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

Zhuo Zhang, Xiong Xiong, Sen Zhang, Yuan Zhao, Xi Yang

TL;DR

This survey addresses the parametric PDE solving challenge by juxtaposing physics-informed neural networks (PINNs) and neural operators as complementary paradigms. PINNs embed governing equations as soft constraints to handle inverse problems with limited data, while neural operators (e.g., DeepONet, FNO) learn parameter-to-solution mappings that generalize across parameter spaces, delivering 10^3–10^5× speedups for multi-query scenarios. The work surveys mathematical foundations, algorithmic advances (ill-conditioning mitigation, separable and Kolmogorov–Arnol'd networks, geo-aware operators), and broad domain applications spanning fluid dynamics, solid mechanics, heat transfer, and electromagnetics, with specific performance benchmarks and case studies. It also outlines theoretical and practical gaps, such as high-dimensional parameter spaces, out-of-distribution generalization, and long-time stability, and proposes future directions including uncertainty quantification, foundation models, and hybrid physics–ML frameworks. Collectively, the paper provides a unified, evolving resource that guides method selection, benchmarking, and community-building toward reliable, scalable neural PDE solvers for parametric problems.

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

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

TL;DR

This survey addresses the parametric PDE solving challenge by juxtaposing physics-informed neural networks (PINNs) and neural operators as complementary paradigms. PINNs embed governing equations as soft constraints to handle inverse problems with limited data, while neural operators (e.g., DeepONet, FNO) learn parameter-to-solution mappings that generalize across parameter spaces, delivering 10^3–10^5× speedups for multi-query scenarios. The work surveys mathematical foundations, algorithmic advances (ill-conditioning mitigation, separable and Kolmogorov–Arnol'd networks, geo-aware operators), and broad domain applications spanning fluid dynamics, solid mechanics, heat transfer, and electromagnetics, with specific performance benchmarks and case studies. It also outlines theoretical and practical gaps, such as high-dimensional parameter spaces, out-of-distribution generalization, and long-time stability, and proposes future directions including uncertainty quantification, foundation models, and hybrid physics–ML frameworks. Collectively, the paper provides a unified, evolving resource that guides method selection, benchmarking, and community-building toward reliable, scalable neural PDE solvers for parametric problems.

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 to 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

Paper Structure

This paper contains 135 sections, 2 theorems, 90 equations, 3 figures, 7 tables, 1 algorithm.

Table of Contents

  1. Introduction and Background
  2. Motivation: The Parametric Challenge in Scientific Computing
  3. Mathematical Formulation of Parametric PDEs
  4. Traditional Approaches: Reduced Order Models
  5. Proper Orthogonal Decomposition and Galerkin Projection
  6. Reduced Basis Methods
  7. Sparse Grid and Multi-Level Methods
  8. Comparative Performance Analysis
  9. Methodologies: Physics-Informed Neural Networks and Neural Operators
  10. Physics-Informed Neural Networks (PINNs)
  11. Foundational Framework
  12. Parametric Enhancement Techniques
  13. Recent Advances in PINN Architecture and Training
  14. Ill-Conditioning Analysis and Solutions
  15. Separable Architectures
  16. ...and 120 more sections

Key Result

Theorem 1

Let $\mathcal{G}: \mathcal{A} \to \mathcal{U}$ be a continuous operator between function spaces. For any $\epsilon > 0$, there exists a neural operator architecture and parameters $\theta$ such that:

Figures (3)

  • Figure 1: Human-AI collaborative workflow for survey generation.
  • Figure 2: Survey organization and reading roadmap.
  • Figure 3: Research landscape of parametric PDE solving methods.

Theorems & Definitions (3)

  • Definition 1: Parametric PDE
  • Theorem 1: Neural Operator Universal Approximation (Informal)
  • Theorem 2: Neural Operator Universal Approximation Kovachki2021