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

TL;DR

PINNacle is a comprehensive benchmarking framework for Physics-Informed Neural Networks (PINNs), evaluating over 20 PDEs across various domains to facilitate systematic comparison and guide future research in solving complex PDEs.

Contribution

This paper introduces PINNacle, the largest diverse benchmark dataset and evaluation toolbox for PINNs, enabling systematic comparison across multiple PDE types and complexities.

Findings

PINN methods vary significantly in performance across different PDEs.

Domain decomposition and loss reweighting improve PINN accuracy on complex problems.

PINNacle reveals key strengths and weaknesses of current PINN approaches.

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.