Conditional Flow Matching¶

← Normalizing Flows | Forward Generative Inversion →

Conditional flow matching (CFM) learns a time-dependent velocity field \(v_\theta(x,t,c)\) that transports Gaussian noise to the conditional target density. It is simulation-free: no ODE solve is needed during training.

For an optimal-transport interpolation,

the objective regresses the target velocity:

Samples are obtained by Euler integration from \(t=0\) to \(1\):

FlowMatchingMLP(time_embedding=k) emits exactly \(k\) time features, including odd widths; positive even widths retain the same sine/cosine layout.

When to use it¶

CFM is a good fit for high-dimensional or multi-target generation, and as a flexible sampler when exact NLL is not required. For tabular regression with calibrated NLL, an exact NSF/MDN is still the simpler default. The companion harness compares CFM against NSFs, cINN, BPF and MDN on shared encoders and multi-target datasets (density_parity), and against noise-aware densities on known-noise targets (noise_aware_density).

import torch
from torchregress.models import FlowMatchingMLP, cfm_loss, sample_cfm

velocity = FlowMatchingMLP(n_features=2, context_dim=64, hidden=(128, 128))
context = torch.randn(32, 64)
y = torch.randn(32, 2)

loss = cfm_loss(velocity, context, y)
loss.backward()

samples = sample_cfm(velocity, context, n_samples=128, steps=32)
# [32, 128, 2]

Comparison with related methods¶

• Forward generative inversion models \(x_{\rm obs}=g(y)+\epsilon\) and inverts it at prediction time; it is aimed at errors-in-variables, not general conditional density estimation.
• Contrastive normalizing flows train a likelihood-ratio classifier and are used for hypothesis ranking/uncertainty-aware parameter inference; they do not replace a calibrated conditional density.
• Noise-aware density losses add the measurement model to an exact likelihood; CFM currently does not have an equivalent exact marginal likelihood.

References¶