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Accelerating PDE Data Generation via Differential Operator Action in Solution Space
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Proposes DiffOAS, a PDE dataset generation method that builds solutions as random combinations of basis solutions and computes right-hand sides by applying discretized differential operators (matrix-vector products), avoiding linear system solves.
Problem
Generating large, high-precision PDE training datasets for neural operators is computationally expensive because each sample requires solving large linear systems iteratively, which is slow and introduces termination error.
Approach
Generate solution functions u as Gaussian-weighted combinations of a few basis PDE solutions (plus boundary-preserving noise), then obtain the right-hand side f by applying the discretized operator A to u, replacing the linear solve with a single matrix-vector product at machine precision.
Key results
- DiffOAS generates 10,000 Darcy samples in about 9.5 s versus 299 s for GMRES on 1000 instances' setting, roughly 30x faster in the main table, and about 300x total speedup in the timing table.
- On DeepONet, DiffOAS-generated data gives lower test error than GMRES data for all three PDEs (e.g., Darcy 5.62e-2 with 10,000 instances vs 6.82e-2 for GMRES).
- With 10,000 instances, FNO trained on DiffOAS data reaches 4.26e-3 relative error on Darcy versus 4.56e-3 for GMRES data, while the wave-equation FNO error is worse (6.52e-4 vs 4.05e-4 for GMRES with 1000 instances).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace GMRES solves with DiffOAS solution generation and operator actions | PDE dataset generation · Darcy Flow, Wave, and Diffusion datasets | generation time: approximately 30 times | Helped |
| Train on DiffOAS-generated data instead of GMRES-generated data | FNO · Darcy Flow, Wave, and Diffusion | error: 4.56e-3 -> 4.26e-3; 4.05e-4 -> 6.52e-4; 4.29e-3 -> 4.79e-3 | Mixed |
| Train on DiffOAS-generated data instead of GMRES-generated data | DeepONet · Darcy Flow, Wave, and Diffusion | error: 6.82e-2 -> 5.62e-2; 6.12e-2 -> 4.72e-2; 7.70e-2 -> 6.04e-2 | Helped |
| Use DiffOAS operator actions instead of GMRES solves | PDE data generation, 10,000 data points · Matrix size 62500 | operator-action speedup: up to approximately 70,000 times | Helped |
| Use DiffOAS for dataset generation instead of GMRES | PDE data generation, 10,000 data points · Multiple matrix sizes | total generation speedup: approximately 300 times | Helped |
| Use physical-distribution solution functions as basis functions instead of GRF basis functions (ablation) | DiffOAS dataset generation · Darcy Flow | FNO error: 31.44% -> 4.260% | Helped |
| Use physical-distribution solution functions as basis functions instead of Fourier basis functions (ablation) | DiffOAS dataset generation · Darcy Flow | FNO error: 93.56% -> 4.260% | Helped |
| Use physical-distribution solution functions as basis functions instead of truncated Chebyshev basis functions (ablation) | DiffOAS dataset generation · Darcy Flow | FNO error: 63.11% -> 4.260% | Helped |
| Use physical-distribution solution functions as basis functions instead of GRF basis functions (ablation) | DiffOAS dataset generation · Darcy Flow | DeepONet error: 88.08% -> 5.629% | Helped |
| Use physical-distribution solution functions as basis functions instead of Fourier basis functions (ablation) | DiffOAS dataset generation · Darcy Flow | DeepONet error: 96.98% -> 5.629% | Helped |
| Use physical-distribution solution functions as basis functions instead of truncated Chebyshev basis functions (ablation) | DiffOAS dataset generation · Darcy Flow | DeepONet error: 68.23% -> 5.629% | Helped |
| Improve GMRES tolerance from 1E-1 to 1E-5 | FNO trained on GMRES-generated Darcy Flow data · Darcy Flow, matrix dimension 10,000, 10,000 training instances | FNO error: 1.20E-1 -> 1.15E-3 | Helped |
Surprising findings
- Data from DiffOAS trained FNO on the wave equation had lower error with GMRES-generated data (4.05e-4) than with DiffOAS data at 10,000 instances (6.52e-4)
- Basis-function choice matters greatly: GRF, Fourier, and Chebyshev bases yield test errors of 31% to 97%, far worse than DiffOAS's 4.26%
Limitations
- Basis functions are still generated with a conventional solver (GMRES), so basis generation dominates the remaining cost.
- Method is not tailored to specific PDE types or symmetric matrices; no optimization of basis selection is performed.
- Some models trained on DiffOAS data show higher test error than GMRES data (e.g., wave equation FNO), attributed to test-set error from GMRES.
Experimental setup
Datasets: Darcy flow (GRF-generated) scalar wave equation (Helmholtz, GRF-generated) solute diffusion in porous media (GRF-generated)
Benchmarks: Darcy flow scalar wave equation solute diffusion
Metrics: generation time (s) relative error of trained FNO and DeepONet speedup ratio
Baselines: GMRES MINRES Conjugate Gradient SKR
Builds on: Fourier Neural Operator DeepONet GMRES
Closest papers by meaning
- Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning (NeurIPS 2024 · similarity 0.86)
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- DiffusionPDE: Generative PDE-Solving under Partial Observation (NeurIPS 2024 · similarity 0.85)
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