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RandNet-Parareal: a time-parallel PDE solver using Random Neural Networks
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Proposes RandNet-Parareal, which learns the fine-minus-coarse discrepancy in Parareal with random neural networks (closed-form ridgeless readout) to speed up time-parallel PDE solving on meshes up to 10^5 points.
Problem
Parareal converges slowly on hyperbolic PDEs, and learned-discrepancy variants (GParareal, nnGParareal) become costly in hyperparameter optimization and scale poorly when the state dimension d greatly exceeds the number of cores N.
Approach
Replace the per-dimension nearest-neighbor Gaussian processes used to learn F-G with a single random neural network (single hidden layer, random fixed weights, ReLU) whose output weights are obtained in closed form by ridgeless least squares on the m nearest neighbours of each query, avoiding backpropagation and hyperparameter tuning and predicting all d outputs jointly.
Key results
- On viscous Burgers' with d=1128, RandNet-Parareal reaches speed-up 29.98 in 38 minutes versus 12h57m for Parareal (S=1.41) and 2h17m for nnGParareal (S=8.26).
- On the shallow water equation with d=105336, RandNet-Parareal achieves speed-up 38.90 versus 29.37 for Parareal, converging in 6 iterations versus 8.
- RandNet model training cost is roughly 700 times lower than nnGP's on the Burgers' example, and nnGParareal fails to converge within 48 hours for the Diffusion-Reaction and SWE cases with d much larger than N.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use RandNet-Parareal instead of Parareal | Parareal · Burgers' d=128, N=128 | speed-up: 1.47 -> 12.61 | Helped |
| Use RandNet-Parareal instead of nnGParareal | nnGParareal · Burgers' d=128, N=128 | speed-up: 7.90 -> 12.61 | Helped |
| Use RandNet-Parareal instead of Parareal | Parareal · Burgers' d=1128, N=128 | speed-up: 1.41 -> 29.98 | Helped |
| Use RandNet-Parareal instead of nnGParareal | nnGParareal · Burgers' d=1128, N=128 | speed-up: 8.26 -> 29.98 | Helped |
| Use RandNet-Parareal instead of Parareal | Parareal · SWE d=105336 | speed-up: 29.37 -> 38.90 | Helped |
| Use RandNet-Parareal instead of Parareal | Parareal · Diffusion-Reaction d=7.2e^{2} | maximum absolute error: 1.83e^{-8} -> 3.56e^{-8} | Hurt |
| Use RandNet-Parareal instead of Parareal | Parareal · SWE d=3.1e^{4} | maximum absolute error: 5.15e^{-8} -> 6.75e^{-8} | Hurt |
| Use RandNet-Parareal instead of Parareal | Parareal · Diffusion-Reaction d=3.3e^{3} | maximum absolute error: 2.45e^{-8} -> 8.56e^{-10} | Helped |
| Use RandNet-Parareal instead of Parareal | Parareal · Diffusion-Reaction d=2.5e^{4} | maximum absolute error: 7.43e^{-9} -> 8.09e^{-11} | Helped |
| Use RandNet-Parareal instead of Parareal | Parareal · SWE d=6.1e^{4} | maximum absolute error: 2.84e^{-8} -> 8.54e^{-9} | Helped |
| Vary the number of nearest neighbors, hidden neurons, or random weight seed (ablation) | RandNet-Parareal · Burgers' and Diffusion-Reaction | iterations to convergence | No clear effect |
Surprising findings
- nnGParareal fails to converge within the 48-hour budget for large d/N, while RandNet-Parareal still converges.
- Robustness study shows iteration counts vary by only a few units across number of neighbours, neurons, and random seeds.
Limitations
- Convergence depends on the accuracy/suitability of the coarse solver; mismatched solvers (e.g., unstable for stiff ODEs) may not converge.
- Robustness and complexity analysis is limited to relatively small systems and the comparison is restricted to Parareal-family methods.
- Speed-up gains over Parareal are smaller for the SWE (x1.3-3.6) and Brusselator (x3.4-4.4) than for Burgers' (up to x21.2).
Experimental setup
Datasets: viscous Burgers' equation (finite difference, generated) Diffusion-Reaction (Fitzhugh-Nagumo, finite volume, generated) shallow water equation (radial dam break, generated) 2D and 3D Brusselator (generated)
Benchmarks: viscous Burgers' equation Diffusion-Reaction equation shallow water equation 2D Brusselator 3D Brusselator
Metrics: speed-up S_alg iterations to convergence K wallclock time maximum absolute error
Baselines: Parareal GParareal nnGParareal fine solver run serially
Builds on: Parareal GParareal nnGParareal Random Feature Networks
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