用神经网络快速求解参数化偏微分方程，速度提升千倍以上。
Physics-Informed Neural Networks and Neural Operators for Parametric PDEs
- 通过物理信息神经网络和神经算子学习解映射，实现跨参数泛化。
- 在多查询场景下比传统求解器快1000至10万倍，精度相当。
- 适合需要快速模拟大量参数变化的工程与科学计算场景。
偏微分方程（PDEs）广泛存在于科学与工程中，其解依赖于参数（如物理属性、边界条件、几何形状）。传统数值方法需对每个参数重新求解，导致参数空间探索代价高昂。近年来，机器学习进展，特别是物理信息神经网络（PINNs）和神经算子（如DeepONet、傅里叶神经算子），通过学习解算子实现了对参数空间的泛化，革新了参数化PDE的求解方式。本文系统分析两类主流范式：(1) PINNs，将物理定律作为软约束嵌入，擅长稀疏数据下的反问题求解；(2) 神经算子（如DeepONet、FNO），学习无限维函数空间间的映射，实现前所未有的泛化能力。在流体动力学、固体力学、传热与电磁学等多个领域对比表明，神经算子在多查询场景下可实现10³至10⁵倍的速度提升，同时保持相近精度。本文提供方法选型建议，讨论理论基础（通用逼近性、收敛性），并指出关键挑战：高维参数、复杂几何及分布外泛化。本工作构建了基于算子学习的统一理解框架，为该快速发展的领域提供全面且持续更新的参考资源。
原文摘要 · Abstract (English)
PDEs arise ubiquitously in science and engineering, where solutions depend on parameters (physical properties, boundary conditions, geometry). Traditional numerical methods require re-solving the PDE for each parameter, making parameter space exploration prohibitively expensive. Recent machine learning advances, particularly physics-informed neural networks (PINNs) and neural operators, have revolutionized parametric PDE solving by learning solution operators that generalize across parameter spaces. We critically analyze two main paradigms: (1) PINNs, which embed physical laws as soft constraints and excel at inverse problems with sparse data, and (2) neural operators (e.g., DeepONet, Fourier Neural Operator), which learn mappings between infinite-dimensional function spaces and achieve unprecedented generalization. Through comparisons across fluid dynamics, solid mechanics, heat transfer, and electromagnetics, we show neural operators can achieve computational speedups of $10^3$ to $10^5$ times faster than traditional solvers for multi-query scenarios, while maintaining comparable accuracy. We provide practical guidance for method selection, discuss theoretical foundations (universal approximation, convergence), and identify critical open challenges: high-dimensional parameters, complex geometries, and out-of-distribution generalization. This work establishes a unified framework for understanding parametric PDE solvers via operator learning, offering a comprehensive, incrementally updated resource for this rapidly evolving field
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