Separable Neural Architectures as Physical World Models: from Mathematical Theory to Applications

The article presents the Separable Neural Architecture (SNA), a model that integrates neural approximation with tensor decomposition to effectively solve partial differential equations (PDEs). Utilizing a variational SNA (VSNA) framework, it achieves well-posedness and stability while significantly reducing computational costs, scaling algebraically in high-dimensional settings. The SNA demonstrates impressive performance, executing a 1,000,000-query Monte Carlo sweep in 102 seconds on a standard CPU, offering a 150,000x speedup compared to traditional finite element methods, making it a valuable tool for real-time simulations and optimizations in engineering applications.