Factorized Neural Operators Decompose Dynamic and Persistent Responses

2026-06-15

AI Summary

Existing neural operators struggle to effectively model the multiscale behavior of physical systems, where transient dynamics and persistent structures coexist, due to their reliance on a single inductive bias. This work proposes the Factorized Neural Operator (FaNO), which, within a unified Green’s function framework, explicitly decouples spectral representations into an equivariant dynamic branch for transient responses and an invariant persistent branch for stable structures. FaNO is the first neural operator to achieve explicit separation of these two distinct physical mechanisms, significantly improving long-term prediction accuracy, cross-resolution generalization, and parameter efficiency across diverse multiphysics systems. By overcoming the limitations of a single inductive bias, this approach establishes a more interpretable and reliable modeling paradigm for scientific computing.

Abstract

Physical systems often exhibit heterogeneous mechanisms, where rapidly evolving dynamics coexist with persistent structures. Capturing such multiscale physical behavior remains challenging for existing neural operators, which typically rely on single dominant inductive bias and therefore couple distinct physical responses into a shared representation. We introduce the Unified Green's Function Framework across domains and propose the Factorized Neural Operators (FaNO), which decompose spectral representations into equivariant dynamic responses and invariant persistent responses, leading to better interpretability and generalization. Mechanistically, we show that the two operator branches spontaneously specialize into distinct physical roles that remain consistent across scales and domains: the equivariant branch captures rapidly varying transient dynamics, whereas the invariant branch extracts coherent persistent structures. This factorized mechanism of FaNO improves prediction accuracy, parameter efficiency and cross-scale generalization across physical systems and domains. In particular, it maintains consistent predictions under long-horizon autoregressive rollout, cross-resolution extrapolation and physical-regime shifts. These findings suggest that scalable physical modeling may benefit from moving beyond single-inductive-bias formulations toward factorized operator representations that better reflect the heterogeneous organization of physical systems, accelerating the reliable deployment of machine learning for scientific computing and discovery.

Problem

• neural operators
• multiscale dynamics
• persistent structures
• inductive bias
• heterogeneous mechanisms

Innovation

• Factorized Neural Operators
• equivariant dynamics
• invariant structures
• multiscale modeling
• neural operators