Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators

TL;DR

This paper presents a unified framework for understanding physics-informed neural networks and neural operators, clarifying their relationships, limitations, and roles in scientific modeling workflows.

Contribution

It introduces a shared design space organizing existing PDE learning methods along three fundamental dimensions, enabling better understanding and development of physics-aware data-driven solvers.

Findings

Unified perspective clarifies relationships between PINNs and Neural Operators

Organizes methods based on what is learned, physical integration, and computational efficiency

Facilitates development of more reliable learning-based PDE solvers

Abstract

Partial differential equations (PDEs) are central to scientific modeling. Modern workflows increasingly rely on learning-based components to support model reuse, inference, and integration across large computational processes. Despite the emergence of various physics-aware data-driven approaches, the field still lacks a unified perspective to uncover their relationships, limitations, and appropriate roles in scientific workflows. To this end, we propose a unifying perspective to place two dominant paradigms: Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs), within a shared design space. We organize existing methods from three fundamental dimensions: what is learned, how physical structures are integrated into the learning process, and how the computational load is amortized across problem instances. In this way, many challenges can be best understood as consequences of these structural properties of learning PDEs. By analyzing advances through this unifying view, our survey aims to facilitate the development of reliable learning-based PDE solvers and catalyze a synthesis of physics and data.