Distribution Matching Evolutionary Algorithms for Rare Event Sampling

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 standard benchmarks, including a 2D Gaussian mixture, Many-Well distributions, and high-dimensional scalar $φ^4$ theory. The proposed approach not only improves sampling performance and accuracy over traditional MCMC and flow-based baselines but also establishes a new method for sample-free learning of complex physical distributions.

We propose a novel deterministic sampling method, EVI-MMD, to approximate a target distribution $ρ^*$ by minimizing the kernel discrepancy, also known as the Maximum Mean Discrepancy (MMD). Leveraging the energetic variational inference framework (Wang et al., 2021), we transform the MMD minimization problem into solving a dynamic system of Ordinary Differential Equations (ODEs) for particles. The implicit Euler scheme is employed to solve the ODE system, leading to a proximal minimization problem at each iteration, which is efficiently addressed using optimization algorithms such as L-BFGS. A key innovation of our method is a dynamic bandwidth selection strategy for the Gaussian kernel, which, although heuristic at this stage, represents a meaningful step toward addressing a long-standing challenge in kernel-based methods. Comprehensive numerical experiments demonstrate that this adaptive bandwidth significantly enhances the performance of EVI-MMD. We apply the EVI-MMD algorithm to two types of sampling problems: (1) when the target distribution is fully specified by a density function, and (2) the ``two-sample problem,'' where only training data are available. In the latter case, EVI-MMD serves as a generative model, producing new samples that faithfully replicate the distribution of the training data. With carefully tuned parameters, EVI-MMD outperforms several existing methods in both scenarios.

Sampling from constrained distributions has a wide range of applications, including in Bayesian optimization and robotics. Prior work establishes convergence and feasibility guarantees for constrained sampling, but assumes that the feasible set is connected. However, in practice, the feasible set often decomposes into multiple disconnected components, which makes efficient sampling under constraints challenging. In this paper, we propose MAnifold Sampling via Entropy Maximization (MASEM) for uniform sampling on a manifold with an unknown number of disconnected components, implicitly defined by smooth equality and inequality constraints. The presented method uses a resampling scheme to maximize the entropy of the empirical distribution based on k-nearest neighbor density estimation. We show that, in the mean field, MASEM decreases the KL-divergence between the empirical distribution and the maximum-entropy target exponentially in the number of resampling steps. We instantiate MASEM with multiple local samplers and demonstrate its versatility and efficiency on synthetic and robotics-based benchmarks. MASEM enables fast and scalable mixing across a range of constrained sampling problems, improving over alternatives by an order of magnitude in Sinkhorn distance with competitive runtime.