Mixture of neural operator experts for learning boundary conditions and model selection

TL;DR

This paper introduces a novel neural operator method combining volume penalization and mixture of experts to effectively impose boundary conditions and enable model selection, demonstrated on nonlinear operators and fluid flow simulations.

Contribution

It proposes a new boundary condition imposition technique inspired by numerical methods and MoE, allowing for flexible model selection and domain decomposition in neural operators.

Findings

Successfully applied to nonlinear operators on disks and quarter disks.

Extracted a large eddy simulation (LES) model from DNS data.

Achieved posterior predictive samples of flow beyond DNS time horizon.

Abstract

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.