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

2025-02-06

AI Summary

Fourier neural operators (FNOs) are inherently constrained by periodicity assumptions, limiting their ability to model complex non-periodic boundary conditions and multi-physics phenomena. Method: This paper proposes a novel neural operator framework integrating volume penalization with a Mixture-of-Experts (MoE) architecture. It is the first to embed volume penalization within a spatially conditioned MoE structure, thereby relaxing periodic constraints and enabling unified learning of nontrivial boundaries and adaptive physics-model selection. The framework further incorporates Modal Operator Regression for Physics (MOR-Physics), Bayesian variational inference, and large-eddy simulation (LES) modeling. Results: The method successfully learns nonlinear operators on non-periodic domains—including disk and quarter-disk geometries—and interpretable LES closure models are extracted from channel-flow DNS data. It generates posterior predictive samples significantly exceeding the temporal extent of the original DNS data.

Technology Category

• Machine Learning: Mixture of Experts (MoE)
• Search and Optimization: Learning to Search
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Learning boundary conditions effectively
• Model selection via competing experts
• Domain decomposition in fluid dynamics

Innovation

• Mixture of Experts for boundary conditions
• Fourier-based Modal Operator Regression
• Bayesian variational inference for LES