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

2025-10-02

AI Summary

This study addresses the interpretability and physical consistency of neural operators—specifically, how they implicitly learn governing physical laws and how prior physical knowledge can be systematically integrated to enhance generalization and scientific discovery. We propose a unified “functional–spatial domain” dual-domain classification framework to characterize the intrinsic dynamical modeling pathways of neural operators. Methodologically, we develop a synergistic approach combining multi-scale dual-space modeling, functional basis expansion, and grid-aware learning, enabling joint data-driven prediction and physical structure identification. Experiments demonstrate that lightweight dual-space models achieve state-of-the-art performance, confirming that neural operators intrinsically capture PDE structures. Our core contribution is the first principled interpretability framework for neural operators tailored to physics discovery, providing both theoretical foundations and practical paradigms for trustworthy scientific machine learning.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

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.

Problem

• Interpreting neural operators' learning mechanisms for dynamics
• Developing generalizable explanation methods for hidden physics
• Incorporating physical principles into neural operator frameworks

Innovation

• Classifying neural operators into spatial and functional domains
• Developing dual-space multi-scale models for complex physics
• Integrating physics principles into neural operator frameworks