Davidson School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907, USA.
J Chem Phys. 2022 May 14;156(18):184108. doi: 10.1063/5.0080540.
A Brownian bridge is a continuous random walk conditioned to end in a given region by adding an effective drift to guide paths toward the desired region of phase space. This idea has many applications in chemical science where one wants to control the endpoint of a stochastic process-e.g., polymer physics, chemical reaction pathways, heat/mass transfer, and Brownian dynamics simulations. Despite its broad applicability, the biggest limitation of the Brownian bridge technique is that it is often difficult to determine the effective drift as it comes from a solution of a Backward Fokker-Planck (BFP) equation that is infeasible to compute for complex or high-dimensional systems. This paper introduces a fast approximation method to generate a Brownian bridge process without solving the BFP equation explicitly. Specifically, this paper uses the asymptotic properties of the BFP equation to generate an approximate drift and determine ways to correct (i.e., re-weight) any errors incurred from this approximation. Because such a procedure avoids the solution of the BFP equation, we show that it drastically accelerates the generation of conditioned random walks. We also show that this approach offers reasonable improvement compared to other sampling approaches using simple bias potentials.
布朗桥是一种连续随机游走，通过在有效漂移中添加附加项来引导路径进入期望的相空间区域，从而对其进行约束。这一思想在化学科学中有许多应用，例如聚合物物理、化学反应途径、热/质量传递和布朗动力学模拟等领域，人们希望控制随机过程的终点。尽管布朗桥技术具有广泛的适用性，但它最大的局限性在于，由于它来自于一个向后福克-普朗克（BFP）方程的解，对于复杂或高维系统来说，计算这个解是不可行的，因此通常很难确定有效漂移。本文介绍了一种快速逼近方法，可以在不明确求解 BFP 方程的情况下生成布朗桥过程。具体来说，本文利用 BFP 方程的渐近性质生成近似漂移，并确定从这种近似中纠正（即重新加权）任何误差的方法。由于这种方法避免了 BFP 方程的求解，因此我们表明它可以极大地加速条件随机游走的生成。我们还表明，与使用简单偏差势的其他采样方法相比，该方法提供了合理的改进。
