Physics-Informed Neural Networks and Neural Operators for Parametric PDEs

TL;DR

This paper reviews physics-informed neural networks and neural operators for efficiently solving parametric PDEs, comparing their capabilities, limitations, and practical applications across various scientific fields.

Contribution

It provides a comprehensive analysis of PINNs and neural operators, highlighting their differences, advantages, and challenges, and offers practical guidance for their use in solving parametric PDEs.

Findings

Neural operators achieve 10^3 to 10^5 times speedup over traditional solvers.

PINNs excel at inverse problems with sparse data.

The paper discusses theoretical foundations and open challenges in the field.

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