PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs

TL;DR

PINNacle addresses the lack of a comprehensive PINN benchmark by assembling a dataset of over 20 PDEs across heat, fluid, electromagnetics, and biology, and by providing a DeepXDE-based toolbox with around 10 state-of-the-art PINN variants. The study evaluates these variants on diverse PDE challenges, including complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality, using standardized metrics such as $L2RE$, $L1RE$, $MSE$, and $fMSE$. Key findings show that vanilla PINNs struggle on hard problems, while strategies like loss reweighting, domain decomposition (FBPINN), and variational formulations offer notable gains, and that hyperparameters and problem parameters strongly influence performance. The results suggest that integrating PINNs with traditional numerical methods (e.g., preconditioning, weak formulations, multigrid) could further close the gap to classical solvers, and provide practical guidance for future PINN research and development.

Abstract

While significant progress has been made on Physics-Informed Neural Networks (PINNs), a comprehensive comparison of these methods across a wide range of Partial Differential Equations (PDEs) is still lacking. This study introduces PINNacle, a benchmarking tool designed to fill this gap. PINNacle provides a diverse dataset, comprising over 20 distinct PDEs from various domains, including heat conduction, fluid dynamics, biology, and electromagnetics. These PDEs encapsulate key challenges inherent to real-world problems, such as complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality. PINNacle also offers a user-friendly toolbox, incorporating about 10 state-of-the-art PINN methods for systematic evaluation and comparison. We have conducted extensive experiments with these methods, offering insights into their strengths and weaknesses. In addition to providing a standardized means of assessing performance, PINNacle also offers an in-depth analysis to guide future research, particularly in areas such as domain decomposition methods and loss reweighting for handling multi-scale problems and complex geometry. To the best of our knowledge, it is the largest benchmark with a diverse and comprehensive evaluation that will undoubtedly foster further research in PINNs.

Figures (27)

• Figure 1: Architecture of PINNacle. It contains a dataset covering more than 20 PDEs, a toolbox that implements about 10 SOTA methods, and an evaluation module. These methods have a wide range of application scenarios like fluid mechanics, electromagnetism, heat conduction, geophysics, and so on.
• Figure 2: Performance of vanilla PINNs under different batch sizes (number of collocation points), which is shown in the left figure; and number of training epochs, which is shown in the right figure.
• Figure 3: Convergence curve of PINNs with different learning rate schedules on Burgers1d, Heat2d-CG, and Poisson2d-C.
• Figure 4: Reference solution of Burgers1d using FEM solver.
• Figure 5: Reference solution of Burgers2d at timesteps $t=0, 0.2, 0.4, 1.0$ using FEM solver.