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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2.368526e-04 86.5%┣█████████████████████████ ┫ 866/1.0k [01:11<00:11, 12it/s]
2.351261e-04 86.8%┣█████████████████████████▏ ┫ 869/1.0k [01:11<00:11, 12it/s]
2.336255e-04 87.1%┣█████████████████████████▎ ┫ 872/1.0k [01:11<00:10, 12it/s]
2.324937e-04 87.4%┣█████████████████████████▍ ┫ 875/1.0k [01:11<00:10, 12it/s]
2.309698e-04 87.7%┣█████████████████████████▍ ┫ 878/1.0k [01:11<00:10, 12it/s]
2.289891e-04 88.0%┣█████████████████████████▌ ┫ 881/1.0k [01:11<00:10, 12it/s]
2.277514e-04 88.2%┣█████████████████████████▋ ┫ 883/1.0k [01:11<00:10, 12it/s]
2.260493e-04 88.5%┣█████████████████████████▊ ┫ 886/1.0k [01:11<00:09, 12it/s]
2.245121e-04 88.7%┣█████████████████████████▊ ┫ 888/1.0k [01:11<00:09, 12it/s]
2.228720e-04 89.0%┣█████████████████████████▉ ┫ 891/1.0k [01:11<00:09, 12it/s]
2.218657e-04 89.2%┣█████████████████████████▉ ┫ 893/1.0k [01:11<00:09, 13it/s]
2.206617e-04 89.5%┣██████████████████████████ ┫ 896/1.0k [01:11<00:08, 13it/s]
2.199168e-04 89.8%┣██████████████████████████ ┫ 899/1.0k [01:11<00:08, 13it/s]
2.184991e-04 90.1%┣██████████████████████████▏ ┫ 902/1.0k [01:11<00:08, 13it/s]
2.169058e-04 90.3%┣██████████████████████████▏ ┫ 904/1.0k [01:12<00:08, 13it/s]
2.151914e-04 90.6%┣██████████████████████████▎ ┫ 907/1.0k [01:12<00:07, 13it/s]
2.142134e-04 90.8%┣██████████████████████████▍ ┫ 909/1.0k [01:12<00:07, 13it/s]
2.130748e-04 91.1%┣██████████████████████████▍ ┫ 912/1.0k [01:12<00:07, 13it/s]
2.114997e-04 91.4%┣██████████████████████████▌ ┫ 915/1.0k [01:12<00:07, 13it/s]
2.096920e-04 91.7%┣██████████████████████████▋ ┫ 918/1.0k [01:12<00:07, 13it/s]
2.076066e-04 92.0%┣██████████████████████████▊ ┫ 921/1.0k [01:12<00:06, 13it/s]
2.061857e-04 92.3%┣██████████████████████████▊ ┫ 924/1.0k [01:12<00:06, 13it/s]
2.055254e-04 92.6%┣██████████████████████████▉ ┫ 927/1.0k [01:12<00:06, 13it/s]
2.050400e-04 92.8%┣███████████████████████████ ┫ 929/1.0k [01:12<00:06, 13it/s]
2.042081e-04 93.0%┣███████████████████████████ ┫ 931/1.0k [01:12<00:05, 13it/s]
2.021609e-04 93.3%┣███████████████████████████ ┫ 934/1.0k [01:12<00:05, 13it/s]
2.006947e-04 93.6%┣███████████████████████████▏ ┫ 937/1.0k [01:12<00:05, 13it/s]
1.991490e-04 93.9%┣███████████████████████████▎ ┫ 940/1.0k [01:12<00:05, 13it/s]
1.978371e-04 94.2%┣███████████████████████████▎ ┫ 943/1.0k [01:12<00:04, 13it/s]
1.968840e-04 94.5%┣███████████████████████████▍ ┫ 946/1.0k [01:12<00:04, 13it/s]
1.958773e-04 94.7%┣███████████████████████████▌ ┫ 948/1.0k [01:12<00:04, 13it/s]
1.946775e-04 95.0%┣███████████████████████████▌ ┫ 951/1.0k [01:13<00:04, 13it/s]
1.941923e-04 95.2%┣███████████████████████████▋ ┫ 953/1.0k [01:13<00:04, 13it/s]
1.937193e-04 95.4%┣███████████████████████████▊ ┫ 955/1.0k [01:13<00:04, 13it/s]
1.931737e-04 95.6%┣███████████████████████████▊ ┫ 957/1.0k [01:13<00:03, 13it/s]
1.919803e-04 95.9%┣███████████████████████████▉ ┫ 960/1.0k [01:13<00:03, 13it/s]
1.911734e-04 96.1%┣███████████████████████████▉ ┫ 962/1.0k [01:13<00:03, 13it/s]
1.904044e-04 96.3%┣████████████████████████████ ┫ 964/1.0k [01:13<00:03, 13it/s]
1.897955e-04 96.5%┣████████████████████████████ ┫ 966/1.0k [01:13<00:03, 13it/s]
1.892028e-04 96.7%┣████████████████████████████ ┫ 968/1.0k [01:13<00:02, 13it/s]
1.884064e-04 96.9%┣████████████████████████████ ┫ 970/1.0k [01:13<00:02, 13it/s]
1.871408e-04 97.2%┣████████████████████████████▏┫ 973/1.0k [01:13<00:02, 13it/s]
1.857775e-04 97.5%┣████████████████████████████▎┫ 976/1.0k [01:13<00:02, 13it/s]
1.844294e-04 97.8%┣████████████████████████████▍┫ 979/1.0k [01:13<00:02, 13it/s]
1.836894e-04 98.1%┣████████████████████████████▌┫ 982/1.0k [01:13<00:01, 13it/s]
1.831779e-04 98.3%┣████████████████████████████▌┫ 984/1.0k [01:13<00:01, 13it/s]
1.826115e-04 98.5%┣████████████████████████████▋┫ 986/1.0k [01:13<00:01, 13it/s]
1.814151e-04 98.8%┣████████████████████████████▋┫ 989/1.0k [01:14<00:01, 13it/s]
1.808344e-04 99.0%┣████████████████████████████▊┫ 991/1.0k [01:14<00:01, 13it/s]
1.804076e-04 99.2%┣████████████████████████████▊┫ 993/1.0k [01:14<00:01, 13it/s]
1.793945e-04 99.5%┣████████████████████████████▉┫ 996/1.0k [01:14<00:00, 14it/s]
1.784290e-04 99.7%┣█████████████████████████████┫ 998/1.0k [01:14<00:00, 14it/s]
1.777598e-04 99.9%┣████████████████████████████┫ 1.0k/1.0k [01:14<00:00, 14it/s]
1.773681e-04 100.0%┣███████████████████████████┫ 1.0k/1.0k [01:14<00:00, 14it/s]