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PINNACLE: PINN Adaptive ColLocation and Experimental points selection
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Introduces PINNACLE, an algorithm that jointly selects experimental and collocation points (PDE and IC/BC) for PINNs using a Nystrom-approximated NTK-based convergence-degree criterion, with theoretical generalization bound.
Problem
Training PINNs requires many collocation points (PDE and IC/BC) and costly experimental points, but prior work selected only one point type at a time, ignoring cross-type interactions under a limited budget.
Approach
Augments the input space with point-type indicators and analyzes PINN training via the eNTK eigenbasis, which captures cross-type interactions; selects points maximizing an NTK-based convergence degree (RKHS norm of residual change), approximated via Nystrom, and periodically re-computes the eNTK. Points are sampled or chosen by K-Means++ with automatic adjustment of CL point type proportions.
Key results
- PINNACLE variants achieve lower predictive loss than RAD and Hammersley on 1D Advection and Burgers forward problems, learning all solution stripes where benchmarks stall near the initial condition.
- On the 2D Navier-Stokes inverse problem, PINNACLE recovers the correct inverse parameters while other point selection methods have difficulty.
- The paper claims PINNACLE reaches target loss faster in transfer learning with perturbed ICs using 5x fewer points than training from scratch.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| jointly select all collocation-point types using PINNACLE | PINN forward problem · 1D Advection | predictive loss | Helped |
| jointly select all collocation-point types using PINNACLE | PINN forward problem · 1D Burgers | predictive loss | Helped |
| jointly select all training point types using PINNACLE | PINN inverse problem · 2D Navier-Stokes | inverse parameter recovery | Helped |
| jointly select all training point types using PINNACLE | pre-trained PINN fine-tuned for a perturbed initial condition · 1D Advection transfer learning | predictive error | Helped |
| use PINNACLE for point selection | PINN with only one collocation-point type to select · hard-constrained PINNs | training performance | Helped |
Surprising findings
- PINNACLE outperforms other methods even when only one CL point type must be selected (hard-constrained PINNs), where no cross-type information exists.
Limitations
- Requires eNTK computation and periodic re-computation, adding cost
- Generalization bound relies on assumptions such as linearized regime and bounded residuals
Experimental setup
Datasets: PDEBench
Benchmarks: 1D Advection 1D Burgers 2D Navier-Stokes inverse problem Eikonal inverse problem PDEBench
Metrics: predictive loss mean prediction error inverse parameter recovery
Baselines: Hammersley sequence sampler RAD RAD-All random sampling
Builds on: physics-informed neural networks RAD neural tangent kernel DeepXDE
Closest papers by meaning
- PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.90)
- Challenges in Training PINNs: A Loss Landscape Perspective (ICML 2024 · similarity 0.89)
- RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.88)
- Per-Loss Adapters for Gradient Conflict in Physics-Informed Neural Networks (NeurIPS 2026 · similarity 0.87)
- ConFIG: Towards Conflict-free Training of Physics Informed Neural Networks (ICLR 2025 · similarity 0.87)
- PINN Balls: Scaling Second-Order Methods for PINNs with Domain Decomposition and Adaptive Sampling (NeurIPS 2025 · similarity 0.87)
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