估计大偏差的随机优化方法
Stochastic optimisation method for estimating large deviations
- Stellenbosch University（斯泰伦博斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文介绍LDSTOP随机优化方法，结合控制理论与机器学习，高效计算马尔可夫过程的大偏差函数，适用于离散链、跳跃和扩散过程，具有简单灵活可扩展的优势。
AI中文摘要：
最近提出了一种随机优化方法，用于高效计算大偏差函数，该方法在统计物理学中用于表征非平衡系统（建模为马尔可夫过程）的涨落。该方法名为LDSTOP，结合了控制理论和机器学习中的数值技术，迭代构建“驱动过程”，即作为非平衡系统模型的马尔可夫过程的一个受控版本，以最优方式实现该系统的给定涨落或大偏差。与其他基于谱近似、重要性采样、克隆或分裂的方法相比，LDSTOP简单、灵活且可扩展，因为它通过模拟逐渐引导至驱动过程的单条轨迹来工作。在此，我们通过将该方法应用于应用中考虑的所有马尔可夫过程类型（即离散时间马尔可夫链、连续时间跳跃过程和扩散过程）来展示这些优势。对于每种类型，我们定义了要优化的目标函数，解释了表示驱动过程的不同选项（例如，使用神经网络），并通过简单应用提供了实现细节。通过这些贡献，我们旨在展示该方法的效率，以及其与现有机器学习包（如PyTorch、TensorFlow和JAX）结合使用时用于模拟随机过程和求解高维优化问题的易用性。
英文摘要：
A stochastic optimisation method was recently proposed to efficiently compute large deviation functions, used in statistical physics to characterise the fluctuations of nonequilibrium systems, modelled as Markov processes. The method, called LDSTOP, combines numerical techniques from control theory and machine learning to iteratively construct the "driven process", a controlled version of the Markov process used as a model of nonequilibrium system that realises a given fluctuation or large deviation of that system in an optimal way. Compared to other methods based on spectral approximations, importance sampling, cloning or splitting, LDSTOP is simple, flexible, and scalable, as it works by simulating single trajectories that are gradually guided towards the driven process. Here, we illustrate these advantages by applying the method on the full range of Markov processes considered in applications, namely, discrete-time Markov chains, continuous-time jump processes, and diffusion processes. For each type, we define the objective function to be optimised, explain different options available for representing the driven process (using, e.g., neural networks), and provide implementation details through simple applications. With these contributions, we aim to showcase the method's efficiency, as well as its ease of use when combined with available machine learning packages, such as PyTorch, TensorFlow and JAX, for simulating stochastic processes and solving high-dimensional optimisation problems.
