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Solving Poisson Equations using Neural Walk-on-Spheres
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Proposes Neural Walk-on-Spheres (NWoS), a gradient-free neural loss for high-dimensional Poisson equations built from recursive Walk-on-Spheres estimates, with a buffer, control variates, and boundary loss.
Problem
Solving high-dimensional Poisson equations on general domains with neural networks; existing PINN, Deep Ritz, BSDE and diffusion losses suffer from high cost, bias, instability, or slow propagation of boundary information.
Approach
Derive a variational loss whose target is a single Walk-on-Spheres trajectory (recursive sphere sampling plus Green's-function source terms), so the network regresses to an unbiased-up-to-epsilon estimate without any spatial derivatives; uses a buffer of cached WoS estimates with partial replacement, control variates from the network gradient, and optional neural-network evaluation at truncated wa
Key results
- On the 10d Laplace benchmark NWoS reaches relative L2 error 4.29e-5 versus 7.42e-4 for PINN and 8.43e-4 for Deep Ritz.
- On the 10d committor problem NWoS achieves 6.56e-4 relative L2 error versus 1.26e-3 for Neural Cache and 4.48e-2 for the diffusion loss.
- On the 100d Poisson benchmark a simple boundary projection (2.92e-4) beats NWoS (6.22e-4), which is reported as a limitation of the benchmark.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use NWoS instead of Diffusion loss | Poisson solvers · Laplace (10d) | relative L2-error: 1.57e^{-4}\pm7.74e^{-6} -> 4.29e^{-5}\pm2.02e^{-6} | Helped |
| use NWoS instead of Neural Cache | Poisson solvers · Committor (10d) | relative L2-error: 1.26^{-3}\pm5.82e^{-5} -> 6.56e^{-4}\pm2.42e^{-5} | Helped |
| use NWoS instead of Deep Ritz | Poisson solvers · Poisson Rect. (10d) | relative L2-error: 1.06e^{-2}\pm6.20e^{-4} -> 2.60e^{-3}\pm9.99e^{-5} | Helped |
| use NWoS instead of Diffusion loss | Poisson solvers · Poisson (50d) | relative L2-error: 5.96e^{-4}\pm1.06e^{-5} -> 4.82e^{-4}\pm1.32e^{-5} | Helped |
| replace Projection with NWoS | Poisson solver · Poisson (100d) | relative L2-error: 2.92e^{-4}\pm5.17e^{-7} -> 6.22e^{-4}\pm1.18e^{-5} | Hurt |
| replace Projection with NWoS | Poisson solver · Poisson (500d) | relative L2-error: 1.19e^{-5}\pm1.67e^{-8} -> 9.14^{-3}\pm6.31e^{-5} | Hurt |
| add control variate (ablation) | NWoS · PDE-constrained optimization | relative L2-error: 2.89\% -> 1.72\% | Helped |
| add neural network terminal evaluation for walks not converged after K steps (ablation) | NWoS · PDE-constrained optimization | relative L2-error: 2.89\% -> 1.70\% | Helped |
| add control variate and neural network terminal evaluation (ablation) | NWoS · PDE-constrained optimization | relative L2-error: 2.89\% -> 1.30\% | Helped |
Surprising findings
- A simple projection to the boundary outperforms all neural methods, including NWoS, on the 100d and 500d Poisson benchmarks.
- Neural Cache, which only updates buffer values without replacing points, appears to overfit in high dimensions, motivating NWoS's partial replacement of buffer points.
Limitations
- Only applicable to Poisson equations with Dirichlet boundary conditions on sufficiently regular domains.
- Requires fixed parametrization of source and boundary functions; extension to neural operators left open.
- On some high-dimensional benchmarks (100d, 500d Poisson) NWoS does not beat a simple projection baseline.
Experimental setup
Benchmarks: Laplace (10d) Committor (10d) Poisson rectangular annulus (10d) Poisson (50d) Poisson (100d) Poisson (500d) PDE-constrained optimization (2D Poisson control)
Metrics: relative L2 error training time GPU memory
Baselines: PINN Deep Ritz Diffusion loss Neural Cache WoS Projection
Builds on: Walk-on-Spheres Deep Ritz method Feynman-Kac / BSDE losses diffusion loss Neural Cache
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