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RoPINN: Region Optimized Physics-Informed Neural Networks
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Proposes region optimization for PINNs, implemented as RoPINN via Monte Carlo sampling of one point per neighborhood with trust-region calibration of the region size, improving diverse PINN backbones across PDE benchmarks without extra gradient computation.
Problem
PINNs are conventionally optimized only on scattered collocation points, which mismatches the goal of approximating PDE solutions over continuous domains; high-order regularization and variational alternatives are costly or unstable.
Approach
Extend the PINN loss from isolated points to their neighborhood regions, approximate the region integral by uniformly sampling one point per region each iteration, and calibrate the region size (trust region) inversely to gradient variance estimated over successive iterations.
Key results
- RoPINN improves canonical PINN rMSE on 1D-Reaction from 0.981 to 0.095 (about 90% promotion), with Loss 2.0e-1 to 4.7e-5.
- On PINNacle (16 tasks), RoPINN improves the vanilla PINN on 93.8% of tasks (rMAE and rMSE proportions) and reaches 100% for PINNsFormer-based and KAN-based variants in some metrics.
- RoPINN adds no extra gradient or backpropagation calls relative to point optimization, and remains compatible with NTK and RAR, further improving them.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace point optimization with trust-calibrated region optimization | PINN · 1D-Reaction | rMSE: 0.981 -> 0.095 | Helped |
| replace point optimization with trust-calibrated region optimization | PINN · 1D-Wave | rMSE: 0.335 -> 0.064 | Helped |
| replace point optimization with trust-calibrated region optimization | PINN · Convection | rMSE: 0.840 -> 0.720 | Helped |
| add region optimization (ablation) | PINN optimization · multiple PDEs and base models | rMSE | Helped |
| add trust-region calibration to region optimization (ablation) | PINN optimization · multiple PDEs and base models | rMSE | Helped |
| increase the number of sampled points in the region (ablation) | RoPINN · 1D-Reaction | final performance | Helped |
| initialize with a larger region size such as 1e-3 rather than a size closer to the balance point (ablation) | RoPINN · 1D-Reaction | convergence speed | Hurt |
Surprising findings
- Fixed-size region optimization already helps in most cases, but trust-region calibration is needed for consistent gains.
- gPINN degrades on 1D-Wave and vPINN fails on several settings, showing prior objective modifications can hurt.
Limitations
- Gradient estimation error from Monte Carlo sampling can destabilize training when region size is too large.
- Theoretical guarantees rely on assumptions such as Lipschitz and smoothness of the loss, and convexity in the convex case.
- Practical benefits are shown on a limited set of PDE benchmarks and base models.
Experimental setup
Datasets: 1D-Reaction 1D-Wave Convection PINNacle
Benchmarks: 1D-Reaction 1D-Wave Convection PINNacle
Metrics: loss rMAE rMSE proportion of improved tasks
Baselines: gPINN vPINN NTK RAR
Builds on: PINNs gPINN vPINN KAN PINNsFormer
Closest papers by meaning
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.90)
- PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.90)
- Dual Cone Gradient Descent for Training Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.90)
- Challenges in Training PINNs: A Loss Landscape Perspective (ICML 2024 · similarity 0.90)
- Sub-Sequential Physics-Informed Learning with State Space Model (ICML 2025 · similarity 0.90)
- Efficient Error Certification for Physics-Informed Neural Networks (ICML 2024 · similarity 0.90)
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