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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2.486044e-03 67.2%┣██████████████████▉ ┫ 1.3k/2.0k [00:44<00:22, 31it/s]
2.391868e-03 67.5%┣███████████████████ ┫ 1.4k/2.0k [00:44<00:21, 31it/s]
2.285475e-03 68.0%┣███████████████████ ┫ 1.4k/2.0k [00:44<00:21, 31it/s]
2.169383e-03 68.4%┣███████████████████▏ ┫ 1.4k/2.0k [00:44<00:20, 31it/s]
2.065262e-03 68.9%┣███████████████████▎ ┫ 1.4k/2.0k [00:44<00:20, 31it/s]
1.976537e-03 69.2%┣███████████████████▍ ┫ 1.4k/2.0k [00:44<00:20, 31it/s]
1.877483e-03 69.6%┣███████████████████▌ ┫ 1.4k/2.0k [00:44<00:19, 31it/s]
1.781967e-03 70.0%┣███████████████████▋ ┫ 1.4k/2.0k [00:44<00:19, 32it/s]
1.706436e-03 70.4%┣███████████████████▊ ┫ 1.4k/2.0k [00:44<00:19, 32it/s]
1.614389e-03 70.9%┣███████████████████▉ ┫ 1.4k/2.0k [00:44<00:18, 32it/s]
1.560269e-03 71.3%┣████████████████████ ┫ 1.4k/2.0k [00:45<00:18, 32it/s]
1.520343e-03 71.8%┣████████████████████ ┫ 1.4k/2.0k [00:45<00:18, 32it/s]
1.494245e-03 72.2%┣████████████████████▏ ┫ 1.4k/2.0k [00:45<00:17, 32it/s]
1.442266e-03 72.6%┣████████████████████▎ ┫ 1.5k/2.0k [00:45<00:17, 32it/s]
1.409813e-03 72.9%┣████████████████████▍ ┫ 1.5k/2.0k [00:45<00:17, 33it/s]
1.384513e-03 73.3%┣████████████████████▌ ┫ 1.5k/2.0k [00:45<00:16, 33it/s]
1.370442e-03 73.6%┣████████████████████▋ ┫ 1.5k/2.0k [00:45<00:16, 33it/s]
1.340131e-03 74.1%┣████████████████████▊ ┫ 1.5k/2.0k [00:45<00:16, 33it/s]
1.309218e-03 74.5%┣████████████████████▉ ┫ 1.5k/2.0k [00:45<00:15, 33it/s]
1.246795e-03 74.9%┣█████████████████████ ┫ 1.5k/2.0k [00:45<00:15, 33it/s]
1.192281e-03 75.4%┣█████████████████████▏ ┫ 1.5k/2.0k [00:45<00:15, 33it/s]
1.150465e-03 75.9%┣█████████████████████▎ ┫ 1.5k/2.0k [00:45<00:14, 34it/s]
1.108601e-03 76.2%┣█████████████████████▍ ┫ 1.5k/2.0k [00:45<00:14, 34it/s]
1.071789e-03 76.7%┣█████████████████████▌ ┫ 1.5k/2.0k [00:45<00:14, 34it/s]
1.035688e-03 77.2%┣█████████████████████▋ ┫ 1.5k/2.0k [00:45<00:13, 34it/s]
9.977512e-04 77.6%┣█████████████████████▊ ┫ 1.6k/2.0k [00:45<00:13, 34it/s]
9.806695e-04 78.0%┣█████████████████████▉ ┫ 1.6k/2.0k [00:45<00:13, 34it/s]
9.493924e-04 78.4%┣██████████████████████ ┫ 1.6k/2.0k [00:45<00:13, 34it/s]
9.357554e-04 78.9%┣██████████████████████ ┫ 1.6k/2.0k [00:46<00:12, 35it/s]
9.185199e-04 79.4%┣██████████████████████▏ ┫ 1.6k/2.0k [00:46<00:12, 35it/s]
9.052021e-04 79.7%┣██████████████████████▎ ┫ 1.6k/2.0k [00:46<00:12, 35it/s]
8.785946e-04 80.1%┣██████████████████████▍ ┫ 1.6k/2.0k [00:46<00:11, 35it/s]
8.608629e-04 80.5%┣██████████████████████▌ ┫ 1.6k/2.0k [00:46<00:11, 35it/s]
8.502895e-04 80.8%┣██████████████████████▋ ┫ 1.6k/2.0k [00:46<00:11, 35it/s]
8.271231e-04 81.3%┣██████████████████████▊ ┫ 1.6k/2.0k [00:46<00:11, 35it/s]
8.091991e-04 81.6%┣██████████████████████▉ ┫ 1.6k/2.0k [00:46<00:10, 36it/s]
7.960719e-04 82.1%┣███████████████████████ ┫ 1.6k/2.0k [00:46<00:10, 36it/s]
7.754686e-04 82.5%┣███████████████████████ ┫ 1.7k/2.0k [00:46<00:10, 36it/s]
7.622053e-04 83.0%┣███████████████████████▎ ┫ 1.7k/2.0k [00:46<00:09, 36it/s]
7.512193e-04 83.3%┣███████████████████████▎ ┫ 1.7k/2.0k [00:46<00:09, 36it/s]
7.307203e-04 83.8%┣███████████████████████▌ ┫ 1.7k/2.0k [00:46<00:09, 36it/s]
7.203863e-04 84.2%┣███████████████████████▋ ┫ 1.7k/2.0k [00:46<00:09, 36it/s]
7.063794e-04 84.7%┣███████████████████████▊ ┫ 1.7k/2.0k [00:46<00:08, 37it/s]
6.931902e-04 85.0%┣███████████████████████▉ ┫ 1.7k/2.0k [00:46<00:08, 37it/s]
6.832654e-04 85.5%┣████████████████████████ ┫ 1.7k/2.0k [00:46<00:08, 37it/s]
6.775846e-04 85.9%┣████████████████████████ ┫ 1.7k/2.0k [00:46<00:08, 37it/s]
6.714514e-04 86.4%┣████████████████████████▏ ┫ 1.7k/2.0k [00:47<00:07, 37it/s]
6.670260e-04 86.7%┣████████████████████████▎ ┫ 1.7k/2.0k [00:47<00:07, 37it/s]
6.544517e-04 87.1%┣████████████████████████▍ ┫ 1.7k/2.0k [00:47<00:07, 37it/s]
6.490860e-04 87.5%┣████████████████████████▌ ┫ 1.8k/2.0k [00:47<00:07, 37it/s]
6.426445e-04 87.9%┣████████████████████████▋ ┫ 1.8k/2.0k [00:47<00:06, 38it/s]
6.362482e-04 88.4%┣████████████████████████▊ ┫ 1.8k/2.0k [00:47<00:06, 38it/s]
6.339272e-04 88.6%┣████████████████████████▉ ┫ 1.8k/2.0k [00:47<00:06, 38it/s]
6.082896e-04 89.1%┣█████████████████████████ ┫ 1.8k/2.0k [00:47<00:06, 38it/s]
5.910454e-04 89.5%┣█████████████████████████ ┫ 1.8k/2.0k [00:47<00:06, 38it/s]
5.778728e-04 89.9%┣█████████████████████████▏ ┫ 1.8k/2.0k [00:47<00:05, 38it/s]
5.688689e-04 90.3%┣█████████████████████████▎ ┫ 1.8k/2.0k [00:47<00:05, 38it/s]
5.581091e-04 90.7%┣█████████████████████████▍ ┫ 1.8k/2.0k [00:47<00:05, 38it/s]
5.411605e-04 91.2%┣█████████████████████████▌ ┫ 1.8k/2.0k [00:47<00:05, 39it/s]
5.203213e-04 91.6%┣█████████████████████████▋ ┫ 1.8k/2.0k [00:47<00:04, 39it/s]
5.022494e-04 92.0%┣█████████████████████████▊ ┫ 1.8k/2.0k [00:47<00:04, 39it/s]
4.766589e-04 92.4%┣█████████████████████████▉ ┫ 1.8k/2.0k [00:47<00:04, 39it/s]
4.584549e-04 92.9%┣██████████████████████████ ┫ 1.9k/2.0k [00:47<00:04, 39it/s]
4.482883e-04 93.3%┣██████████████████████████ ┫ 1.9k/2.0k [00:47<00:03, 39it/s]
4.373828e-04 93.6%┣██████████████████████████▏ ┫ 1.9k/2.0k [00:48<00:03, 39it/s]
4.254914e-04 94.0%┣██████████████████████████▎ ┫ 1.9k/2.0k [00:48<00:03, 40it/s]
4.201819e-04 94.4%┣██████████████████████████▍ ┫ 1.9k/2.0k [00:48<00:03, 40it/s]
4.136522e-04 94.8%┣██████████████████████████▌ ┫ 1.9k/2.0k [00:48<00:03, 40it/s]
4.018640e-04 95.3%┣██████████████████████████▊ ┫ 1.9k/2.0k [00:48<00:02, 40it/s]
3.946360e-04 95.6%┣██████████████████████████▊ ┫ 1.9k/2.0k [00:48<00:02, 40it/s]
3.872557e-04 96.1%┣███████████████████████████ ┫ 1.9k/2.0k [00:48<00:02, 40it/s]
3.821473e-04 96.4%┣███████████████████████████ ┫ 1.9k/2.0k [00:48<00:02, 40it/s]
3.699913e-04 96.8%┣███████████████████████████ ┫ 1.9k/2.0k [00:48<00:02, 40it/s]
3.611223e-04 97.2%┣███████████████████████████▏┫ 1.9k/2.0k [00:48<00:01, 40it/s]
3.415626e-04 97.6%┣███████████████████████████▎┫ 2.0k/2.0k [00:48<00:01, 41it/s]
3.274656e-04 98.1%┣███████████████████████████▌┫ 2.0k/2.0k [00:48<00:01, 41it/s]
3.188943e-04 98.5%┣███████████████████████████▋┫ 2.0k/2.0k [00:48<00:01, 41it/s]
3.130764e-04 98.9%┣███████████████████████████▊┫ 2.0k/2.0k [00:48<00:01, 41it/s]
3.040417e-04 99.3%┣███████████████████████████▉┫ 2.0k/2.0k [00:48<00:00, 41it/s]
2.979800e-04 99.7%┣████████████████████████████┫ 2.0k/2.0k [00:48<00:00, 41it/s]
2.913693e-04 100.0%┣███████████████████████████┫ 2.0k/2.0k [00:48<00:00, 41it/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}:
8.740902437741377e-8