0] \\). DiffSL uses the same broadcasting rules as NumPy, and you can read more about this in the NumPy documentation .","breadcrumbs":"Tensor Operations » Broadcasting","id":"8","title":"Broadcasting"},"9":{"body":"The tensor indexing notation can also matrix-vector multiplications and any other contraction operations. Any indices that do not appear in the output tensor will be summed over. For example, the following will define a new vector $v$ that is the result of a matrix-vector multiplication: v_i { A_ij * u_j } Here the j index is summed over, so the i index of the output vector v is the sum of the element-wise product of the ith row of A and the vector u. Another way to think about this matrix-vector multiplication is by considering it as a broadcasted element-wise multiplication followed by a sum or contraction over the j index. The A_ij * u_j expression broadcasts the vector u to the same shape as A, forming a new 2D tensor where each row is the element-wise product of the ith row of A and the vector u. When we specify the output vector v_i, we implicitly then sum over the missing j index to form the final output vector. 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Language The DSL is designed to be an easy and flexible language for specifying DAE systems and is based on the idea that a DAE system can be specified by a set of equations of the form: $$ M(t) \\frac{d\\mathbf{u}}{dt} = F(\\mathbf{u}, t) $$ where \\( \\mathbf{u}$ \\) is the vector of state variables and \\( t \\) is the time. The DSL allows the user to specify the state vector \\( \\mathbf{u} \\) and the RHS function \\( F \\). Optionally, the user can also define the derivative of the state vector \\( d\\mathbf{u}/dt \\) and the mass matrix \\( M \\) as a function of \\( d\\mathbf{u}/dt \\) (note that this function should be linear!). The user is also free to define an an arbitrary number of intermediate scalars and vectors of the users that are required to calculate \\( F \\) and \\( M \\). To illustrate the language, consider the following simple example of a logistic growth model: $$ \\frac{dN}{dt} = r N (1 - N/K) $$ where \\( N \\) is the population, \\( r \\) is the growth rate, and \\( K \\) is the carrying capacity. To specify this model in DiffSL, we can write: in = [r, k] u_i { N = 0.0 } F_i { r * N * (1 - N/k) } out_i { N } Here, we define the input parameters for our model as a vector in with the growth rate r and the carrying capacity k. We then define the state vector u_i with the population N initialized to 0.0. Next, we define the RHS function F_i as the logistic growth equation. Finally, we define the output vector out_i with the population N.","breadcrumbs":"DiffSL Language » A Simple Example","id":"0","title":"A Simple Example"},"1":{"body":"The DiffSL language only has a single type of variable, which is a n-dimensional tensor filled with double precision floating point numbers. These tensors can be dense, sparse or diagonal, and the compiler will try to choose the representation that is most efficient, preferring diagonal and then sparse matrices over dense. The simplest tensor is a 0th dimensional scalar. For example, to define a scalar variable \\( k )\\$ with value 1.0, we write: k { 1.0 } Here k is the label for the tensor, which we can later use to refer to it in expressions. In the curly brackets we have one or more elements of the tensor, and here k only has a single element, which is a constant value 1. Lets now define a 1-dimensional vector variable $\\mathbf{v}$ with 3 elements: v_i { 1.0, 2.0, 3.0,\n} The list of elements within a tensor are deliniated with a comma , and the trailing comma at the end of the list is optional. Whitespace is ignored so you can also write this tensor all on the one line: v_i { 1.0, 2.0, 3.0 }","breadcrumbs":"Tensor Variables » Defining tensor variables","id":"1","title":"Defining tensor variables"},"10":{"body":"The primary goal of the DiffSL language is to define a system of ODEs in the following form: $$ \\begin{align*} M(t) \\frac{d\\mathbf{u}}{dt} &= F(\\mathbf{u}, t) \\\\ \\mathbf{u}(0) &= \\mathbf{u}_0 \\end{align*} $$ where \\( \\mathbf{u} \\) is the vector of state variables, \\( \\mathbf{u}_0 \\) is the initial condition, \\( F \\) is the RHS function, and \\( M \\) is the mass matrix. The DSL allows the user to specify the state vector \\( \\mathbf{u} \\) and the RHS function \\( F \\). Optionally, the user can also define the derivative of the state vector \\( d\\mathbf{u}/dt \\) and the mass matrix \\( M \\) as a function of \\( d\\mathbf{u}/dt \\) (note that this function should be linear!). The user is also free to define an arbitrary number of intermediate tensors that are required to calculate \\( F \\) and \\( M \\).","breadcrumbs":"Defining ODEs » Defining a system of ODEs","id":"10","title":"Defining a system of ODEs"},"11":{"body":"To define the state variables \\(\\mathbf{u} \\), we create a special vector variable u_i. Note the particular name u is used to indicate that this is the state vector. The values that we use for u_i are the initial values of the state variables at \\( t=0 \\), so an initial condition \\( \\mathbf{u|(t=0)} = [x(0), y(0), z(0)] = [1, 0, 0] \\) is defined as: u_i { x = 1, y = 0, z = 0,\n} Since we will often use the individual elements of the state vector in the RHS function, it is useful to define them as separate variables as well. We can optionally define the time derivatives of the state variables, \\( \\mathbf{\\dot{u}} \\) as well: dudt_i { dxdt = 1, dydt = 0, dzdt = 0,\n} Here the initial values of the time derivatives are given. In many cases any values can be given here as the time derivatives of the state variables are calculated from the RHS. However, if there are any algebraic variables in the equations then these values can be used used as a starting point to calculate a set of consistent initial values for the state variables. Note that there is no need to define dudt if you do not define a mass matrix $M$.","breadcrumbs":"Defining ODEs » Defining state variables","id":"11","title":"Defining state variables"},"12":{"body":"We now define the right-hand-side function \\( F \\) that we want to solve, using the variables that we have defined earlier. We do this by defining a vector variable F_i that corresponds to the RHS of the equations. For example, to define a simple system of ODEs: $$ \\begin{align*} \\frac{dx}{dt} &= y \\\\ \\frac{dy}{dt} &= -x \\\\ x(0) &= 1 \\\\ y(0) &= 0 \\\\ \\end{align*} $$ We write: u_i { x = 1, y = 0 }\nF_i { y, -x }","breadcrumbs":"Defining ODEs » Defining the ODE system equations","id":"12","title":"Defining the ODE system equations"},"13":{"body":"We can also define a mass matrix $M$ by defining a vector variable M_i which is the product of the mass matrix with the time derivative of the state vector \\( M \\mathbf{\\dot{u}} \\). 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In this section we will show how to specify inputs and outputs in the DiffSL language.","breadcrumbs":"Inputs and Outputs » Inputs & Outputs","id":"14","title":"Inputs & Outputs"},"15":{"body":"We can override the values of any scalar variables by specifying them as input variables. To do this, we add a line at the top of the code to specify that these are input variables: in = [k]\nk { 1.0 }\nu { 0.1 }\nF { k * u } Here we have specified a single input parameter k that is used in the RHS function F. The value of k is set to 1.0 in the code, but this value is only a default, and can be overridden by passing in a value at solve time. We can use input parameters anywhere in the code, including in the definition of other input parameters. in = [k]\nk { 1.0 }\ng { 2 * k }\nF { g * u } or in the intial conditions of the state variables: in = [k]\nk { 1.0 }\nu_i { x = k,\n}\nF { u }","breadcrumbs":"Inputs and Outputs » Specifying inputs","id":"15","title":"Specifying inputs"},"16":{"body":"We can also specify the outputs of the system. These might be the state variables themselves, or they might be other variables that are calculated from the state variables. Here is an example where we simply output the elements of the state vector: u_i { x = 1.0, y = 2.0, z = 3.0,\n}\nout_i { x, y, z } or we can derive additional outputs from the state variables: u_i { x = 1.0, y = 2.0, z = 3.0,\n}\nout { x + y + z }","breadcrumbs":"Inputs and Outputs » Specifying outputs","id":"16","title":"Specifying outputs"},"2":{"body":"In the previous vector v_i, the subscript _i indicates that this is a 1D vector. Each subscript is a single character, and the number of subscripts indicates the number of dimensions of the tensor. You can use any character for the subscript, v_x { 1.0, 2.0, 3.0 }\nw_a { 2.0, 3.0 }\nv_i { 1.0, 2.0, 3.0, 4.0 }","breadcrumbs":"Tensor Variables » Subscripts","id":"2","title":"Subscripts"},"3":{"body":"Each element of a tensor can optionally give a range of index numbers, which is used by the compiler to determine the extent of each element. This is useful when defining higher dimensional tensors, such as matrices. For example, to define a 2x3 matrix $A$ with all elements set to 1.0, we write: A_ij { (0:2, 0:3) = 1.0,\n} Note the two subscript to indicate that this is a 2D tensor. The size of the single element is given in the brackets, we have two ranges 0:2 and 0:3 that correspond to the two dimensions of the matrix. Here is another example of a 4x2 matrix $B$ with rows 0 to 2 set to 1.0 and rows 3 to 4 set to 2.0: A_ij { (0:2, 0:3) = 1.0, (3:4, 0:3) = 2.0,\n} For specifying a single index, you can simply write the index number without the colon, for example to define a 3x3 identity matrix $I$: I_ij { (0, 0) = 1.0, (1, 1) = 1.0, (2, 2) = 1.0,\n} Note that the compiler will automatically infer the size of the tensor from the ranges you provide, so you don't need to specify the size of the tensor explicitly. Since the maximum index in the range is 2, the compiler will infer that the size of the tensor is 3x3. Notice also that we have not defined all the elements of the matrix, only the non-zero elements. The compiler will assume that all other elements are zero. Finally, you can also use the .. operator to specify a diagonal range of indices. For example, to define a 3x3 matrix $D$ with the diagonal elements set to 1.0: D_ij { (0..2, 0..2) = 1.0,\n}","breadcrumbs":"Tensor Variables » Ranges","id":"3","title":"Ranges"},"4":{"body":"We can automatically define a sparse matrix $B$ by simply specifying the non-zero elements: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 1) = 3.0,\n} The compiler will infer that this is a 2x2 matrix, and will automatically represent it as a sparse matrix. We can force the compiler to use a dense representation by specifying the zeros explicitly: B_ij { (0, 0) = 1.0, (0, 1) = 2.0, (1, 0) = 0.0, (1, 1) = 3.0,\n} As well as specifying a sparse matrix, we can also define a diagonal matrix by specifying the diagonal elements: D_ij { (0, 0) = 1.0, (1, 1) = 2.0, (2, 2) = 3.0,\n} The compiler will infer that this is a 3x3 matrix, and will automatically represent it as a diagonal matrix.","breadcrumbs":"Tensor Variables » Sparse and diagonal matrices","id":"4","title":"Sparse and diagonal matrices"},"5":{"body":"Each element of a tensor can optionally be given a name or label that can be used to refer to the element in expressions. For example, to define a vector with two named elements: v_i { x = 1.0, y = 2.0,\n} Here we have defined a single tensor v_i with two named elements x and y. We can then refer to the individual elements in expressions, where they will act as if they were separate variables: v_i { x = 1.0, y = 2.0 }\nw_i { 2 * y, 3 * x }","breadcrumbs":"Tensor Variables » Labels","id":"5","title":"Labels"},"6":{"body":"We can use standard algebraic operations on variables. To refer to previously defined variables, we use the variable name, making sure to use the correct subscript if it is a vector or tensor. For example, to define a scalar variable $a$ as the sum of two other scalar variables $b$ and $c$, we write: b { 1.0 }\nc { 2.0 }\na { b + c } The scalar a will therefore be equal to 3.0. To define a vector variable \\( \\mathbf{v} \\) as the sum of two other vector variables \\( \\mathbf{u} \\) and \\( \\mathbf{w} \\), we write: u_i { 1.0, 2.0 }\nw_i { 3.0, 4.0 }\nv_i { u_i + w_i } Notice that the index of the vectors within the expression must match the index of the output vector v_i. So if we defined v_a instead of v_i, the expression would be: v_a { u_a + w_a }","breadcrumbs":"Tensor Operations » Tensor Operations","id":"6","title":"Tensor Operations"},"7":{"body":"For higher-dimensional tensors, the order of the indices in the expression relative to the output tensor is important. For example, we can use the indices to define a translation of a matrix. Here we define a new matrix \\( C \\) that is the sum of \\( A \\) and \\( B^T \\)$, where \\( B^T \\) is the transpose of \\( B \\) C_ij { A_ij + B_ji } Notice that the indices of \\( B^T \\) are reversed in the expression compared to the output tensor \\( C \\), indicating that we are indexing the rows of \\( B \\) with j and the columns with i when we calculate the (i, j) element of \\( C \\).","breadcrumbs":"Tensor Operations » Translations","id":"7","title":"Translations"},"8":{"body":"Broadcasting is supported in the language, so you can perform element-wise operations on tensors of different shapes. For example, the following will define a new vector \\( \\mathbf{d} \\) that is the sum of \\( \\mathbf{a} \\) and a scalar \\( k \\): a_i { 1.0, 2.0 }\nk { 3.0 }\nd_i { a_i + k } Here the scalar \\( k \\) is broadcast to the same shape as \\( \\mathbf{a} \\) before the addition. The output vector \\( \\mathbf{d} \\) will be \\( [4.0, 5.0] \\). DiffSL uses the same broadcasting rules as NumPy, and you can read more about this in the NumPy documentation .","breadcrumbs":"Tensor Operations » Broadcasting","id":"8","title":"Broadcasting"},"9":{"body":"The tensor indexing notation can also matrix-vector multiplications and any other contraction operations. Any indices that do not appear in the output tensor will be summed over. 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\ No newline at end of file diff --git a/tensors.html b/tensors.html new file mode 100644 index 0000000..ebdf443 --- /dev/null +++ b/tensors.html @@ -0,0 +1,322 @@ + + + + + + Tensor Variables - DiffSL + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 1. DiffSL Language2. Tensor Variables3. Tensor Operations4. Defining ODEs5. Inputs and Outputs + + + + + + + + + + + + + + + + + + + + + + + Light + Rust + Coal + Navy + Ayu + + + + + + + DiffSL + + + + + + + + + + + + + + + + + + + + + + + + + + Defining tensor variables +The DiffSL language only has a single type of variable, which is a n-dimensional tensor filled with double precision floating point numbers. +These tensors can be dense, sparse or diagonal, and the compiler will try to choose the representation that is most efficient, preferring diagonal and then sparse matrices over dense. +The simplest tensor is a 0th dimensional scalar. For example, to define a scalar variable \( k )\$ with value 1.0, we write: +k { 1.0 } + +Here k is the label for the tensor, which we can later use to refer to it in expressions. In the curly brackets we have one or more elements of the tensor, +and here k only has a single element, which is a constant value 1. +Lets now define a 1-dimensional vector variable $\mathbf{v}$ with 3 elements: +v_i { + 1.0, + 2.0, + 3.0, +} + +The list of elements within a tensor are deliniated with a comma , and the trailing comma at the end of the list is optional. +Whitespace is ignored so you can also write this tensor all on the one line: +v_i { 1.0, 2.0, 3.0 } + +Subscripts +In the previous vector v_i, the subscript _i indicates that this is a 1D vector. Each subscript is a single character, +and the number of subscripts indicates the number of dimensions of the tensor. You can use any character for the subscript, +v_x { 1.0, 2.0, 3.0 } +w_a { 2.0, 3.0 } +v_i { 1.0, 2.0, 3.0, 4.0 } + +Ranges +Each element of a tensor can optionally give a range of index numbers, which is used by the compiler to determine the extent of each element. +This is useful when defining higher dimensional tensors, such as matrices. For example, to define a 2x3 matrix $A$ with all elements set to 1.0, we write: +A_ij { + (0:2, 0:3) = 1.0, +} + +Note the two subscript to indicate that this is a 2D tensor. The size of the +single element is given in the brackets, we have two ranges 0:2 and 0:3 that correspond to the two dimensions of the matrix. +Here is another example of a 4x2 matrix $B$ with rows 0 to 2 set to 1.0 and rows 3 to 4 set to 2.0: +A_ij { + (0:2, 0:3) = 1.0, + (3:4, 0:3) = 2.0, +} + +For specifying a single index, you can simply write the index number without the colon, for example to define a 3x3 identity matrix $I$: +I_ij { + (0, 0) = 1.0, + (1, 1) = 1.0, + (2, 2) = 1.0, +} + +Note that the compiler will automatically infer the size of the tensor from the ranges you provide, so you don't need to specify the size of the tensor explicitly. +Since the maximum index in the range is 2, the compiler will infer that the size of the tensor is 3x3. +Notice also that we have not defined all the elements of the matrix, only the non-zero elements. The compiler will assume that all other elements are zero. +Finally, you can also use the .. operator to specify a diagonal range of indices. For example, to define a 3x3 matrix $D$ with the diagonal elements set to 1.0: +D_ij { + (0..2, 0..2) = 1.0, +} + +Sparse and diagonal matrices +We can automatically define a sparse matrix $B$ by simply specifying the non-zero elements: +B_ij { + (0, 0) = 1.0, + (0, 1) = 2.0, + (1, 1) = 3.0, +} + +The compiler will infer that this is a 2x2 matrix, and will automatically represent it as a sparse matrix. +We can force the compiler to use a dense representation by specifying the zeros explicitly: +B_ij { + (0, 0) = 1.0, + (0, 1) = 2.0, + (1, 0) = 0.0, + (1, 1) = 3.0, +} + +As well as specifying a sparse matrix, we can also define a diagonal matrix by specifying the diagonal elements: +D_ij { + (0, 0) = 1.0, + (1, 1) = 2.0, + (2, 2) = 3.0, +} + +The compiler will infer that this is a 3x3 matrix, and will automatically represent it as a diagonal matrix. +Labels +Each element of a tensor can optionally be given a name or label that can be used to refer to the element in expressions. +For example, to define a vector with two named elements: +v_i { + x = 1.0, + y = 2.0, +} + +Here we have defined a single tensor v_i with two named elements x and y. We can then refer to the individual elements in expressions, where they will act as if they were separate variables: +v_i { x = 1.0, y = 2.0 } +w_i { 2 * y, 3 * x } + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
u_i { + x = 1.0, + y = 2.0, + z = 3.0, } +out { x + y + z }
k { 1.0 } +
v_i { + 1.0, + 2.0, + 3.0, +} +
A_ij { + (0:2, 0:3) = 1.0, +} +
Note the two subscript to indicate that this is a 2D tensor. The size of the +single element is given in the brackets, we have two ranges 0:2 and 0:3 that correspond to the two dimensions of the matrix.
A_ij { + (0:2, 0:3) = 1.0, + (3:4, 0:3) = 2.0, +} +
B_ij { + (0, 0) = 1.0, + (0, 1) = 2.0, + (1, 1) = 3.0, +} +
D_ij { + (0, 0) = 1.0, + (1, 1) = 2.0, + (2, 2) = 3.0, +} +