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Given their ability to effectively learn non-linear mappings and perform fastinference, deep neural networks (NNs) have been proposed as a viablealternative to traditional simulation-driven approaches for solvinghigh-dimensional eigenvalue equations (HDEs), which are the foundation for manyscientific applications. Unfortunately, for the learned models in thesescientific applications to achieve generalization, a large, diverse, andpreferably annotated dataset is typically needed and is computationallyexpensive to obtain. Furthermore, the learned models tend to be memory- andcompute-intensive primarily due to the size of the output layer. Whilegeneralization, especially extrapolation, with scarce data has been attemptedby imposing physical constraints in the form of physics loss, the problem ofmodel scalability has remained.  In this paper, we alleviate the compute bottleneck in the output layer byusing physics knowledge to decompose the complex regression task of predictingthe high-dimensional eigenvectors into multiple simpler sub-tasks, each ofwhich are learned by a simple "expert" network. We call the resultingarchitecture of specialized experts Physics-Guided Mixture-of-Experts (PG-MoE).We demonstrate the efficacy of such physics-guided problem decomposition forthe case of the Schr\"{o}dinger's Equation in Quantum Mechanics. Our proposedPG-MoE model predicts the ground-state solution, i.e., the eigenvector thatcorresponds to the smallest possible eigenvalue. The model is 150x smaller thanthe network trained to learn the complex task while being competitive ingeneralization. To improve the generalization of the PG-MoE, we also employ aphysics-guided loss function based on variational energy, which by quantummechanics principles is minimized iff the output is the ground-state solution.

Physics-Guided Problem Decomposition for Scaling Deep Learning of High-dimensional Eigen-Solvers: The Case of Schr\"{o}dinger's Equation