FUND: Flow Matching for Sampling Unnormalized Distributions

Abstract:

Efficient sampling from Boltzmann distributions is central to modelling complex physical systems. Markov Chain Monte Carlo (MCMC) methods suffer from critical slowing down, high autocorrelation, and poor mode-mixing, limiting their scalability. Recent advances, like Boltzmann Generators, offer a promising alternative but remain constrained by costly MCMC-based training, inefficient sampling, and poor ergodicity. We introduce an algorithm for learning Boltzmann distributions that does not require any true samples for training. Our approach draws inspiration from flow matching but departs fundamentally from sample-trajectory matching to distribution-trajectory matching. The algorithm iteratively reshapes the target distribution, using model generated samples to guide learning and ensure comprehensive mode coverage. We validate our method on challenging benchmarks, including a 2D Gaussian mixture, Many-Well distributions, and high-dimensional scalar $\phi^4$ theory. Our approach not only outperforms traditional MCMC and flow-based methods in efficiency and
accuracy but also establishes a new paradigm for sample-free learning of complex physical distributions.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Baoxiang_Wang1

Submission Number: 7713