Mixture of neural operator experts for learning boundary conditions and model selection
This work addresses a bottleneck in applying neural operators to real-world physics problems with non-trivial boundaries, offering a novel method for domain decomposition and model selection, though it is incremental in combining existing techniques.
The authors tackled the problem of learning boundary conditions for neural operators on non-periodic domains by introducing a mixture of experts approach inspired by volume penalization, which also enables model selection. They demonstrated the method by recovering a nonlinear operator on complex geometries and extracting a large eddy simulation model from channel flow data, achieving posterior predictive samples beyond the simulation time horizon.
While Fourier-based neural operators are best suited to learning mappings between functions on periodic domains, several works have introduced techniques for incorporating non trivial boundary conditions. However, all previously introduced methods have restrictions that limit their applicability. In this work, we introduce an alternative approach to imposing boundary conditions inspired by volume penalization from numerical methods and Mixture of Experts (MoE) from machine learning. By introducing competing experts, the approach additionally allows for model selection. To demonstrate the method, we combine a spatially conditioned MoE with the Fourier based, Modal Operator Regression for Physics (MOR-Physics) neural operator and recover a nonlinear operator on a disk and quarter disk. Next, we extract a large eddy simulation (LES) model from direct numerical simulation of channel flow and show the domain decomposition provided by our approach. Finally, we train our LES model with Bayesian variational inference and obtain posterior predictive samples of flow far past the DNS simulation time horizon.