发表机构
Sun Yat-sen University; National Key Laboratory of Aerospace Physics in Fluids（中山大学; 国家航空航天流体物理重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文系统比较了物理信息神经网络中自动微分与基于离散化的约束，发现随非线性增强后者精度优势更显著，且GNN优于MLP，为工程应用提供配置指南。
AI 中文摘要
物理信息神经网络（PINNs）代表了利用人工智能求解偏微分方程（PDEs）的一个新兴前沿。自动微分（AD）在这一范式中扮演核心角色，该方法无需网格，并在连续空间中以基于梯度的优化取代传统迭代求解器。然而，AD固有的局限性，特别是在处理高阶导数和间断解方面，对复杂问题构成了重大挑战。这促使越来越多的研究者探索基于离散化的约束作为替代路径。然而，这两种范式各自的适用性在很大程度上仍未得到探索。在本工作中，我们在从简单线性泊松方程到具有强间断的高马赫数高超声速流动的广泛问题谱系上进行了系统性实验。通过对近似误差、优化误差和截断误差的严格分解，我们系统地阐明了两种范式的基本权衡和误差控制机制，以及两种代表性网络架构：多层感知机（MLP）和图神经网络（GNN）。我们的结果揭示了一个一致的趋势：随着非线性增强，基于离散化的约束的精度优势变得越来越显著，较小的优化误差补偿了截断误差。此外，非线性和边界条件越复杂，GNN相对于MLP的优势越大。这些见解为在要求苛刻的工程应用中配置神经PDE求解器提供了稳健的实用指南。我们的源数据和代码可在以下网址获取：此https URL。
英文摘要
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.
