Merge pull request #32 from bowenszhu/update-readme

Add package intro and an example in README

Author: ChrisRackauckas

README.md

@@ -1,5 +1,69 @@
 # ModelOrderReduction.jl
 
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+
+ModelOrderReduction.jl is a package for automatically reducing the computational complexity
+of mathematical models, while keeping expected fidelity within a controlled error bound. 
+These methods function a submodel with a projection
+where solving the smaller model gives approximation information about the full model. 
+MOR.jl uses [ModelingToolkit.jl](https://github.com/SciML/ModelingToolkit.jl)
+as a system description and automatically transforms equations
+to the subform, defining the observables to automatically lazily reconstruct the full
+model on-demand in a fast and stable form.
+
+## Example
+#### Proper Orthogonal Decomposition and Discrete Empirical Interpolation Method (POD-DEIM) on the FitzHugh-Nagumo system
+```julia
+using ModelingToolkit, MethodOfLines, DifferentialEquations, ModelOrderReduction
+
+# firstly construct a ModelingToolkit.PDESystem for the FitzHugh-Nagumo model
+@variables x t v(..) w(..)
+Dx = Differential(x)
+Dxx = Dx^2
+Dt = Differential(t)
+const L = 1.0
+const ε = 0.015
+const b = 0.5
+const γ = 2.0
+const c = 0.05
+f(v) = v * (v - 0.1) * (1.0 - v)
+i₀(t) = 50000.0t^3 * exp(-15.0t)
+eqs = [ε * Dt(v(x, t)) ~ ε^2 * Dxx(v(x, t)) + f(v(x, t)) - w(x, t) + c,
+    Dt(w(x, t)) ~ b * v(x, t) - γ * w(x, t) + c]
+bcs = [v(x, 0.0) ~ 0.0, w(x, 0) ~ 0.0, Dx(v(0, t)) ~ -i₀(t), Dx(v(L, t)) ~ 0.0]
+domains = [x ∈ (0.0, L), t ∈ (0.0, 14.0)]
+ivs = [x, t]
+dvs = [v(x, t), w(x, t)]
+pde_sys = PDESystem(eqs, bcs, domains, ivs, dvs; name = Symbol("FitzHugh-Nagumo"))
+
+# transfer to a ModelingToolkit.ODESystem by automated discretization via MethodOfLines
+N = 15 # equidistant discretization intervals
+dx = (L - 0.0) / N
+dxs = [x => dx]
+discretization = MOLFiniteDifference(dxs, t)
+ode_sys, tspan = symbolic_discretize(pde_sys, discretization)
+simp_sys = structural_simplify(ode_sys)
+ode_prob = ODEProblem(simp_sys, nothing, tspan)
+
+# solve the full-order model to get snapshots
+sol = solve(ode_prob, Tsit5())
+snapshot_simpsys = Array(sol.original_sol)
+
+# set POD and DEIM dimensions
+# apply POD-DEIM to obtain the reduced-order model
+pod_dim = deim_dim = 5
+deim_sys = deim(simp_sys, snapshot_simpsys, pod_dim; deim_dim = deim_dim)
+deim_prob = ODEProblem(deim_sys, nothing, tspan)
+deim_sol = solve(deim_prob, Tsit5())
+
+# retrieve the approximate solution of the original full-order model
+sol_deim_x = deim_sol[x]
+sol_deim_v = deim_sol[v(x, t)]
+sol_deim_w = deim_sol[w(x, t)]
+```
+
+The following figure shows the comparison of the solutions of the 32-dimension full-order model and the POD5-DEIM5 reduced-order model.
+
+![comparison](https://user-images.githubusercontent.com/45696147/195765614-df9092a2-4fca-4602-bb15-81e65b2b572e.svg)