Recent Advances of NeuroDiffEq -- An Open-Source Library for Physics-Informed Neural Networks

TL;DR

The paper introduces NeuroDiffEq, an open-source library that uses physics-informed neural networks to solve differential equations with a PyTorch backend, enabling complex boundary value problems in arbitrary dimensions and at infinity. It presents a comprehensive feature set including a generic high-dimensional PDE solver, a BundleSolver for parameterized ODEs, dynamic behavior injection via runtime callbacks, built-in operators and bases, and expanded reparameterizations and loss functions, reinforced by a community-sharing platform NeuroDiffHub. The implementation centers on a BaseSolver with customizable residuals, conditions, collocation generators, networks, and optimizers, trained via fit() and evaluated through get_solution(), with robust quality control and CI. The work demonstrates scalable capabilities (e.g., D-dimensional heat equations up to $D=10$ and Poisson problems with periodic boundaries) and emphasizes extensibility and performance optimizations, positioning NeuroDiffEq as a practical tool for PINN research and applications.

Abstract

Solving differential equations is a critical challenge across a host of domains. While many software packages efficiently solve these equations using classical numerical approaches, there has been less effort in developing a library for researchers interested in solving such systems using neural networks. With PyTorch as its backend, NeuroDiffEq is a software library that exploits neural networks to solve differential equations. In this paper, we highlight the latest features of the NeuroDiffEq library since its debut. We show that NeuroDiffEq can solve complex boundary value problems in arbitrary dimensions, tackle boundary conditions at infinity, and maintain flexibility for dynamic injection at runtime.

Figures (6)

• Figure 1: Time evolution of the 3-dimensional heat equation at $z=0.5$
• Figure 2: Comparison of ANN solution and True Solution
• Figure 3: Bundle solvers are trained on multiple initial conditions simultaneously and are therefore robust at inference to new initial conditions.
• Figure 4: Comparison of MSE against True Solution
• Figure 5: An equirectangular projection of the solution to Eq. \ref{['eq:gaussian-charge']} averaged across radius $r$. The solution is concatenated with a copy of itself to the right, showing that the solution is periodic in the azimuth.