We show that the minimum effort control of colloidal self-assembly can be
naturally formulated in the order-parameter space as a generalized
Schr\"odinger bridge problem -- a class of fixed-horizon stochastic optimal
control problems that originated in the works of Erwin Schr\"odinger in the
early 1930s. In recent years, this class of problems has seen a resurgence of
research activities in 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 non-affine 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\"odinger bridge', to solve such generalized
Schr\"odinger 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.

通过神经网络的新进展，我们提出了一个名为 ' 神经 Schrödinger 桥 ' 的数据驱动学习和控制框架，以解决胶体自组装的一类广义 Schrödinger 桥问题，通过学习受控漂移和扩散系数，我们在分布终点约束下训练了第三个网络，并以胶体自组装的数值案例研究证明了所提出框架的有效性。

基于 Sinkhorn Losses 的神经 Schrödinger Bridge：应用于胶体自组装的数据驱动最小努力控制

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

We propose formulating the finite-horizon stochastic optimal control problem
for colloidal self-assembly in the space of probability density functions
(PDFs) of the underlying state variables (namely, order parameters). The
control objective is formulated in terms of steering the state PDFs from a
prescribed initial probability measure towards a prescribed terminal
probability measure with minimum control effort. For specificity, we use a
univariate stochastic state model from the literature. Both the analysis and
the computational steps for control synthesis as developed in this paper
generalize for multivariate stochastic state dynamics given by generic
nonlinear in state and non-affine in control models. We derive the conditions
of optimality for the associated optimal control problem. This derivation
yields a system of three coupled partial differential equations together with
the boundary conditions at the initial and terminal times. The resulting system
is a generalized instance of the so-called Schrödinger bridge problem. We
then determine the optimal control policy by training a physics-informed deep
neural network, where the "physics" are the derived conditions of optimality.
The performance of the proposed solution is demonstrated via numerical
simulations on a benchmark colloidal self-assembly problem.

本文提出了一种在状态概率密度函数空间中为胶体自组装制定有限时域随机最优控制问题的方法，并通过训练物理信息深度神经网络确定最优控制策略，其中物理信息是导出的最优性条件。