FaNO 把物理系统的快慢变分开建模，比传统算子更准更省参数，科学计算和仿真场景值得一试。
物理系统常包含快变动态与持久结构共存的多尺度行为，现有神经网络算子依赖单一归纳偏置，难以解耦。因子化神经算子 FaNO 将频谱分解为等变动态响应和不变持久响应，两个分支分别专注于瞬态动力学和相干结构提取。在 Navier-Stokes 方程等系统上，FaNO 相比 FNO 参数效率提升 50%，跨分辨率外推误差降低 30%。长自回归 rollout 下 FaNO 能保持 1000 步稳定预测，而基线发散。该方法在跨域迁移和物理条件偏移时保持泛化能力。
Factorized Neural Operators Decompose Dynamic and Persistent Responses
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.