1D Poisson's Equation
This example is taken from [4]. Consider a simple 1D Poisson’s equation with Dirichlet boundary conditions. The solution is given by
\[u(x)=\sin (2 \pi x)+0.1 \sin (50 \pi x)\]
using ModelingToolkit, IntervalSets, Sophon
using Optimization, OptimizationOptimJL, Zygote
using CairoMakie
@parameters x
@variables u(..)
Dₓ² = Differential(x)^2
f(x) = -4 * π^2 * sin(2 * π * x) - 250 * π^2 * sin(50 * π * x)
eq = Dₓ²(u(x)) ~ f(x)
domain = [x ∈ 0 .. 1]
bcs = [u(0) ~ 0, u(1) ~ 0]
@named poisson = PDESystem(eq, bcs, domain, [x], [u(x)])\[ \begin{align} \frac{\mathrm{d}}{\mathrm{d}x} \frac{\mathrm{d} u\left( x \right)}{\mathrm{d}x} =& - 39.478 \sin\left( 6.2832 x \right) - 2467.4 \sin\left( 157.08 x \right) \end{align} \]
chain = Siren(1, 16, 32, 16, 1)
pinn = PINN(chain)
sampler = QuasiRandomSampler(200, 1)
strategy = NonAdaptiveTraining(1 , 50)
prob = Sophon.discretize(poisson, pinn, sampler, strategy)
@showprogress res = Optimization.solve(prob, BFGS(); maxiters=2000)
phi = pinn.phi
xs = 0:0.001:1
u_true = @. sin(2 * pi * xs) + 0.1 * sin(50 * pi * xs)
us = phi(xs', res.u)
fig = Figure()
axis = Axis(fig[1, 1])
lines!(xs, u_true; label="Ground Truth")
lines!(xs, vec(us); label="Prediction")
axislegend(axis)
fig
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1.460022e+00 16.2%┣████▊ ┫ 325/2.0k [00:33<02:53, 10it/s]
1.366123e+00 16.4%┣████▊ ┫ 328/2.0k [00:34<02:52, 10it/s]
1.308952e+00 16.5%┣████▉ ┫ 330/2.0k [00:34<02:51, 10it/s]
1.187019e+00 16.6%┣████▉ ┫ 333/2.0k [00:34<02:49, 10it/s]
1.128470e+00 16.7%┣████▉ ┫ 335/2.0k [00:34<02:48, 10it/s]
1.054182e+00 16.9%┣█████ ┫ 338/2.0k [00:34<02:47, 10it/s]
9.891104e-01 17.0%┣█████ ┫ 341/2.0k [00:34<02:45, 10it/s]
9.404362e-01 17.2%┣█████ ┫ 344/2.0k [00:34<02:44, 10it/s]
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5.870873e-01 19.1%┣█████▌ ┫ 383/2.0k [00:35<02:27, 11it/s]
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4.316363e-01 20.2%┣█████▉ ┫ 405/2.0k [00:35<02:18, 12it/s]
4.075348e-01 20.5%┣██████ ┫ 410/2.0k [00:35<02:16, 12it/s]
3.821375e-01 20.7%┣██████ ┫ 415/2.0k [00:35<02:14, 12it/s]
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2.189178e-01 22.9%┣██████▋ ┫ 458/2.0k [00:36<02:00, 13it/s]
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1.877473e-01 23.5%┣██████▉ ┫ 471/2.0k [00:36<01:56, 13it/s]
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1.648065e-01 24.1%┣███████ ┫ 482/2.0k [00:36<01:53, 13it/s]
1.540131e-01 24.4%┣███████ ┫ 488/2.0k [00:36<01:52, 14it/s]
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1.053258e-01 26.0%┣███████▌ ┫ 521/2.0k [00:36<01:43, 14it/s]
9.560892e-02 26.4%┣███████▋ ┫ 528/2.0k [00:36<01:41, 15it/s]
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3.122103e-02 32.3%┣█████████▍ ┫ 646/2.0k [00:37<01:19, 17it/s]
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1.542027e-03 62.6%┣█████████████████▌ ┫ 1.3k/2.0k [00:42<00:25, 30it/s]
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1.089076e-03 67.8%┣███████████████████ ┫ 1.4k/2.0k [00:43<00:20, 32it/s]
1.049461e-03 68.3%┣███████████████████▏ ┫ 1.4k/2.0k [00:43<00:20, 32it/s]
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9.417840e-04 69.1%┣███████████████████▍ ┫ 1.4k/2.0k [00:43<00:19, 32it/s]
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8.177395e-04 70.0%┣███████████████████▋ ┫ 1.4k/2.0k [00:43<00:19, 32it/s]
7.719005e-04 70.5%┣███████████████████▊ ┫ 1.4k/2.0k [00:43<00:18, 33it/s]
7.154437e-04 71.0%┣███████████████████▉ ┫ 1.4k/2.0k [00:43<00:18, 33it/s]
6.787786e-04 71.3%┣████████████████████ ┫ 1.4k/2.0k [00:43<00:17, 33it/s]
6.225880e-04 71.8%┣████████████████████ ┫ 1.4k/2.0k [00:43<00:17, 33it/s]
5.769212e-04 72.3%┣████████████████████▎ ┫ 1.4k/2.0k [00:44<00:17, 33it/s]
5.344773e-04 72.8%┣████████████████████▍ ┫ 1.5k/2.0k [00:44<00:16, 33it/s]
5.097298e-04 73.2%┣████████████████████▌ ┫ 1.5k/2.0k [00:44<00:16, 34it/s]
4.930512e-04 73.6%┣████████████████████▋ ┫ 1.5k/2.0k [00:44<00:16, 34it/s]
4.769815e-04 74.1%┣████████████████████▊ ┫ 1.5k/2.0k [00:44<00:15, 34it/s]
4.625357e-04 74.6%┣████████████████████▉ ┫ 1.5k/2.0k [00:44<00:15, 34it/s]
4.398070e-04 75.0%┣█████████████████████ ┫ 1.5k/2.0k [00:44<00:15, 34it/s]
4.155922e-04 75.4%┣█████████████████████ ┫ 1.5k/2.0k [00:44<00:14, 34it/s]
3.859251e-04 75.9%┣█████████████████████▎ ┫ 1.5k/2.0k [00:44<00:14, 35it/s]
3.631654e-04 76.4%┣█████████████████████▍ ┫ 1.5k/2.0k [00:44<00:14, 35it/s]
3.543712e-04 76.9%┣█████████████████████▌ ┫ 1.5k/2.0k [00:44<00:13, 35it/s]
3.438153e-04 77.2%┣█████████████████████▋ ┫ 1.5k/2.0k [00:44<00:13, 35it/s]
3.246423e-04 77.7%┣█████████████████████▊ ┫ 1.6k/2.0k [00:44<00:13, 35it/s]
3.087461e-04 78.2%┣█████████████████████▉ ┫ 1.6k/2.0k [00:44<00:12, 35it/s]
2.925008e-04 78.7%┣██████████████████████ ┫ 1.6k/2.0k [00:44<00:12, 36it/s]
2.834174e-04 79.0%┣██████████████████████ ┫ 1.6k/2.0k [00:44<00:12, 36it/s]
2.746756e-04 79.4%┣██████████████████████▎ ┫ 1.6k/2.0k [00:44<00:12, 36it/s]
2.673456e-04 79.9%┣██████████████████████▍ ┫ 1.6k/2.0k [00:44<00:11, 36it/s]
2.619896e-04 80.4%┣██████████████████████▌ ┫ 1.6k/2.0k [00:44<00:11, 36it/s]
2.566898e-04 80.8%┣██████████████████████▋ ┫ 1.6k/2.0k [00:45<00:11, 36it/s]
2.506119e-04 81.3%┣██████████████████████▊ ┫ 1.6k/2.0k [00:45<00:10, 36it/s]
2.434811e-04 81.8%┣███████████████████████ ┫ 1.6k/2.0k [00:45<00:10, 37it/s]
2.345284e-04 82.3%┣███████████████████████ ┫ 1.6k/2.0k [00:45<00:10, 37it/s]
2.279708e-04 82.7%┣███████████████████████▏ ┫ 1.7k/2.0k [00:45<00:09, 37it/s]
2.252035e-04 83.0%┣███████████████████████▎ ┫ 1.7k/2.0k [00:45<00:09, 37it/s]
2.194848e-04 83.5%┣███████████████████████▍ ┫ 1.7k/2.0k [00:45<00:09, 37it/s]
2.118448e-04 84.0%┣███████████████████████▌ ┫ 1.7k/2.0k [00:45<00:09, 37it/s]
2.030400e-04 84.5%┣███████████████████████▋ ┫ 1.7k/2.0k [00:45<00:08, 38it/s]
2.009093e-04 84.7%┣███████████████████████▊ ┫ 1.7k/2.0k [00:45<00:08, 38it/s]
1.981349e-04 85.2%┣███████████████████████▉ ┫ 1.7k/2.0k [00:45<00:08, 38it/s]
1.947511e-04 85.7%┣████████████████████████ ┫ 1.7k/2.0k [00:45<00:08, 38it/s]
1.910733e-04 86.2%┣████████████████████████▏ ┫ 1.7k/2.0k [00:45<00:07, 38it/s]
1.877767e-04 86.6%┣████████████████████████▎ ┫ 1.7k/2.0k [00:45<00:07, 38it/s]
1.847318e-04 86.9%┣████████████████████████▍ ┫ 1.7k/2.0k [00:45<00:07, 38it/s]
1.796687e-04 87.4%┣████████████████████████▌ ┫ 1.7k/2.0k [00:45<00:07, 39it/s]
1.749959e-04 87.9%┣████████████████████████▋ ┫ 1.8k/2.0k [00:45<00:06, 39it/s]
1.715932e-04 88.4%┣████████████████████████▊ ┫ 1.8k/2.0k [00:45<00:06, 39it/s]
1.667290e-04 88.8%┣████████████████████████▉ ┫ 1.8k/2.0k [00:46<00:06, 39it/s]
1.607875e-04 89.3%┣█████████████████████████ ┫ 1.8k/2.0k [00:46<00:05, 39it/s]
1.573171e-04 89.8%┣█████████████████████████▏ ┫ 1.8k/2.0k [00:46<00:05, 39it/s]
1.522889e-04 90.3%┣█████████████████████████▎ ┫ 1.8k/2.0k [00:46<00:05, 40it/s]
1.495920e-04 90.6%┣█████████████████████████▍ ┫ 1.8k/2.0k [00:46<00:05, 40it/s]
1.471885e-04 91.0%┣█████████████████████████▌ ┫ 1.8k/2.0k [00:46<00:05, 40it/s]
1.451792e-04 91.5%┣█████████████████████████▋ ┫ 1.8k/2.0k [00:46<00:04, 40it/s]
1.434688e-04 92.0%┣█████████████████████████▊ ┫ 1.8k/2.0k [00:46<00:04, 40it/s]
1.407036e-04 92.5%┣██████████████████████████ ┫ 1.9k/2.0k [00:46<00:04, 40it/s]
1.370180e-04 93.0%┣██████████████████████████ ┫ 1.9k/2.0k [00:46<00:03, 40it/s]
1.345419e-04 93.4%┣██████████████████████████▏ ┫ 1.9k/2.0k [00:46<00:03, 41it/s]
1.332631e-04 93.9%┣██████████████████████████▎ ┫ 1.9k/2.0k [00:46<00:03, 41it/s]
1.320952e-04 94.4%┣██████████████████████████▍ ┫ 1.9k/2.0k [00:46<00:03, 41it/s]
1.309526e-04 94.7%┣██████████████████████████▌ ┫ 1.9k/2.0k [00:46<00:03, 41it/s]
1.284015e-04 95.1%┣██████████████████████████▋ ┫ 1.9k/2.0k [00:46<00:02, 41it/s]
1.275730e-04 95.6%┣██████████████████████████▊ ┫ 1.9k/2.0k [00:46<00:02, 41it/s]
1.264767e-04 96.1%┣███████████████████████████ ┫ 1.9k/2.0k [00:46<00:02, 41it/s]
1.257943e-04 96.4%┣███████████████████████████ ┫ 1.9k/2.0k [00:46<00:02, 42it/s]
1.245031e-04 96.9%┣███████████████████████████▏┫ 1.9k/2.0k [00:46<00:02, 42it/s]
1.231696e-04 97.3%┣███████████████████████████▎┫ 1.9k/2.0k [00:47<00:01, 42it/s]
1.228674e-04 97.7%┣███████████████████████████▍┫ 2.0k/2.0k [00:47<00:01, 42it/s]
1.219406e-04 98.1%┣███████████████████████████▌┫ 2.0k/2.0k [00:47<00:01, 42it/s]
1.209161e-04 98.4%┣███████████████████████████▌┫ 2.0k/2.0k [00:47<00:01, 42it/s]
1.205982e-04 98.8%┣███████████████████████████▋┫ 2.0k/2.0k [00:47<00:01, 42it/s]
1.200536e-04 99.2%┣███████████████████████████▊┫ 2.0k/2.0k [00:47<00:00, 42it/s]
1.181568e-04 99.7%┣████████████████████████████┫ 2.0k/2.0k [00:47<00:00, 43it/s]
1.178800e-04 100.0%┣███████████████████████████┫ 2.0k/2.0k [00:47<00:00, 43it/s]
Compute the relative L2 error
using Integrals
u_analytical(x,p) = sin.(2 * pi .* x) + 0.1 * sin.(50 * pi .* x)
error(x,p) = abs2.(vec(phi([x;;],res.u)) .- u_analytical(x,p))
relative_L2_error = solve(IntegralProblem(error,0,1),HCubatureJL(),reltol=1e-3,abstol=1e-3) ./ solve(IntegralProblem((x,p) -> abs2.(u_analytical(x,p)),0, 1),HCubatureJL(),reltol=1e-3,abstol=1e-3)1-element Vector{Float64}:
2.724438778516777e-9