A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly

TL;DR

This paper introduces a physics-informed deep learning method to solve minimum effort stochastic control problems for colloidal self-assembly, using a Schrödinger bridge formulation and neural networks for control synthesis.

Contribution

It formulates the control problem in the space of probability density functions and develops a neural network-based solution leveraging optimality conditions.

Findings

Successfully applied to a benchmark colloidal self-assembly problem.

Demonstrates effectiveness of physics-informed neural networks in stochastic control.

Provides a general framework extendable to multivariate nonlinear models.

Abstract

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 Schr\"{o}dinger 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.