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.

利用生成模型构建了一个深度学习框架，可以估计活性物质系统中的熵产生速率和概率流量，并将其分解为个体粒子、空间区域和自由度的本地贡献。通过将该方法应用于基于运动引起的相分离的活性粒子的高维系统，验证了该方法的广泛应用性和可扩展性。

主动物质中的深度学习概率流和熵产生速率

Deep learning probability flows and entropy production rates in active  matter

We present a novel yet simple deep learning approach, dubbed EPR-Net, for
constructing the potential landscape of high-dimensional non-equilibrium steady
state (NESS) systems. The key idea of our approach is to utilize the fact that
the negative potential gradient is the orthogonal projection of the driving
force in a weighted Hilbert space with respect to the steady-state
distribution. The constructed loss function also coincides with the entropy
production rate (EPR) formula in NESS theory. This approach can be extended to
dealing with dimensionality reduction and state-dependent diffusion
coefficients in a unified fashion. The robustness and effectiveness of the
proposed approach are demonstrated by numerical studies of several
high-dimensional biophysical models with multi-stability, limit cycle, or
strange attractor with non-vanishing noise.

本文提出了一种称为 EPR-Net 的新型深度学习方法，用于构建高维非平衡稳态系统的势能景观。该方法利用了负势梯度是带权 Hilbert 空间中稳态分布关于驱动力的正交投影的事实，同时提出了与 NESS 理论中的熵产率公式相一致的损失函数。该方法可以统一地处理降维和状态相关扩散系数，并通过多稳定性、极限环或具有非零噪声的奇异吸引子的几个高维生物物理模型的数值研究证明了其鲁棒性和有效性。