Physics informed neural networks compared with discretization methods for pde solving
A Systematic Analysis of Automatic Differentiation versus Discretization-based Constraints for Physics-Informed PDE Solvers
Machine Learning
Summary
Solving complex partial differential equations (PDEs) accurately is important for many scientific and engineering problems. The authors studied two main ways neural networks can solve these PDEs: one using automatic differentiation (AD), and another using discretization-based constraints. They tested these approaches across many problems, from simple to very complex, and found that as problems become more nonlinear and involve tricky boundaries, discretization-based methods usually do better. They also showed that graph neural networks outperform simpler models when problems get more complex.
What this means in practice
- •For engineering simulation teams: Configure neural PDE solvers optimally for complex nonlinear and discontinuous flow problems by choosing appropriate differentiation and network methods.
- •For computational fluid dynamics developers: Enhance CFD models for high-speed flows with strong discontinuities by integrating graph neural networks and discretization-based constraints.
Authors
Xing Guo, Hongwei Tang, Zewei Meng, Yidong Zhang, Shaoqiu Xiao, Feng Liu
Abstract
Physics-informed neural networks (PINNs) represent a growing frontier in using artificial intelligence to solve partial differential equations (PDEs). Automatic differentiation (AD) plays a central role in this paradigm, which is mesh-free and replaces traditional iterative solvers with gradient-based optimization in continuous space. However, the inherent limitations of AD, particularly in handling higher-order derivatives and discontinuous solutions, pose significant challenges for complex problems. This has motivated a growing number of researchers to explore discretization-based constraints as an alternative path. Yet, the respective applicability of these two paradigms remains largely unexplored. In this work, we conduct systematic experiments across a wide spectrum of problems, from simple linear Poisson to high-Mach hypersonic flows with strong discontinuities. Through a rigorous decomposition of approximation, optimization, and truncation errors, we systematically elucidate the fundamental trade-offs and error-governing mechanisms of both paradigms, as well as two representative network architectures: multi-layer perceptron (MLP) and graph neural network (GNN). Our results reveal a consistent trend: as nonlinearity strengthens, the accuracy advantage of discretization-based constraints becomes increasingly pronounced, with smaller optimization errors compensating for the truncation errors. Moreover, the more complex the nonlinearity and boundary conditions, the greater the advantage of GNN over MLP. These insights offer a robust practical guideline for configuring neural PDE solvers in demanding engineering applications. Our source data and code are available at https://github.com/guoxing0809/neuropde_analysis.