Solving PDEs with ModelingToolkit and NeuralPDE#
Solving Poisson PDE Systems (https://docs.sciml.ai/NeuralPDE/stable/tutorials/pdesystem/)
\[
\partial^{2}_{x}u(x,y) + \partial^{2}_{y}u(x,y) = -\sin (\pi x) \sin (\pi y)
\]
with boundary conditions
\[\begin{split}
\begin{align}
u(0, y) &= 0 \\
u(1, y) &= 0 \\
u(x, 0) &= 0 \\
u(x, 1) &= 0 \\
\end{align}
\end{split}\]
where
\(x ∈ [0, 1], y ∈ [0, 1]\)
using NeuralPDE
using Lux
using Optimization
using OptimizationOptimJL
using ModelingToolkit
using DomainSets
using LineSearches
using Plots
2D PDE
@parameters x y
@variables u(..)
Dxx = Differential(x)^2
Dyy = Differential(y)^2
eq = Dxx(u(x, y)) + Dyy(u(x, y)) ~ -sinpi(x) * sinpi(y)
\[ \begin{equation}
\frac{\mathrm{d}}{\mathrm{d}y} \frac{\mathrm{d}}{\mathrm{d}y} u\left( x, y \right) + \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d}}{\mathrm{d}x} u\left( x, y \right) = - sinpi\left( x \right) sinpi\left( y \right)
\end{equation}
\]
Boundary conditions
bcs = [
u(0, y) ~ 0.0,
u(1, y) ~ 0.0,
u(x, 0) ~ 0.0,
u(x, 1) ~ 0.0
]
\[\begin{split} \begin{align}
u\left( 0, y \right) &= 0 \\
u\left( 1, y \right) &= 0 \\
u\left( x, 0 \right) &= 0 \\
u\left( x, 1 \right) &= 0
\end{align}
\end{split}\]
Space domains
domains = [
x ∈ DomainSets.Interval(0.0, 1.0),
y ∈ DomainSets.Interval(0.0, 1.0)
]
2-element Vector{Symbolics.VarDomainPairing}:
Symbolics.VarDomainPairing(x, 0.0 .. 1.0)
Symbolics.VarDomainPairing(y, 0.0 .. 1.0)
Build a neural network for the PDE solver.
Input: 2 dimensions.
Hidden layers: 16 neurons * 2 layers.
Output: single output u(x, y)
dim = 2
chain = Lux.Chain(Dense(dim, 16, Lux.σ), Dense(16, 16, Lux.σ), Dense(16, 1))
Chain(
layer_1 = Dense(2 => 16, σ), # 48 parameters
layer_2 = Dense(16 => 16, σ), # 272 parameters
layer_3 = Dense(16 => 1), # 17 parameters
) # Total: 337 parameters,
# plus 0 states.
Discretization method usesPhysicsInformedNN() (PINN).
dx = 0.05
discretization = PhysicsInformedNN(chain, QuadratureTraining(; batch = 200, abstol = 1e-6, reltol = 1e-6))
NeuralPDE.PhysicsInformedNN{Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, NeuralPDE.QuadratureTraining{Float64, Integrals.CubatureJLh}, Nothing, Nothing, NeuralPDE.Phi{LuxCore.StatefulLuxLayerImpl.StatefulLuxLayer{Val{true}, Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, Nothing, @NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}}}, typeof(NeuralPDE.numeric_derivative), Bool, Nothing, Nothing, Nothing, Base.RefValue{Int64}, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}}(Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}((layer_1 = Dense(2 => 16, σ), layer_2 = Dense(16 => 16, σ), layer_3 = Dense(16 => 1)), nothing), NeuralPDE.QuadratureTraining{Float64, Integrals.CubatureJLh}(Integrals.CubatureJLh(0), 1.0e-6, 1.0e-6, 1000, 200), nothing, nothing, NeuralPDE.Phi{LuxCore.StatefulLuxLayerImpl.StatefulLuxLayer{Val{true}, Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, Nothing, @NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}}}(LuxCore.StatefulLuxLayerImpl.StatefulLuxLayer{Val{true}, Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}, Nothing, @NamedTuple{layer_1::@NamedTuple{}, layer_2::@NamedTuple{}, layer_3::@NamedTuple{}}}(Lux.Chain{@NamedTuple{layer_1::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_2::Lux.Dense{typeof(NNlib.σ), Int64, Int64, Nothing, Nothing, Static.True}, layer_3::Lux.Dense{typeof(identity), Int64, Int64, Nothing, Nothing, Static.True}}, Nothing}((layer_1 = Dense(2 => 16, σ), layer_2 = Dense(16 => 16, σ), layer_3 = Dense(16 => 1)), nothing), nothing, (layer_1 = NamedTuple(), layer_2 = NamedTuple(), layer_3 = NamedTuple()), nothing, Val{true}())), NeuralPDE.numeric_derivative, false, nothing, nothing, nothing, NeuralPDE.LogOptions(50), Base.RefValue{Int64}(1), true, false, Base.Pairs{Symbol, Union{}, Tuple{}, @NamedTuple{}}())
Build the PDE system and discretize it.
@named pde_system = PDESystem(eq, bcs, domains, [x, y], [u(x, y)])
prob = discretize(pde_system, discretization)
OptimizationProblem. In-place: true
u0: ComponentVector{Float64}(layer_1 = (weight = [0.7798402309417725 0.7714139223098755; 0.5802998542785645 0.4550359845161438; … ; -0.22006142139434814 -1.0936412811279297; -0.44520440697669983 1.166462779045105], bias = [0.24288538098335266, -0.40192458033561707, -0.1888885200023651, -0.04649345204234123, 0.1177700087428093, -0.6978550553321838, -0.21471956372261047, -0.05960913002490997, -0.10941161960363388, 0.2531987130641937, -0.2104482352733612, 0.6805310249328613, -0.23738285899162292, -0.18501876294612885, -0.21167142689228058, -0.2395436316728592]), layer_2 = (weight = [-0.2388668805360794 -0.22025632858276367 … -0.36523765325546265 -0.4286521077156067; 0.08630724996328354 0.3409085273742676 … 0.20255060493946075 0.025577591732144356; … ; 0.2902018129825592 0.35351452231407166 … 0.12296261638402939 0.11770133674144745; -0.2693678140640259 0.23862911760807037 … -0.40705570578575134 0.31215471029281616], bias = [0.2464446723461151, 0.046215057373046875, 0.24930480122566223, -0.06035134196281433, 0.1833418309688568, 0.06522560119628906, 0.23567911982536316, -0.07572835683822632, 0.22704532742500305, 0.18679052591323853, -0.1929524540901184, 0.07832619547843933, 0.23260295391082764, -0.07068678736686707, -0.01337626576423645, 0.20062783360481262]), layer_3 = (weight = [-0.4003361165523529 -0.29924434423446655 … -0.10808081924915314 -0.1426621824502945], bias = [0.1945304274559021]))
Callback function to record the loss
lossrecord = Float64[]
callback = function (p, l)
push!(lossrecord, l)
return false
end
#1 (generic function with 1 method)
Solve the problem. It may take a long time.
opt = OptimizationOptimJL.LBFGS(linesearch = LineSearches.BackTracking())
@time res = Optimization.solve(prob, opt, callback = callback, maxiters=1000)
997.075587 seconds (4.02 G allocations: 303.315 GiB, 3.75% gc time, 12.42% compilation time: <1% of which was recompilation)
retcode: Success
u: ComponentVector{Float64}(layer_1 = (weight = [0.26232464973338465 0.45880425204512687; 0.6402837101508196 0.6885308355711968; … ; 0.7000506750430829 -1.221748535803418; -2.1676108500802376 2.1361678333075034], bias = [-0.3931375953884019, -1.4123692283671643, 0.4663467292936297, 0.9741867884322727, 0.06064107505830261, -1.5540049311895578, -0.2023143336724759, 0.33300629288458816, 0.504492617124244, -0.24810627729663345, -1.0137417889097518, 2.5598155532673754, -0.3928532136610881, -0.11818131573229143, -0.6029112699149126, 0.77503355292077]), layer_2 = (weight = [-0.8448703226322946 -0.6961241440177792 … 0.09709474848929818 -0.20822342517911596; -0.08893327843668902 0.7155123964061884 … 0.6984127788038346 -0.17877451740860179; … ; 0.2437570216951347 0.3313208967151344 … 0.11317643576506745 0.08290354402867939; -0.46287790251631306 0.19442458435302676 … -0.3052662088921988 0.27852324650322224], bias = [0.3087754780251707, 0.10675688667129767, 0.32040996859416043, -0.141869038034641, 0.1666846386641946, 0.3711362822496281, 0.3011103898933208, -0.08294763579608508, 0.2516236207180197, 0.12512516124176928, -0.674660686064573, 0.01237463947186731, 0.31935495731511215, -0.03972503421392855, 0.06057960765538829, 0.14948313695291815]), layer_3 = (weight = [-1.2624623292702672 0.8472557931995872 … -0.2583310074587382 -0.06218957716564262], bias = [0.604627391779455]))
plot(lossrecord, xlabel="Iters", yscale=:log10, ylabel="Loss", lab=false)
Plot the predicted solution of the PDE and compare it with the analytical solution to see the relative error.
xs, ys = [DomainSets.infimum(d.domain):dx/10:DomainSets.supremum(d.domain) for d in domains]
analytic_sol_func(x,y) = (sinpi(x)*sinpi(y))/(2pi^2)
phi = discretization.phi
u_predict = reshape([first(phi([x, y], res.u)) for x in xs for y in ys], (length(xs), length(ys)))
u_real = reshape([analytic_sol_func(x, y) for x in xs for y in ys], (length(xs), length(ys)))
diff_u = abs.(u_predict .- u_real)
201×201 Matrix{Float64}:
0.00246386 0.00242635 0.00238977 … 0.00480502 0.00485943 0.00491334
0.00243076 0.00239399 0.00235814 0.00467159 0.00472378 0.00477548
0.00239742 0.0023614 0.00232626 0.00453856 0.00458854 0.00463802
0.00236384 0.00232857 0.00229414 0.00440595 0.00445373 0.00450099
0.00233003 0.00229548 0.00226177 0.00427379 0.00431935 0.0043644
0.00229597 0.00226215 0.00222913 … 0.00414209 0.00418544 0.00422827
0.00226166 0.00222856 0.00219624 0.00401087 0.00405202 0.00409263
0.00222709 0.00219471 0.00216308 0.00388014 0.00391909 0.00395749
0.00219227 0.00216059 0.00212966 0.00374994 0.00378669 0.00382288
0.00215718 0.00212622 0.00209597 0.00362028 0.00365482 0.0036888
⋮ ⋱ ⋮
0.00205923 0.00192203 0.00178713 0.00300274 0.00301729 0.00302986
0.0019893 0.0018514 0.00171581 0.00315484 0.00317152 0.00318621
0.00191685 0.00177827 0.001642 0.00330762 0.00332643 0.00334324
0.00184181 0.00170258 0.00156567 … 0.00346101 0.00348197 0.00350091
0.00176417 0.00162431 0.00148677 0.00361498 0.00363809 0.00365915
0.00168388 0.00154341 0.00140528 0.00376946 0.00379472 0.00381791
0.0016009 0.00145986 0.00132115 0.00392441 0.00395181 0.00397714
0.0015152 0.00137361 0.00123435 0.00407976 0.00410932 0.00413678
0.00142674 0.00128463 0.00114485 … 0.00423547 0.00426718 0.00429677
p1 = plot(xs, ys, u_real, linetype=:contourf, title = "analytic");
p2 = plot(xs, ys, u_predict, linetype=:contourf, title = "predicted");
p3 = plot(xs, ys, diff_u, linetype=:contourf, title = "error");
plot(p1, p2, p3)
This notebook was generated using Literate.jl.