Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators
This work provides a unifying view to help researchers understand relationships and limitations in learning-based PDE solvers, facilitating more reliable integration of physics and data in scientific workflows, though it is incremental as it synthesizes existing paradigms rather than introducing new methods.
The paper tackles the lack of a unified perspective in physics-aware data-driven approaches for solving partial differential equations (PDEs) by proposing a framework that organizes methods like Physics-Informed Neural Networks (PINNs) and Neural Operators (NOs) into a shared design space based on three dimensions: what is learned, how physics is integrated, and computational amortization.
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.