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Stochastic Process Learning via Operator Flow Matching
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Proposes operator flow matching (OFM), extending optimal-transport flow matching with neural operators to learn stochastic process priors over point collections, enabling exact densities and Bayesian functional regression via SGLD.
Problem
Learning general stochastic process priors from data (beyond Gaussian processes) with exact density over any finite collection of points, enabling universal functional regression with mean and density estimation.
Approach
Generalizes marginal optimal-transport flow matching to infinite-dimensional function spaces using a Fourier neural operator parameterized vector field that transports a GP reference measure to the data process; likelihoods come from change of variables with a Skilling-Hutchinson divergence estimator, and posterior sampling uses SGLD in the GP latent space.
Key results
- On 1D GP regression, OFM achieves SMSE 4.1e-1 and MSLL 5.5e-2, versus OpFlow SMSE 5.0e-1 and MSLL 2.0e-1.
- On 1D truncated GP regression, OFM reaches mean MSE 5.2e-3 and standard-deviation MSE 9.5e-4, versus 1.3e-2 and 3.9e-3 for OpFlow.
- On 2D GP regression, OFM attains SMSE 1.3e-1 but MSLL 1.6e-1, while OpFlow achieves the best MSLL of 1.1e-1.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use OFM instead of DGP | 1D GP regression · 1D GP | MSLL: 6.8 \cdot 10^{-2} -> 5.5 \cdot 10^{-2} | Helped |
| Use OFM instead of OpFlow | 2D GP regression · 2D GP | SMSE: 1.4 \cdot 10^{-1} -> 1.3 \cdot 10^{-1} | Helped |
| Use OFM instead of OpFlow | 2D GP regression · 2D GP | MSLL: 1.1 \cdot 10^{-1} -> 1.6 \cdot 10^{-1} | Hurt |
| Use OFM instead of DSPP | 1D TGP regression · 1D TGP | mean squared error of predicted mean: 1.1 \cdot 10^{-2} -> 5.2 \cdot 10^{-3} | Helped |
| Use OFM instead of ConvCNP | 1D TGP regression · 1D TGP | mean squared error of predicted standard deviation: 2.1 \cdot 10^{-3} -> 9.5 \cdot 10^{-4} | Helped |
Surprising findings
- On 2D GP regression, OFM does not win on MSLL, where OpFlow is best.
Limitations
- Limited to low-dimensional domains.
- Demands larger datasets and more compute than GP-based baselines.
- Likelihood cannot be meaningfully evaluated for non-GP posteriors because the evidence term is intractable.
Experimental setup
Datasets: 1D GP samples 2D GP samples 1D truncated GP samples Navier-Stokes simulations black hole simulations MNIST-SDF
Benchmarks: 1D GP regression 2D GP regression 1D truncated GP regression Navier-Stokes regression black hole regression MNIST-SDF regression
Metrics: SMSE MSLL mean squared error of mean and standard deviation
Baselines: GP regression OpFlow NP ANP ConvCNP DGP DSPP
Builds on: flow matching Fourier neural operator OpFlow stochastic gradient Langevin dynamics Hutchinson trace estimator
Closest papers by meaning
- Flow Matching Neural Processes (NeurIPS 2025 · similarity 0.87)
- Improving Function Space Flow Matching with Kernel Optimal Transport (NeurIPS 2026 · similarity 0.85)
- Multitask Learning with Stochastic Interpolants (NeurIPS 2025 · similarity 0.84)
- Efficient Training of Neural Stochastic Differential Equations by Matching Finite Dimensional Distributions (ICLR 2025 · similarity 0.84)
- Stream-level Flow Matching with Gaussian Processes (ICML 2025 · similarity 0.84)
- Sequence Modeling with Spectral Mean Flows (NeurIPS 2025 · similarity 0.83)
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