BNEM: A Boltzmann Sampler Based on Bootstrapped Noised Energy Matching

Developing an efficient sampler capable of generating independent and identically distributed (IID) samples from a Boltzmann distribution is a crucial challenge in scientific research, e.g. molecular dynamics. In this work, we intend to learn neural samplers given energy functions instead of data sampled from the Boltzmann distribution. By learning the energies of the noised data, we propose a diffusion-based sampler, Noised Energy Matching, which theoretically has lower variance and more complexity compared to related works. Furthermore, a novel bootstrapping technique is applied to NEM to balance between bias and variance. We evaluate NEM and BNEM on a 2-dimensional 40 Gaussian Mixture Model (GMM) and a 4-particle double-well potential (DW-4). The experimental results demonstrate that BNEM can achieve state-of-the-art performance while being more robust.

💡 Research Summary

The paper tackles the problem of generating independent and identically distributed (IID) samples from a Boltzmann distribution µ_target(x) ∝ exp(−E(x)) when only the energy function E(x) is available and no samples from the target distribution can be drawn directly. Traditional Monte‑Carlo methods (e.g., AIS, HMC, SMC) become prohibitively expensive in high dimensions, while recent amortised samplers rely on data drawn from the target distribution, which is unavailable in many scientific settings such as molecular dynamics or protein folding.

Key Idea – Noised Energy Matching (NEM).
Instead of learning the conditional score ∇log p_t(x_t|x_0) as in standard diffusion models, the authors propose to learn the “noised energy” E_t(x_t) = −log E_{x_0∼µ_target}