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[2601.14517] Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators
Summary
This paper presents a unified perspective on learning PDE solvers, integrating Physics-Informed Neural Networks and Neural Operators to enhance scientific modeling and computational efficiency.
Why It Matters
The integration of physics and data in modeling partial differential equations (PDEs) is crucial for advancing scientific workflows. This paper addresses the existing gaps in understanding the relationships and limitations of current approaches, which can lead to more reliable and efficient PDE solvers.
Key Takeaways
- The paper proposes a unifying framework for Physics-Informed Neural Networks and Neural Operators.
- It categorizes existing methods based on what is learned, integration of physical structures, and computational load management.
- Understanding structural properties can help address challenges in learning PDEs.
- The insights aim to facilitate the development of more reliable learning-based PDE solvers.
- This work encourages a synthesis of physics and data in scientific modeling.
Computer Science > Machine Learning arXiv:2601.14517 (cs) [Submitted on 20 Jan 2026 (v1), last revised 18 Feb 2026 (this version, v2)] Title:Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators Authors:Yilong Dai, Shengyu Chen, Ziyi Wang, Xiaowei Jia, Yiqun Xie, Vipin Kumar, Runlong Yu View a PDF of the paper titled Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators, by Yilong Dai and 6 other authors View PDF HTML (experimental) 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 propert...
