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

Yilong Dai, Shengyu Chen, Ziyi Wang, Xiaowei Jia, Yiqun Xie, Vipin Kumar, Runlong Yu

TL;DR

Partial differential equations (PDEs) underpin scientific modeling, and learning-based solvers promise reusable representations across varying coefficients, geometries, and data. The paper proposes a unifying view of PINNs and Neural Operators within a common design space defined by what is learned, how physics is incorporated, and where computation is amortized. It introduces a unifying taxonomy, analyzes structural properties that affect reliability under deployment, and discusses practical guidance for method choice and deployment, including open challenges like controlled extrapolation and cost transparency. The work aims to bridge physics and data to enable reliable, reusable PDE solvers in diverse scientific workflows, supporting design optimization, uncertainty quantification, and data assimilation.

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.

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

TL;DR

Partial differential equations (PDEs) underpin scientific modeling, and learning-based solvers promise reusable representations across varying coefficients, geometries, and data. The paper proposes a unifying view of PINNs and Neural Operators within a common design space defined by what is learned, how physics is incorporated, and where computation is amortized. It introduces a unifying taxonomy, analyzes structural properties that affect reliability under deployment, and discusses practical guidance for method choice and deployment, including open challenges like controlled extrapolation and cost transparency. The work aims to bridge physics and data to enable reliable, reusable PDE solvers in diverse scientific workflows, supporting design optimization, uncertainty quantification, and data assimilation.

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.
Paper Structure (67 sections, 7 equations, 1 figure)

This paper contains 67 sections, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Scope and contributions.
  3. Organization.
  4. Problem Setup and Design Space
  5. PDE problems.
  6. Well-posedness and learning.
  7. Observations.
  8. Workflow regimes.
  9. Solution learning (PINN baseline).
  10. Operator learning (NO baseline).
  11. A unifying taxonomy.
  12. (T1) Object of learning.
  13. (T2) Supervision interface.
  14. (T3) Amortization locus.
  15. Workflow contracts.
  16. ...and 52 more sections

Figures (1)

  • Figure 1: A unifying view of learning-based PDE solvers. Top: three structural properties (C1--C3) that challenge all learning paradigms (Section \ref{['sec:crosscut']}). Middle: a continuum shaped by what is learned and how physics enters supervision, organized within a unified design space taxonomy. Bottom: paradigm-specific innovations for PINNs (left, Section \ref{['sec:pinn']}) and neural operators (right, Section \ref{['sec:no']}).