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PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs
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Introduces PINNacle, a benchmark of 22 PDE cases and about 10 PINN variants in a unified DeepXDE-based toolbox, evaluated with L2RE and related metrics.
Problem
A comprehensive, standardized comparison of physics-informed neural network (PINN) methods across a wide range of PDEs with challenges such as complex geometry, multi-scale behavior, nonlinearity, and high dimensionality is lacking.
Approach
A hierarchical benchmark combining 22 diverse PDE cases (forward, inverse, parametric) with a unified toolbox implementing about 10 PINN variants spanning loss reweighting, optimizers, loss functions, and architectures, plus standardized error metrics and hyperparameter ablations.
Key results
- Vanilla PINN solves only 10 of 22 tasks at a 10% L2RE threshold.
- FBPINN achieves the best error on the chaotic Gray-Scott equation (7.99E-2 L2RE), and LAAF gives the best result on Heat-2d-CG (2.39E-2).
- vPINN reaches 1.19E-2 L2RE on the HInv inverse heat problem, showing variational losses help inverse problems, while few methods handle nonlinear cases such as KS and 2D Burgers well.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use loss reweighting or residual-based resampling | PINN · PDEs with complex geometry and multi-scale phenomena | L2RE: 1.43% | Helped |
| Use a variational formulation (vPINN) | PINN · Inverse problems, including HInv | L2RE: 1.19% | Helped |
| Use a variational formulation (vPINN) | PINN · Forward PDE problems | L2RE | No clear effect |
| Use the FBPINN domain-decomposition architecture | PINN · Gray-Scott | L2RE: 7.99% | Helped |
| Use adaptive activation (LAAF) | PINN · Heat 2d-CG | L2RE: 2.39% | Helped |
| Use PINN-NTK loss balancing | PINN · Parameterized PDE tasks | L2RE | Helped |
| Use MultiAdam | PINN · High-dimensional heat equations | L2RE | Helped |
| Increase the batch size (ablation) | Vanilla PINN · Burgers1d, Gray-Scott, Heat2d-CG, and Poisson2d-C | L2RE | Helped |
| Increase the number of training epochs (ablation) | Vanilla PINN · Burgers1d, Gray-Scott, Heat2d-CG, and Poisson2d-C | L2RE | Helped |
| Use a moderate learning rate (10^{-3} or 10^{-4}) or step-decay schedule instead of a high or low rate (ablation) | Vanilla PINN · Burgers1d, Heat2d-CG, and Poisson2d-C | L2RE | Helped |
Surprising findings
- Only PINN with LBFGS solves the high-dimensional heat equation (HNd), suggesting a second-order optimizer matters there.
- Loss reweighting and domain decomposition methods do not uniformly dominate; several methods such as the vanilla PINN with larger boundary weights are competitive on some cases.
Limitations
- Overall PINN performance remains below traditional numerical methods on high-dimensional and nonlinear problems.
- Most methods fail on long-time, nonlinear, and complex-geometry cases such as Heat2d-LT and Poisson3d-CG.
- Hyperparameters are fixed for the main experiments (lr 1e-3, 20,000 epochs), which may favor some methods.
Experimental setup
Datasets: Burgers Poisson Heat Navier-Stokes Wave Gray-Scott Kuramoto-Sivashinsky high-dimensional Poisson high-dimensional heat inverse Poisson inverse heat
Benchmarks: PINNacle
Metrics: L2RE L1RE MSE max error Fourier error runtime
Baselines: vanilla PINN PINN-LRA PINN-NTK RAR MultiAdam gPINN hp-VPINN LAAF GAAF FBPINN
Builds on: PINN DeepXDE FBPINN PINN-NTK MultiAdam
Closest papers by meaning
- RoPINN: Region Optimized Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.90)
- PINNACLE: PINN Adaptive ColLocation and Experimental points selection (ICLR 2024 · similarity 0.90)
- Dual Cone Gradient Descent for Training Physics-Informed Neural Networks (NeurIPS 2024 · similarity 0.89)
- Efficient Error Certification for Physics-Informed Neural Networks (ICML 2024 · similarity 0.89)
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.89)
- PINNsAgent: Automated PDE Surrogation with Large Language Models (ICML 2025 · similarity 0.89)
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