support ConstrainedEquation for PINNs

[KirillZubov]

Some examples of how this can be:

using ModelingToolkit

@parameters x y
@variables u(..)
@derivatives Dy'~y
@derivatives Dxx''~x

some_f(x,y) = x^2+y^2
eq = Dy(u(x,y)) - Dxx(u(x,y)) ~ some_f(x,y)

x_domain = IntervalDomain(0.0,3.0)
left_x_domain = IntervalDomain(0.0,1.0)
right_x_domain = IntervalDomain(2.0,3.0)
y_domain = IntervalDomain(0.0,1.0)

con_eq = ConstrainedEquation([x ∈ left_x_domain ⊗ right_x_domain, y ∈ y_domain],eq) #separate domain
con2_eq = ConstrainedEquation([x ~ 0,y < 1/2], u(x,y) ~ x + y^2)
con3_eq = ConstrainedEquation(x^2+y^2<0,  eq)

[dangirsh]

I understood the ProductDomain (⊗) to be intended for multidimensional variables - see the comment here. A new abstraction is probably needed for non-contiguous domains as like in this example.

Note that for the P2D model, I've worked-around this issue by using a symbolic IfElse (see here) to branch the equations for each battery domain. That's implemented here.

I think explicitly specifying the non-contiguous domain would be cleaner than the symbolic branch, so this should probably be left open.

[Mrzhang-hub]

I understood the ProductDomain (⊗) to be intended for multidimensional variables - see the comment here. A new abstraction is probably needed for non-contiguous domains as like in this example.

Note that for the P2D model, I've worked-around this issue by using a symbolic IfElse (see here) to branch the equations for each battery domain. That's implemented here.

I think explicitly specifying the non-contiguous domain would be cleaner than the symbolic branch, so this should probably be left open.

The implementation of P2D model using neuralPDE seems missing. Could you please share it again?