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.
| Comments: | 27 pages, 2 figures, 7 tables. Preprint |
| Subjects: | Machine Learning (cs.LG) |
| Cite as: | arXiv:2605.16118 [cs.LG] |
| (or arXiv:2605.16118v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.16118 arXiv-issued DOI via DataCite (pending registration) |
From: Sipeng Chen [view email]
[v1]
Fri, 15 May 2026 16:02:18 UTC (1,392 KB)
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