Scaling physics-informed hard constraints with mixture-of-experts
Nithin Chalapathi Yiheng Du Aditi Krishnapriyan
Imposing known physical constraints, such as conservation laws, during neural
network training introduces an inductive bias that can improve accuracy,
reliability, convergence, and data efficiency for modeling physical dynamics.
While such constraints can be softly imposed via loss function penalties,
recent advancements in differentiable physics and optimization improve
performance by incorporating PDE-constrained optimization as individual layers
in neural networks. This enables a stricter adherence to physical constraints.
However, imposing hard constraints significantly increases computational and
memory costs, especially for complex dynamical systems. This is because it
requires solving an optimization problem over a large number of points in a
mesh, representing spatial and temporal discretizations, which greatly
increases the complexity of the constraint. To address this challenge, we
develop a scalable approach to enforce hard physical constraints using
Mixture-of-Experts (MoE), which can be used with any neural network
architecture. Our approach imposes the constraint over smaller decomposed
domains, each of which is solved by an "expert" through differentiable
optimization. During training, each expert independently performs a localized
backpropagation step by leveraging the implicit function theorem; the
independence of each expert allows for parallelization across multiple GPUs.
Compared to standard differentiable optimization, our scalable approach
achieves greater accuracy in the neural PDE solver setting for predicting the
dynamics of challenging non-linear systems. We also improve training stability
and require significantly less computation time during both training and
inference stages.