# Can Data-Driven Dynamics Reveal Hidden Physics? There Is A Need for Interpretable Neural Operators

arXiv: https://arxiv.org/abs/2510.02683 · HTML: https://mlsanity.com/paper/2510.02683/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 45 references that are in the map.
- First posted on arXiv 3 Oct 2025.

## Builds on (most cited first, 27 of 27)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Understanding the Effective Receptive Field in Deep Convolutional Neural Networks](https://mlsanity.com/paper/1701.04128/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Group Equivariant Fourier Neural Operators for Partial Differential Equations](https://mlsanity.com/paper/2306.05697/) (2023)
- [Equivariant Frames and the Impossibility of Continuous Canonicalization](https://mlsanity.com/paper/2402.16077/) (2024)
- [INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservation](https://mlsanity.com/paper/2212.14365/) (2022)
- [Discretization Error of Fourier Neural Operators](https://mlsanity.com/paper/2405.02221/) (2024)
- [Transform Once: Efficient Operator Learning in Frequency Domain](https://mlsanity.com/paper/2211.14453/) (2022)
- [Position: Optimization in SciML Should Employ the Function Space Geometry](https://mlsanity.com/paper/2402.07318/) (2024)
- [Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery](https://mlsanity.com/paper/2408.07307/) (2024)
- [Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning](https://mlsanity.com/paper/2405.09285/) (2024)

## Built on by (most cited first, 1 of 1)

- [Towards a Foundation Model for Partial Differential Equations Across Physics Domains](https://mlsanity.com/paper/2511.21861/) (2025)

## Abstract (arXiv)

Recently, neural operators have emerged as powerful tools for learning mappings between function spaces, enabling data-driven simulations of complex dynamics. Despite their successes, a deeper understanding of their learning mechanisms remains underexplored. In this work, we classify neural operators into two types: (1) Spatial domain models that learn on grids and (2) Functional domain models that learn with function bases. We present several viewpoints based on this classification and focus on learning data-driven dynamics adhering to physical principles. Specifically, we provide a way to explain the prediction-making process of neural operators and show that neural operator can learn hidden physical patterns from data. However, this explanation method is limited to specific situations, highlighting the urgent need for generalizable explanation methods. Next, we show that a simple dual-space multi-scale model can achieve SOTA performance and we believe that dual-space multi-spatio-scale models hold significant potential to learn complex physics and require further investigation. Lastly, we discuss the critical need for principled frameworks to incorporate known physics into neural operators, enabling better generalization and uncovering more hidden physical phenomena.