Physics-Informed Neural Networks and Neural Operators for Parametric PDEs: A Human-AI Collaborative Analysis

• URL: http://arxiv.org/abs/2511.04576v2
• Date: Fri, 07 Nov 2025 06:01:34 GMT
• Title: Physics-Informed Neural Networks and Neural Operators for Parametric PDEs: A Human-AI Collaborative Analysis
• Authors: Zhuo Zhang, Xiong Xiong, Sen Zhang, Yuan Zhao, Xi Yang,
• Abstract summary: PDEs arise ubiquitously in science and engineering, where solutions depend on parameters.<n>Recent machine learning advances have revolutionized PDE solving by learning solution operators that generalize across parameter spaces.<n>We show neural operators can achieve computational speedups of $103$ to $105$ times faster than traditional solvers.
• Score: 18.201079606404978
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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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