Active matter systems, from self-propelled colloids to motile bacteria, are characterized by the conversion of free energy into useful work at the microscopic scale. These systems generically involve physics beyond the reach of equilibrium statistical mechanics, and a persistent challenge has been to understand the nature of their nonequilibrium states. The entropy production rate and the magnitude of the steady-state probability current provide quantitative ways to do so by measuring the breakdown of time-reversal symmetry and the strength of nonequilibrium transport of measure. Yet, their efficient computation has remained elusive, as they depend on the system's unknown and high-dimensional probability density. Here, building upon recent advances in generative modeling, we develop a deep learning framework that estimates the score of this density. We show that the score, together with the microscopic equations of motion, gives direct access to the entropy production rate, the probability current, and their decomposition into local contributions from individual particles, spatial regions, and degrees of freedom. To represent the score, we introduce a novel, spatially-local transformer-based network architecture that learns high-order interactions between particles while respecting their underlying permutation symmetry. We demonstrate the broad utility and scalability of the method by applying it to several high-dimensional systems of interacting active particles undergoing motility-induced phase separation (MIPS). We show that a single instance of our network trained on a system of 4096 particles at one packing fraction can generalize to other regions of the phase diagram, including systems with as many as 32768 particles. We use this observation to quantify the spatial structure of the departure from equilibrium in MIPS as a function of the number of particles and the packing fraction.

翻译：活性物质系统——从自驱动胶体到运动细菌——其特征是在微观尺度上将自由能转化为有用功。这些系统普遍涉及超越平衡态统计力学范畴的物理学问题，理解其非平衡态的本质一直是一个持久挑战。熵产生率和稳态概率流的大小通过测量时间反演对称性的破缺以及测度的非平衡输运强度，提供了实现这一目标的定量方法。然而，由于这些量依赖于系统未知且高维的概率密度，因此其高效计算一直难以实现。在此，基于生成建模领域的最新进展，我们开发了一个深度学习框架来估计该密度的分数。我们证明，该分数与微观运动方程相结合，可直接给出熵产生率、概率流，以及它们在单个粒子、空间区域和自由度等局部贡献上的分解。为了表示该分数，我们引入了一种新颖的、基于空间局域化Transformer的网络架构，该架构在学习粒子间高阶相互作用的同时，尊重其固有的置换对称性。我们通过将该方法应用于多个经历运动诱导相分离（MIPS）的高维相互作用活性粒子系统，展示了该方法的广泛适用性和可扩展性。我们证明，在一个填充率下、包含4096个粒子的系统上训练好的单个网络实例，可以推广至相图中的其他区域，包括多达32768个粒子的系统。我们利用这一观察结果，定量研究了MIPS中偏离平衡态的空间结构随粒子数和填充率的变化情况。