On the flow matching interpretability

TL;DR

This paper introduces a physically constrained flow matching framework for generative models, making their intermediate steps interpretable by aligning them with known physical distributions, demonstrated using the 2D Ising model.

Contribution

It proposes a novel method to embed physical semantics into flow-based generative models, enhancing interpretability of the intermediate steps.

Findings

Preserves physical fidelity in generated configurations.

Outperforms Monte Carlo in generation speed for larger lattices.

Enables meaningful interpretation of flow steps as physical states.

Abstract

Generative models based on flow matching have demonstrated remarkable success in various domains, yet they suffer from a fundamental limitation: the lack of interpretability in their intermediate generation steps. In fact these models learn to transform noise into data through a series of vector field updates, however the meaning of each step remains opaque. We address this problem by proposing a general framework constraining each flow step to be sampled from a known physical distribution. Flow trajectories are mapped to (and constrained to traverse) the equilibrium states of the simulated physical process. We implement this approach through the 2D Ising model in such a way that flow steps become thermal equilibrium points along a parametric cooling schedule. Our proposed architecture includes an encoder that maps discrete Ising configurations into a continuous latent space, a flow-matching network that performs temperature-driven diffusion, and a projector that returns to discrete Ising states while preserving physical constraints. We validate this framework across multiple lattice sizes, showing that it preserves physical fidelity while outperforming Monte Carlo generation in speed as the lattice size increases. In contrast with standard flow matching, each vector field represents a meaningful stepwise transition in the 2D Ising model's latent space. This demonstrates that embedding physical semantics into generative flows transforms opaque neural trajectories into interpretable physical processes.