Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866913132829474816 |
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| author | Chen, Sipeng Liu, Junliang Tang, Hewei Li, Shibo |
| author_facet | Chen, Sipeng Liu, Junliang Tang, Hewei Li, Shibo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16118 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions Chen, Sipeng Liu, Junliang Tang, Hewei Li, Shibo Machine Learning 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. |
| title | Multi-Fidelity Flow Matching: Cascaded Refinement of PDE Solutions |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2605.16118 |