Factorized Neural Operators Decompose Dynamic and Persistent Responses
This work addresses the challenge of modeling multiscale physical systems with heterogeneous mechanisms, offering a more interpretable and generalizable neural operator for scientific computing.
The authors propose Factorized Neural Operators (FaNO) to decompose spectral representations into equivariant dynamic and invariant persistent responses, improving prediction accuracy, parameter efficiency, and cross-scale generalization across physical systems. FaNO maintains consistent predictions under long-horizon autoregressive rollout, cross-resolution extrapolation, and physical-regime shifts.
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.