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Improving Function Space Flow Matching with Kernel Optimal Transport

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Flow matching and flow sampling · 0 citations (OpenAlex)

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Proposes kFFM, which replaces the independent endpoint pairing in Functional Flow Matching with entropic OT under a bounded kernel-induced cost (signature, Sobolev-RBF, Euclidean RBF), keeping the FNO backbone unchanged, with theory on well-posedness, error decomposition, and discretization invariance.

Problem

Functional flow matching uses independent prior-data endpoint pairing, forcing the velocity field to learn both global dataset structure and instance residuals; OT on function spaces is hard to formulate and a flat L2 surrogate ignores function-space geometry.

Approach

Replace independent minibatch pairing in FFM with entropic OT under a kernel-induced cost c_kappa (RKHS distance), so each prior sample is paired with a geometrically close data function; the FNO and conditional path are unchanged.

Key results

• kFFM lowers MMD-RBF versus FFM with statistically significant paired-seed gains (e.g., Gene Expression 17.8 to 7.11 x10^-3, paired t p=2e-5).
• Versus flat-L2 minibatch OT (CFM-OT(L2)), kFFM reduces distributional error by 75% on Gene Expression and 66% on Economy.
• On turbulent Navier-Stokes, kFFM improves MMD 73x over FFM (p=3e-4) and vorticity-PDF W1 is 23-32x better than score-based and diffusion baselines.

Surprising findings

• A bounded RBF kernel on Euclidean distances (no function-space geometry) already outperforms raw L2 Sinkhorn coupling on smooth fields.
• On turbulent Navier-Stokes, most of the gain over FFM comes from the white-noise base measure rather than the coupling.

Limitations

• Theory certifies the surrogate coupling only; no straightness or sampling-efficiency guarantee in function space.
• Theory is least informative in low-regularity regimes where the kernel cost saturates and covering terms blow up.
• Signature-kernel memory grows with path length, forcing RBF or L2 fallback for very long paths.

Experimental setup

Datasets: AEMET gene expression economy Heston KdV stochastic KdV Navier-Stokes stochastic Navier-Stokes turbulent Navier-Stokes (nu=1e-5)

Benchmarks: AEMET Gene Expression Economy Heston KdV Stochastic KdV Navier-Stokes Stochastic NS turbulent Navier-Stokes

Metrics: MMD-RBF sliced Wasserstein distance marginal W1 autocorrelation error log-spectral distance enstrophy W1 vorticity PDF W1 skewness and kurtosis error

Baselines: FFM CFM-OT(L2) DDO/NCSN functional DDPM GANO

Builds on: Functional Flow Matching Conditional Flow Matching Hilbert Sinkhorn Divergence OT-CFM minibatch coupling signature kernel

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