Separable Neural Architectures Model Physical Worlds Efficiently

This paper presents Separable Neural Architectures (SNA), a novel class of function representation that merges neural approximation with tensor decomposition. The SNA design effectively separates localized coordinate functions from global interactions, which are governed by a sparse, low-rank interaction object. This architecture inherently possesses a compact and smooth inductive bias, making it exceptionally well-suited for solving complex partial differential equations (PDEs). Under the Variational SNA (VSNA) framework, the formulation provides robust mathematical guarantees, including well-posedness, quasi-optimality, convergence, and stability. Crucially, the VSNA addresses the curse of dimensionality in high-dimensional spatiotemporal-parametric PDEs by scaling algebraically rather than exponentially. An entirely factorized, tensor-native alternating least squares (ALS) optimization framework further reduces computational cost to linear in dimension. The SNA is validated across various PDE systems, demonstrating alignment with predicted scaling rates. In practical engineering case studies, such as a 7D manufacturing simulation, the VSNA achieved a 150,000x speedup over traditional methods and enabled real-time generative inverse-mode reconstructions under 100ms, proving its utility as a "solve once, query anywhere" physical world model.