Time-Aware Operator Learning with Multi-Fidelity Solver-Embedded Mixture of Experts

In numerically solving PDEs, achieving both generalization and efficiency is crucial but challenging. While recent advancements in neural operators provide significant speed-ups, they often struggle to generalize to out-of-distribution scenarios. Hybrid models that integrate neural networks with conventional numerical solvers offer improved generalization ability, but they incur high computational costs. To address this efficiency-generalization trade-off, we propose the Multi-Fidelity Mixture-of-Experts (MF-MoE) framework. MF-MoE dynamically allocates resources based on physical parameters and an optimal routing solver across expert models of varying fidelity, optimizing both computational cost and predictive accuracy. This innovative design enables faster inference for in-distribution inputs while ensuring better generalization for out-of-distribution cases. Extensive experiments on fluid flow prediction governed by the incompressible Navier-Stokes equations demonstrate that MF-MoE consistently outperforms baseline approaches, offering an efficient solution for PDE surrogate modeling.