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We show that the minimum effort control of colloidal self-assembly can benaturally formulated in the order-parameter space as a generalizedSchr\"{o}dinger bridge problem -- a class of fixed-horizon stochastic optimalcontrol problems that originated in the works of Erwin Schr\"{o}dinger in theearly 1930s. In recent years, this class of problems has seen a resurgence ofresearch activities in the control and machine learning communities. Differentfrom the existing literature on the theory and computation for such problems,the controlled drift and diffusion coefficients for colloidal self-assembly aretypically nonaffine in control, and are difficult to obtain from physics-basedmodeling. We deduce the conditions of optimality for such generalized problems,and show that the resulting system of equations is structurally very differentfrom the existing results in a way that standard computational approaches nolonger apply. Thus motivated, we propose a data-driven learning and controlframework, named `neural Schr\"{o}dinger bridge', to solve such generalizedSchr\"{o}dinger bridge problems by innovating on recent advances in neuralnetworks. We illustrate the effectiveness of the proposed framework using anumerical case study of colloidal self-assembly. We learn the controlled driftand diffusion coefficients as two neural networks using molecular dynamicssimulation data, and then use these two to train a third network with Sinkhornlosses designed for distributional endpoint constraints, specific for thisclass of control problems.

Neural Schr\"{o}dinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly