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A Theory on Flow Matching with Neural Networks

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

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Gives theoretical guarantees for neural-network flow matching: GD convergence in an over-parameterized two-layer ReLU network, L2 generalization bounds for conditional velocity-field matching via a multi-task unbounded-loss Rademacher bound, and W1 sampling error bounds, validated on synthetic Gaussian mixtures and PCA-reduced MNIST/Fashion-MNIST.

Problem

Flow matching with neural-network velocity fields lacks theory covering optimization error, statistical error of velocity-field learning, and distributional guarantees of generated samples.

Approach

Analyzes gradient descent on a two-layer ReLU network with both layers trainable under NTK-style arguments, derives an unbounded-loss multi-task Rademacher generalization bound, and upgrades L1/L2 velocity error to L-infinity on a compact set to obtain Wasserstein-1 sampling guarantees.

Key results

• Terminal training loss grows with dimension d and decreases with step size eta across a 150-cell sweep on the synthetic Gaussian mixture (n_train=500, n_test=5000).
• Sliced W1 surrogate grows with d and shrinks with eta; the paper reports increasing width m yields only marginal improvement, consistent with the NTK regime.
• On MNIST and Fashion-MNIST in 7-dimensional PCA space, loss and sliced W1 decay geometrically in t and group by eta rather than m; the paper gives no quantitative baseline comparison.

Surprising findings

• Increasing network width m at fixed step size and dimension yields only marginal improvement in the reported experiments.

Limitations

• Analysis restricted to a two-layer ReLU network trained with plain gradient descent; deeper nets and SGD/Adam not covered.
• Wasserstein bound's dimension dependence (d=o(log n) for the L-infinity lemma) is restrictive and the bounds are not claimed to be sharp.
• Requires strong assumptions: bounded support for sampling bound, anti-concentration, and conditional coupling x0 ~ P(x0|x1).

Experimental setup

Datasets: MNIST Fashion-MNIST synthetic two-component Gaussian mixture

Benchmarks: MNIST Fashion-MNIST synthetic Gaussian mixture

Metrics: training loss sliced Wasserstein-1 surrogate

Builds on: flow matching neural tangent kernel rectified flow

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