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.

翻译：物理系统往往展现异质性机制，快速演化的动态过程与持久性结构共存。现有神经算子通常依赖单一主导归纳偏置，将不同物理响应耦合至共享表示中，因而难以捕捉此类多尺度物理行为。我们提出跨域统一的格林函数框架，并引入因子化神经算子(FaNO)，通过将谱表示分解为等变动态响应与不变持久响应，实现更强的可解释性与泛化能力。从机理层面看，我们证明两个算子分支会自发特化为不同物理角色，且该角色在尺度与域间保持一致：等变分支捕获快速变化的瞬态动力学，不变分支则提取相干的持久结构。FaNO的分解机制提升了物理系统与跨域场景下的预测精度、参数效率及跨尺度泛化能力。特别地，该机制能在长时程自回归展开、跨分辨率外推及物理体制迁移下保持一致的预测表现。这些发现表明，可扩展物理建模或需超越单一归纳偏置范式，转向更准确反映物理系统异质性组织的因子化算子表示，从而加速机器学习在科学计算与科学发现中的可靠部署。