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Drifting to Boltzmann: Million-Fold Acceleration in Boltzmann Sampling with Force-Guided Drifting

Pipi Hu

TL;DR

Both approaches achieve one-step generation with over 1000x speedup relative to recent score-matching methods with Boltzmann guiding, providing more than million-fold acceleration over traditional molecular dynamics, while ensuring perfect structural validity and distributional accuracy rivaling multi-step methods.

Abstract

Sampling molecular conformations from the Boltzmann distribution is essential for computational chemistry, but iterative diffusion methods are prohibitively slow. Drifting Models offer one-step generation, yet their equilibrium matches the \emph{training} distribution, which may deviate from the true Boltzmann distribution due to sampling bias. We introduce Drifting Models to molecular conformation generation for the first time, establishing a theoretical bridge via the \emph{Drifting Score Identity}: for Gaussian kernels, the drifting field's attraction equals a kernel-weighted average of \emph{any} distribution's score function. Substituting molecular force labels -- which directly encode the Boltzmann score -- yields the \emph{Drifting Force Identity} and decomposes the field into standard drift plus a Boltzmann correction. We further discover a striking phenomenon unique to molecular systems: force incorporation's effectiveness \emph{reverses across representations}. In coordinate space, Force-Interpolated Drifting (FI) dominates by blending physical force directions with data displacements. In distance feature space, Force-Aligned Kernel (FK) achieves superior accuracy by modifying only kernel weights, thereby preserving the manifold of geometrically valid molecules. On MD17 Ethanol, both approaches achieve one-step generation with over 1000x speedup relative to recent score-matching methods with Boltzmann guiding, providing more than million-fold acceleration over traditional molecular dynamics, while ensuring perfect structural validity and distributional accuracy rivaling multi-step methods.

Drifting to Boltzmann: Million-Fold Acceleration in Boltzmann Sampling with Force-Guided Drifting

TL;DR

Both approaches achieve one-step generation with over 1000x speedup relative to recent score-matching methods with Boltzmann guiding, providing more than million-fold acceleration over traditional molecular dynamics, while ensuring perfect structural validity and distributional accuracy rivaling multi-step methods.

Abstract

Sampling molecular conformations from the Boltzmann distribution is essential for computational chemistry, but iterative diffusion methods are prohibitively slow. Drifting Models offer one-step generation, yet their equilibrium matches the \emph{training} distribution, which may deviate from the true Boltzmann distribution due to sampling bias. We introduce Drifting Models to molecular conformation generation for the first time, establishing a theoretical bridge via the \emph{Drifting Score Identity}: for Gaussian kernels, the drifting field's attraction equals a kernel-weighted average of \emph{any} distribution's score function. Substituting molecular force labels -- which directly encode the Boltzmann score -- yields the \emph{Drifting Force Identity} and decomposes the field into standard drift plus a Boltzmann correction. We further discover a striking phenomenon unique to molecular systems: force incorporation's effectiveness \emph{reverses across representations}. In coordinate space, Force-Interpolated Drifting (FI) dominates by blending physical force directions with data displacements. In distance feature space, Force-Aligned Kernel (FK) achieves superior accuracy by modifying only kernel weights, thereby preserving the manifold of geometrically valid molecules. On MD17 Ethanol, both approaches achieve one-step generation with over 1000x speedup relative to recent score-matching methods with Boltzmann guiding, providing more than million-fold acceleration over traditional molecular dynamics, while ensuring perfect structural validity and distributional accuracy rivaling multi-step methods.
Paper Structure (48 sections, 5 theorems, 37 equations, 4 tables, 5 algorithms)

This paper contains 48 sections, 5 theorems, 37 equations, 4 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Generative Models for Molecular Sampling.
  3. Debiasing with Force Labels.
  4. One-Step Generation via Drifting Models.
  5. Our Contribution.
  6. Preliminaries
  7. Boltzmann Distribution and Force Labels
  8. Drifting Models
  9. Anti-Symmetry and Equilibrium.
  10. Method
  11. Drifting Score Identity
  12. Force-Interpolated Drifting (FI)
  13. Interpretation.
  14. Variance.
  15. Force-Aligned Kernel (FK)
  16. ...and 33 more sections

Key Result

Theorem 3.1

Let $k(\mathbf{x},\mathbf{y}) = \exp(-\|\mathbf{x}-\mathbf{y}\|^2/(2\tau^2))$ be a Gaussian kernel. For any sufficiently regular distribution $p$ (i.e., $p(\mathbf{y})k(\mathbf{x},\mathbf{y}) \to 0$ as $\|\mathbf{y}\|\to\infty$), the attraction term of the drifting field satisfies, in the population where $\bar{k}(\mathbf{x},\mathbf{y}) = k(\mathbf{x},\mathbf{y})/\mathbb{E}_p[k(\mathbf{x},\mathbf{

Theorems & Definitions (14)

  • Theorem 3.1: Drifting Score Identity
  • proof
  • Remark 3.1
  • Corollary 3.2: Drifting Force Identity
  • Definition 3.1: Force-Interpolated Attraction
  • Proposition 3.3: $\omega$ Controls Boltzmann Correction
  • proof
  • Definition 3.2: Force-Aligned Kernel
  • Proposition 3.4: Boltzmann Reweighting
  • Remark 3.2: Metric correction
  • ...and 4 more