标题： 神经薛定谔桥与Sinkhorn损失：应用于胶体自组装的数据驱动最小努力控制 显示英文标题

标题： Neural Schrödinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly

摘要： 我们证明，胶体自组装的最小努力控制可以自然地在序参量空间中表述为广义的薛定谔桥问题——一类起源于20世纪30年代初埃里希·薛定谔工作的固定时间范围随机最优控制问题。近年来，这类问题在控制和机器学习领域重新引起了研究兴趣。与现有文献中关于此类问题的理论和计算研究不同，胶体自组装的受控漂移和扩散系数通常在控制上是非仿射的，并且难以通过基于物理的建模获得。我们推导了此类广义问题的最优条件，并表明所得方程组在结构上与现有结果有很大差异，以至于标准计算方法不再适用。因此，我们提出了一种数据驱动的学习和控制框架，名为“神经薛定谔桥”，通过在神经网络最新进展上的创新来解决此类广义薛定谔桥问题。我们通过胶体自组装的数值案例研究展示了所提出框架的有效性。我们使用分子动力学模拟数据学习受控漂移和扩散系数作为两个神经网络，然后使用这两个网络训练第三个网络，该网络使用针对分布端点约束设计的Sinkhorn损失，专门适用于此类控制问题。 显示英文摘要

摘要： We show that the minimum effort control of colloidal self-assembly can be naturally formulated in the order-parameter space as a generalized Schr\"{o}dinger bridge problem -- a class of fixed-horizon stochastic optimal control problems that originated in the works of Erwin Schr\"{o}dinger in the early 1930s. In recent years, this class of problems has seen a resurgence of research activities in the control and machine learning communities. Different from the existing literature on the theory and computation for such problems, the controlled drift and diffusion coefficients for colloidal self-assembly are typically nonaffine in control, and are difficult to obtain from physics-based modeling. We deduce the conditions of optimality for such generalized problems, and show that the resulting system of equations is structurally very different from the existing results in a way that standard computational approaches no longer apply. Thus motivated, we propose a data-driven learning and control framework, named `neural Schr\"{o}dinger bridge', to solve such generalized Schr\"{o}dinger bridge problems by innovating on recent advances in neural networks. We illustrate the effectiveness of the proposed framework using a numerical case study of colloidal self-assembly. We learn the controlled drift and diffusion coefficients as two neural networks using molecular dynamics simulation data, and then use these two to train a third network with Sinkhorn losses designed for distributional endpoint constraints, specific for this class of control problems.