Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions

2026-05-15

AI Summary

This work addresses the challenges of high computational cost in residual refinement and poor training geometry in multi-fidelity modeling of parametric partial differential equations. To this end, it proposes a cascaded refinement framework that, for the first time, integrates residual-calibrated source distributions with conditional flow matching to construct a learnable, multi-resolution correction flow analogous to multigrid methods. The approach generates high-fidelity solutions progressively conditioned on low-fidelity inputs, leveraging pretraining followed by end-to-end fine-tuning and employing single-step deterministic inference. This significantly simplifies unconditional field generation and improves optimization geometry. Evaluated on eight benchmark tasks—including two super-resolution and six spatiotemporal prediction problems—the method achieves or surpasses state-of-the-art accuracy while maintaining extremely low inference costs.

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.

Problem

• Multi-Fidelity
• PDE Solutions
• Flow Matching
• Cascaded Refinement
• Spatiotemporal Forecasting

Innovation

• Multi-Fidelity Flow Matching
• conditional flow matching
• PDE solution refinement
• residual-calibrated source
• deterministic cascade inference