Poisson equation over an L-shaped domain
The example is taken from here. We showcase define a PDE on an L-shaped domain
using ModelingToolkit, DomainSets, Optimization, OptimizationOptimJL, Zygote
using DomainSets: ×
using Sophon
@parameters x y
@variables u(..)
Dxx = Differential(x)^2
Dyy = Differential(y)^2
eq = Dxx(u(x,y)) + Dyy(u(x,y)) ~ -1.0
eqs = [eq => (-1..0) × (-1..0),
eq => (-1..0) × (0..1),
eq => (0..1) × (-1..0)]
bc = u(x,y) ~ 0.0
boundaries = [(-1 .. -1) × (-1..1),
(-1..0) × (1..1),
(0..0) × (0..1),
(0..1) × (0..0),
(1..1) × (-1..0),
(-1..1) × (-1 .. -1)]
bcs = [bc => boundary for boundary in boundaries]
pde_system = Sophon.PDESystem(eqs, bcs, [x,y], [u(x,y)])PDESystem
Equations:
Differential(y)(Differential(y)(u(x, y))) + Differential(x)(Differential(x)(u(x, y))) ~ -1.0 on (-1 .. 0) × (-1 .. 0)
Differential(y)(Differential(y)(u(x, y))) + Differential(x)(Differential(x)(u(x, y))) ~ -1.0 on (-1 .. 0) × (0 .. 1)
Differential(y)(Differential(y)(u(x, y))) + Differential(x)(Differential(x)(u(x, y))) ~ -1.0 on (0 .. 1) × (-1 .. 0)
Boundary Conditions:
u(x, y) ~ 0.0 on (-1 .. -1) × (-1 .. 1)
u(x, y) ~ 0.0 on (-1 .. 0) × (1 .. 1)
u(x, y) ~ 0.0 on (0 .. 0) × (0 .. 1)
u(x, y) ~ 0.0 on (0 .. 1) × (0 .. 0)
u(x, y) ~ 0.0 on (1 .. 1) × (-1 .. 0)
u(x, y) ~ 0.0 on (-1 .. 1) × (-1 .. -1)
Dependent Variables: Symbolics.Num[u(x, y)]
Independent Variables: Symbolics.Num[x, y]
chain = FullyConnected((2,16,16,16,1), tanh)
pinn = PINN(chain)
sampler = QuasiRandomSampler(300, 30)
strategy = NonAdaptiveTraining()
prob = Sophon.discretize(pde_system, pinn, sampler, strategy)
@showprogress res = Optimization.solve(prob, BFGS(); maxiters=1000)
using CairoMakie
xs = -1:0.01:1
ys = -1:0.01:1
u_pred = [ifelse(x>0.0 && y>0.0, NaN, pinn.phi([x,y], res.u)[1]) for x in xs, y in ys]
fig, ax, hm = heatmap(xs, ys, u_pred, colormap=:jet)
Colorbar(fig[:, end+1], hm)
fig
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2.555891e-04 87.0%┣█████████████████████████▎ ┫ 871/1.0k [01:18<00:12, 11it/s]
2.552870e-04 87.2%┣█████████████████████████▎ ┫ 873/1.0k [01:18<00:11, 11it/s]
2.549278e-04 87.4%┣█████████████████████████▍ ┫ 875/1.0k [01:18<00:11, 11it/s]
2.543165e-04 87.6%┣█████████████████████████▍ ┫ 877/1.0k [01:18<00:11, 11it/s]
2.531438e-04 87.8%┣█████████████████████████▌ ┫ 879/1.0k [01:18<00:11, 11it/s]
2.516911e-04 88.1%┣█████████████████████████▌ ┫ 882/1.0k [01:18<00:11, 11it/s]
2.510346e-04 88.3%┣█████████████████████████▋ ┫ 884/1.0k [01:19<00:10, 11it/s]
2.497775e-04 88.6%┣█████████████████████████▊ ┫ 887/1.0k [01:19<00:10, 11it/s]
2.488437e-04 88.9%┣█████████████████████████▉ ┫ 890/1.0k [01:19<00:10, 11it/s]
2.481546e-04 89.1%┣█████████████████████████▉ ┫ 892/1.0k [01:19<00:10, 11it/s]
2.470068e-04 89.4%┣██████████████████████████ ┫ 895/1.0k [01:19<00:09, 11it/s]
2.458490e-04 89.7%┣██████████████████████████ ┫ 898/1.0k [01:19<00:09, 11it/s]
2.448544e-04 90.0%┣██████████████████████████ ┫ 901/1.0k [01:19<00:09, 11it/s]
2.440837e-04 90.3%┣██████████████████████████▏ ┫ 904/1.0k [01:19<00:08, 11it/s]
2.435639e-04 90.6%┣██████████████████████████▎ ┫ 907/1.0k [01:19<00:08, 11it/s]
2.428145e-04 90.8%┣██████████████████████████▍ ┫ 909/1.0k [01:19<00:08, 11it/s]
2.422054e-04 91.0%┣██████████████████████████▍ ┫ 911/1.0k [01:19<00:08, 11it/s]
2.414574e-04 91.2%┣██████████████████████████▌ ┫ 913/1.0k [01:19<00:08, 12it/s]
2.404756e-04 91.4%┣██████████████████████████▌ ┫ 915/1.0k [01:19<00:07, 12it/s]
2.396114e-04 91.6%┣██████████████████████████▋ ┫ 917/1.0k [01:19<00:07, 12it/s]
2.387859e-04 91.9%┣██████████████████████████▋ ┫ 920/1.0k [01:19<00:07, 12it/s]
2.383670e-04 92.1%┣██████████████████████████▊ ┫ 922/1.0k [01:19<00:07, 12it/s]
2.373818e-04 92.4%┣██████████████████████████▉ ┫ 925/1.0k [01:20<00:07, 12it/s]
2.363511e-04 92.7%┣██████████████████████████▉ ┫ 928/1.0k [01:20<00:06, 12it/s]
2.356841e-04 93.0%┣███████████████████████████ ┫ 931/1.0k [01:20<00:06, 12it/s]
2.348257e-04 93.2%┣███████████████████████████ ┫ 933/1.0k [01:20<00:06, 12it/s]
2.339134e-04 93.4%┣███████████████████████████ ┫ 935/1.0k [01:20<00:06, 12it/s]
2.327948e-04 93.7%┣███████████████████████████▏ ┫ 938/1.0k [01:20<00:05, 12it/s]
2.318046e-04 93.9%┣███████████████████████████▎ ┫ 940/1.0k [01:20<00:05, 12it/s]
2.300128e-04 94.2%┣███████████████████████████▎ ┫ 943/1.0k [01:20<00:05, 12it/s]
2.289113e-04 94.5%┣███████████████████████████▍ ┫ 946/1.0k [01:20<00:05, 12it/s]
2.281867e-04 94.8%┣███████████████████████████▌ ┫ 949/1.0k [01:20<00:04, 12it/s]
2.277954e-04 95.0%┣███████████████████████████▌ ┫ 951/1.0k [01:20<00:04, 12it/s]
2.270726e-04 95.2%┣███████████████████████████▋ ┫ 953/1.0k [01:20<00:04, 12it/s]
2.263099e-04 95.4%┣███████████████████████████▊ ┫ 955/1.0k [01:20<00:04, 12it/s]
2.254371e-04 95.6%┣███████████████████████████▊ ┫ 957/1.0k [01:20<00:04, 12it/s]
2.244820e-04 95.8%┣███████████████████████████▉ ┫ 959/1.0k [01:20<00:04, 12it/s]
2.232432e-04 96.0%┣███████████████████████████▉ ┫ 961/1.0k [01:20<00:03, 12it/s]
2.218690e-04 96.3%┣████████████████████████████ ┫ 964/1.0k [01:20<00:03, 12it/s]
2.207063e-04 96.6%┣████████████████████████████ ┫ 967/1.0k [01:21<00:03, 12it/s]
2.199899e-04 96.9%┣████████████████████████████ ┫ 970/1.0k [01:21<00:03, 12it/s]
2.191179e-04 97.2%┣████████████████████████████▏┫ 973/1.0k [01:21<00:02, 12it/s]
2.178662e-04 97.5%┣████████████████████████████▎┫ 976/1.0k [01:21<00:02, 12it/s]
2.163974e-04 97.8%┣████████████████████████████▍┫ 979/1.0k [01:21<00:02, 12it/s]
2.155143e-04 98.1%┣████████████████████████████▌┫ 982/1.0k [01:21<00:02, 12it/s]
2.147665e-04 98.4%┣████████████████████████████▌┫ 985/1.0k [01:21<00:01, 12it/s]
2.142016e-04 98.6%┣████████████████████████████▋┫ 987/1.0k [01:21<00:01, 12it/s]
2.132858e-04 98.9%┣████████████████████████████▊┫ 990/1.0k [01:21<00:01, 12it/s]
2.126221e-04 99.2%┣████████████████████████████▊┫ 993/1.0k [01:21<00:01, 12it/s]
2.121577e-04 99.4%┣████████████████████████████▉┫ 995/1.0k [01:21<00:00, 12it/s]
2.116197e-04 99.6%┣████████████████████████████▉┫ 997/1.0k [01:21<00:00, 12it/s]
2.108262e-04 99.8%┣█████████████████████████████┫ 999/1.0k [01:21<00:00, 12it/s]
2.102220e-04 100.0%┣███████████████████████████┫ 1.0k/1.0k [01:21<00:00, 12it/s]