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Towards General Neural Surrogate Solvers with Specialized Neural Accelerators
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Proposes SNAP-DDM, which solves large arbitrary-size PDE problems by tiling overlapping subdomains, each solved by specialized SM-FNO neural operators inside an optimized Schwarz iteration, demonstrated on 2D electromagnetics and steady fluid flow.
Problem
Neural PDE surrogate solvers are limited to fixed domain sizes, geometries, and boundary conditions, and end-to-end neural operators do not scale to large arbitrary-size domains.
Approach
Decompose a global domain into overlapping fixed-size 64x64 subdomains solved by specialized neural operators (PML, source, material) with Robin boundary inputs, iterated via an optimized Schwarz DDM; the SM-FNO adds a self-modulation encoder that multiplies Fourier layer weights and residual connections, trained with a hybrid data-physics loss.
Key results
- SM-FNO-v2 subdomain material model reaches 1.01% data loss versus 2.93% for UNet-v2 and 5.34% for FNO-v2 at similar scale (1M training data).
- SNAP-DDM solves large EM domains (e.g., 40x40 subdomain arrays) with absolute field error near 5%.
- For SiO2 (n=1.5), SNAP-DDM outperforms the FDFD solver in time at 600x600 to 2100x2100 grids; for TiO2 (n=2.48) it requires many more iterations and is slower than FDFD.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Add self-modulation to FNO (ablation) | SM-FNO subdomain solver · Electromagnetics subdomain test | L_data: 8.04 -> 3.85; 6.11 -> 3.85 | Helped |
| Add residual connections within Fourier layers (ablation) | SM-FNO subdomain solver · Electromagnetics subdomain test | L_data: 7.61 -> 3.85 | Helped |
| Add hybrid physics loss to data-only training (ablation) | SM-FNO-v2 · Electromagnetics subdomain test | L_pde: 2.76 -> 0.30 | Helped |
| Add hybrid physics loss to data-only training (ablation) | SM-FNO-v2 · Electromagnetics subdomain test | L_data: 1.36 -> 1.00 | Helped |
| Increase subdomain size from 64 by 64 to 128 by 128 (ablation) | SNAP-DDM · TiO2, 604 by 604 | Total time (s): $2.052\pm0.006$ -> $0.900\pm0.001$ | Helped |
| Increase subdomain size from 64 by 64 to 128 by 128 (ablation) | SM-FNO material subdomain solver · Electromagnetics subdomain training | L_data (%): 2.06% | Hurt |
| Use 128 by 128 instead of 64 by 64 subdomain models (ablation) | SNAP-DDM · TiO2 electromagnetics, multiple domain sizes | Total time (s): $11.623\pm0.114$ -> $5.252\pm0.029$; $26.788\pm0.164$ -> $11.526\pm0.064$ | Helped |
Surprising findings
- For TiO2 (high refractive index), SNAP-DDM is significantly slower than conventional FDFD despite its speed advantage for low-index materials.
- An end-to-end SM-FNO trained on full 960x960 problems (262M parameters, 1000 samples) overfits and fails to generalize to test data.
Limitations
- Accuracy of DDM for TiO2 (high refractive index) requires many iterations, making it slower than FDFD.
- Fluid flow results reach errors under 15%, lower than electromagnetics, attributed to Robin-boundary Schwarz convergence issues.
- End-to-end training on full-size problems is data-hungry and fails to generalize, motivating subdomain decomposition.
Experimental setup
Datasets: FDFD-generated electromagnetic subdomain data (ceviche solver) Lattice-Boltzmann steady flow simulations
Benchmarks: 10k held-out subdomain test data (EM material, PML, source models) steady state flow subdomain test set large-domain EM DDM simulations
Metrics: L_data (%) L_pde (a.u.) L_bc (a.u.) parameters (M) FLOPs (G) mean absolute error iterations and wall time
Baselines: FNO F-FNO UNet Swin Transformer PINN
Builds on: Fourier Neural Operator Optimized Schwarz Method physics-informed neural networks
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