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

2026-01-20

AI Summary

This work addresses the lack of a unified understanding of the design principles, applicability, and performance differences between Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs), which hinders the development of reliable data-driven PDE solvers. It proposes the first unified analytical framework that systematically characterizes the design space of both approaches along three dimensions: learning objectives, mechanisms for embedding physical structure, and strategies for computational load distribution. By elucidating the intrinsic connections and fundamental distinctions between these methods, the study not only clarifies the positioning and performance origins of existing techniques but also provides theoretical guidance and novel pathways for designing efficient and robust PDE solvers that effectively integrate physical priors with data-driven learning.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Solvers and Tools

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• PDE solvers
• physics-informed neural networks
• neural operators
• unified perspective
• scientific modeling

Innovation

• Physics-Informed Neural Networks
• Neural Operators
• PDE solvers
• unified framework
• physics-data integration