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Number Theoretic Accelerated Learning of Physics-Informed Neural Networks
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Proposes good lattice training (GLT), which selects PINN collocation points via number-theoretic good lattice rules plus periodization and randomization tricks, reducing collocation points needed by 2-7x on PDE benchmarks.
Problem
PINN training relies on a finite set of collocation points to approximate the integral of the physics-informed loss; choosing points poorly causes large discretization error and slow training, and common choices (LHS, Monte Carlo, uniform grids, Sobol) need many points.
Approach
Treat the physics-informed loss as a quasi-Monte Carlo approximation of an integral over the periodized domain and choose collocation points as a good lattice (points j/N * z mod 1, with z optimized number-theoretically), with a random shift resampled each iteration; periodization maps (folding time, cosine mapping of periodic coordinates, multiplying by boundary factors) make the integrand period
Key results
- GLT reached competitive relative error with 987 points for NLS and KdV versus 2,048-4,181 for baselines, and 610 vs 4,181 for Poisson s=2 (roughly 2-7x fewer points).
- At standardized N, GLT gave the lowest relative error on all PINN benchmarks, e.g., NLS 1.22e-3 vs 2.05e-3 for Sobol, Poisson s=2 0.76e-3 vs 14.74e-3 for Sobol.
- In CPINNs on NLS and Burgers, GLT achieved progress with roughly 3-4x fewer collocation points than LHS (e.g., 6,765 vs about 20,000 for NLS).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use GLT instead of uniformly spaced sampling | PINN · NLS | collocation points at saturated relative error: 2,601 -> 987 | Helped |
| Use GLT instead of uniformly spaced sampling | PINN · KdV | collocation points at saturated relative error: 4,225 -> 987 | Helped |
| Use GLT instead of uniformly random sampling | PINN · AC | collocation points at saturated relative error: 4,181 -> 1,597 | Helped |
| Use GLT instead of LHS | PINN · Poisson s=2 | collocation points at saturated relative error: 4,181 -> 610 | Helped |
| Use GLT instead of LHS | PINN · Poisson s=4 | collocation points at saturated relative error: 701 -> 307 | Helped |
| Use GLT instead of the strongest listed baseline | PINN · NLS | relative error (×10⁻³): 2.15 -> 1.22 | Helped |
| Use GLT instead of Sobol | PINN · KdV | relative error (×10⁻³): 2.52 -> 2.19 | Helped |
| Use GLT instead of Sobol | PINN · AC | relative error (×10⁻³): 1.22 -> 0.93 | Helped |
| Use GLT instead of Sobol | PINN · Poisson s=2 | relative error (×10⁻³): 14.74 -> 0.76 | Helped |
| Use GLT instead of LHS | PINN · Poisson s=4 | relative error (×10⁻³): 0.24 -> 0.15 | Helped |
| Add periodization tricks (ablation) | GLT PINN · NLS | relative error (×10⁻³): 1.31 -> 1.22 | Helped |
| Add periodization tricks (ablation) | GLT PINN · KdV | relative error (×10⁻³): 12.30 -> 2.19 | Helped |
Surprising findings
- Without the periodization tricks, relative error deteriorated significantly for all sampling methods, and GLT was not always best in the high-dimensional Poisson case.
- For s=6 and s=8 Poisson, uniformly random Monte Carlo sampling was as effective as or better than GLT.
Limitations
- Performance depends on smoothness of the integrand; may be poor for PDEs with discontinuities or sharp gradients.
- Periodization requires domain transformation and is not directly applicable to non-rectangular domains; no known generator for good lattices when s>2 except prime-based constructions.
- Advantage diminishes in higher dimensions (s=6, 8), where Monte Carlo is comparable or better.
Experimental setup
Datasets: NLS equation numerical solution (PINNs repository) KdV equation numerical solution (PINNs repository) Allen-Cahn equation numerical solution (PINNs repository) Poisson's equation analytical solutions (synthetic, s=2,4) Burgers' equation numerical solution
Benchmarks: NLS equation KdV equation Allen-Cahn equation Poisson's equation (s=2, s=4) Burgers' equation system identification of KdV and AC parameters
Metrics: relative L2 error number of collocation points needed for saturation standard deviation of physics-informed loss as discretizatio relative error of identified parameters
Baselines: uniformly random sampling uniformly spaced sampling Latin hypercube sampling Sobol sequence competitive PINNs with LHS
Builds on: PINNs competitive PINNs Korobov space quasi-Monte Carlo theory Sobol sequence
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