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

摘要

Partial differential equations (PDEs) are central to scientific modeling. Modern workflows increasingly rely on machine learning-based components. 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 that places 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 practical challenges can be best understood as consequences of these structural properties of learning PDEs. By analyzing recent advances through this unifying view, our survey aims to facilitate the development of reliable learning-based PDE solvers and help catalyze a deeper synthesis of physics and data.