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Efficient Learning of PDEs via Taylor Expansion and Sparse Decomposition into Value and Fourier Domains
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Proposes REEL, which accelerates learning of PDE models by decomposing updates into sparse value- and Fourier-domain components, Taylor-expanding nonlinear terms into decomposable form, and compressing with random projection.
Problem
Learning PDE parameters from data via simulation-based gradient training is expensive, and prior random-projection acceleration (RAPID-PDE) only applies to decomposable PDEs with value-domain sparsity.
Approach
Split dense system updates into a sparse value-domain part and a sparse frequency-domain part via a Fourier threshold, use Taylor expansion to make non-decomposable PDE terms decomposable, then randomly project both parts to compress learning; theory gives a constant-factor loss approximation with poly-logarithmic projected dimensions.
Key results
- REEL reduces sintering training time by 70%-98% when data is compressed to 1% of original size.
- For nanovoid, about 70% training time reduction without compression, and REEL at 10% compression captures void shrinking while RAPID-PDE disintegrates the void.
- MSE of phi and T for REEL at 1% compression is small and comparable to the uncompressed baseline.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use REEL for PDE learning | Sintering PDE model · Solid-state selective laser sintering | training time: 70−98% reduction | Helped |
| Use REEL without compression | Nanovoid PDE model · Nanovoid evolution | training time: ≈70% reduction | Helped |
| Compress data to 1% of original dimension (ablation) | REEL · Solid-state selective laser sintering | MSE | No clear effect |
| Compress data to 10% of original dimension (ablation) | REEL · Nanovoid evolution | void evolution dynamics | Helped |
| Use RAPID-PDE with compression to 10% instead of REEL | Nanovoid PDE model · Nanovoid evolution | void evolution dynamics | Hurt |
Limitations
- Code stated to be released only in future; no public release at submission.
- Decomposition threshold chosen empirically on few signal samples.
- Theoretical guarantee relies on sparsity after decomposition and sub-exponential assumptions.
Experimental setup
Datasets: synthetic sintering phase-field simulations synthetic nanovoid phase-field simulations
Benchmarks: solid-state laser sintering phase-field simulation nanovoid evolution phase-field simulation
Metrics: training time mean squared error of phase field and temperature fields simulation visual comparison
Baselines: RAPID-PDE REEL without random projection
Builds on: RAPID-PDE physics-informed neural networks Taylor series expansion
Closest papers by meaning
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