  Given their ability to effectively learn non-linear mappings and perform fast
inference, deep neural networks (NNs) have been proposed as a viable
alternative to traditional simulation-driven approaches for solving
high-dimensional eigenvalue equations (HDEs), which are the foundation for many
scientific applications. Unfortunately, for the learned models in these
scientific applications to achieve generalization, a large, diverse, and
preferably annotated dataset is typically needed and is computationally
expensive to obtain. Furthermore, the learned models tend to be memory- and
compute-intensive primarily due to the size of the output layer. While
generalization, especially extrapolation, with scarce data has been attempted
by imposing physical constraints in the form of physics loss, the problem of
model scalability has remained.
  In this paper, we alleviate the compute bottleneck in the output layer by
using physics knowledge to decompose the complex regression task of predicting
the high-dimensional eigenvectors into multiple simpler sub-tasks, each of
which are learned by a simple "expert" network. We call the resulting
architecture of specialized experts Physics-Guided Mixture-of-Experts (PG-MoE).
We demonstrate the efficacy of such physics-guided problem decomposition for
the case of the Schr\"{o}dinger's Equation in Quantum Mechanics. Our proposed
PG-MoE model predicts the ground-state solution, i.e., the eigenvector that
corresponds to the smallest possible eigenvalue. The model is 150x smaller than
the network trained to learn the complex task while being competitive in
generalization. To improve the generalization of the PG-MoE, we also employ a
physics-guided loss function based on variational energy, which by quantum
mechanics principles is minimized iff the output is the ground-state solution.
