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PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations
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Proposes Physics-Informed Gaussians (PIG), a PINN-style PDE solver whose feature embedding is a weighted sum of learnable Gaussians (trainable means and covariances) followed by a small tanh MLP, outperforming parametric grid and large-MLP PINN baselines on several PDEs with fewer parameters.
Problem
Solving PDEs with neural networks is limited by MLP spectral bias and by fixed-position parametric grids that need high resolution and many collocation points to avoid overfitting.
Approach
Replace fixed grid vertices with learnable Gaussian basis functions whose means and covariances are optimized by the PDE residual loss, so basis functions migrate to regions of high residual or singular behavior, followed by a lightweight MLP.
Key results
- PIG reaches relative L2 error 4.13e-5 on Helmholtz (best 2.12e-5 with Adam) versus 8.63e-4 for PIXEL, about four times lower.
- On Flow Mixing, PIG achieves 2.67e-4 best relative L2 error versus 2.90e-3 for SPINN.
- On Allen-Cahn, PIG uses about 20K parameters versus over 250K for JAX-PI and runs at 7.25e-3 s/it versus 1.67e-2 s/it, though JAX-PI and PirateNet reach lower best errors (5.37e-5 and 2.24e-5 vs 5.93e-5).
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add the MLP to PIG with learnable Gaussian positions (ablation) | PIG · Allen-Cahn | relative L^2 error: 7.29e-05 -> 7.27e-05 | No clear effect |
| add the MLP to PIG with learnable Gaussian positions (ablation) | PIG · Helmholtz | relative L^2 error: 1.86e-04 -> 2.12e-05 | Helped |
| add the MLP to PIG with learnable Gaussian positions (ablation) | PIG · Nonlinear Diffusion | relative L^2 error: 5.26e-03 -> 1.44e-03 | Helped |
| add the MLP to PIG with learnable Gaussian positions (ablation) | PIG · Flow Mixing | relative L^2 error: 7.93e-04 -> 4.51e-04 | Helped |
| add the MLP to PIG with learnable Gaussian positions (ablation) | PIG · Klein-Gordon | relative L^2 error: 8.51e-03 -> 2.76e-03 | Helped |
| make Gaussian positions learnable instead of fixed (ablation) | PIG without MLP · Allen-Cahn | relative L^2 error: 4.72e-03 -> 7.29e-05 | Helped |
| make Gaussian positions learnable instead of fixed (ablation) | PIG without MLP · Helmholtz | relative L^2 error: 3.97e-04 -> 1.86e-04 | Helped |
| make Gaussian positions learnable instead of fixed (ablation) | PIG without MLP · Nonlinear Diffusion | relative L^2 error: 6.32e-03 -> 5.26e-03 | Helped |
| make Gaussian positions learnable instead of fixed (ablation) | PIG without MLP · Flow Mixing | relative L^2 error: 4.33e-03 -> 7.93e-04 | Helped |
| make Gaussian positions learnable instead of fixed (ablation) | PIG without MLP · Klein-Gordon | relative L^2 error: 6.44e-02 -> 8.51e-03 | Helped |
| use dense instead of diagonal covariance, initialized from pretrained diagonal covariance (ablation) | PIG · Helmholtz | relative L^2 error: 2.12e-05 -> 5.17e-05 | Hurt |
| use dense instead of diagonal covariance, initialized from pretrained diagonal covariance (ablation) | PIG · Klein-Gordon | relative L^2 error: 2.76e-03 -> 1.81e-03 | Helped |
Surprising findings
- Training with dense covariance matrices from scratch did not yield accurate solutions; initializing from a trained diagonal model was needed.
- SIREN as the feature embedding performed poorly, while SIREN as the small MLP gave the best Helmholtz result.
Limitations
- Dynamic Gaussian parameter adjustment adds computational overhead, especially for large problems.
- The number of Gaussians is fixed at the start of training and cannot be adaptively increased.
- No complete convergence analysis is provided.
Experimental setup
Datasets: Allen-Cahn reference solution (Chebfun) manufactured solutions for Helmholtz, Klein-Gordon, flow mixing, nonlinear diffusion
Benchmarks: Allen-Cahn Helmholtz Nonlinear Diffusion Flow Mixing Klein-Gordon PIRBN benchmarks lid-driven cavity 2D Burgers
Metrics: relative L2 error time per iteration
Baselines: PINN Learning Rate Annealing PIXEL SPINN JAX-PI PirateNet PIRBNs PGCAN PI-GS SIREN
Builds on: Physics-Informed Neural Networks PIXEL SPINN Gaussian splatting NTK-based loss balancing
Closest papers by meaning
- PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs (NeurIPS 2024 · similarity 0.88)
- Parameterized Physics-informed Neural Networks for Parameterized PDEs (ICML 2024 · similarity 0.88)
- Graph Neural PDE Solvers with Conservation and Similarity-Equivariance (ICML 2024 · similarity 0.87)
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.87)
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.87)
- Efficient Error Certification for Physics-Informed Neural Networks (ICML 2024 · similarity 0.86)
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