Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions

TL;DR

This paper introduces Multi-Fidelity Flow Matching (MFFM), a cascade refinement framework for solving parametric PDEs by calibrating source distribution and conditioning velocity networks, achieving efficient multilevel resolution refinement.

Contribution

MFFM presents a novel cascade refinement approach that calibrates source distribution and conditions velocity networks, enabling efficient multilevel PDE solution refinement.

Findings

Validated on eight benchmarks including super-resolution and forecasting tasks.

Achieved multigrid-like refinement with only one network evaluation per cascade level.

Improved training geometry through residual-calibrated source noise.

Abstract

The source distribution in conditional flow matching is a design parameter that can be calibrated to data, not a default isotropic prior. We exploit this in Multi-Fidelity Flow Matching (MFFM), a cascade refinement framework for parametric PDE solutions: the source is calibrated to the empirical low-to-high-fidelity residual scale with local Gaussian-blur correlation, and the velocity network is conditioned on the low-fidelity solution. Conditioning makes the residual refinement problem substantially easier than unconditional field generation, while residual-calibrated source noise improves the flow-matching training geometry. A multi-resolution cascade applies the same construction independently between adjacent fidelities. After level-wise flow-matching pretraining, we fine-tune the composed cascade end-to-end with a deterministic one-step rollout, which makes one velocity evaluation per cascade level the optimized operating point at inference. The result is a learned analog of multigrid refinement that reaches the finest grid in $L$ deterministic network evaluations per query. We validate MFFM on eight benchmarks: two super-resolution problems and six spatiotemporal forecasting tasks from PDEBench, The Well, and the FNO Navier--Stokes dataset.