UC Santa Cruz

We consider the problem of directly controlling (i.e., reshaping) stochastic uncertainties subject to continuous-time controlled nonlinear dynamics and fixed deadline constraints. This problem is known as the generalized Schr ̈odinger bridge problem. Historically, a version of the problem originated in the works of Erwin Schr ̈odinger in 1931-32, and can be seen as the stochastic version of the optimal mass transport problem with prior nonlinear dynamics. From a control-theoretic viewpoint, this is an atypical stochastic optimal control problem–“atypical” because instead of boundary conditions on a finite dimensional state space, the problem in- volves endpoint boundary conditions on the space of joint state probability density functions, or probability measures in general. In recent years, significant strides have been made in the control community to solve such problems for some spe- cial classes of prior nonlinear dynamics including control-affine gradient and mixed conservative-dissipative systems. However, the existing algorithms in the literature exploit special structures of such prior nonlinear dynamics, but in the absence of such structures, no general computational framework is available.In this work, we propose to leverage recent advances in machine learning, specifically physics-informed neural networks (PINNs), to numerically solve general- ized Schr ̈odinger bridge problems. We introduce a variant of the standard PINN to account for the endpoint joint distributional constraints via the Sinkhorn divergence that exploits the geometry on the space of probability measures. We explain how this architecture can be implemented as differentiable layers. Our proposed frame- work allows numerically solving variants of Schr ̈odinger bridge problems for which no algorithms are available otherwise. This includes systems with control non-affine as well as nonlinear non-autonomous (i.e., explicit time dependent) drifts and dif- fusions, as well as situations where the controlled dynamics may only be available in a data-driven manner (e.g., in the form of neural networks). We demonstrate the efficacy of our computational framework using two en- gineering case studies. The first case study involves optimally steering the stochastic angular velocity of a rotating rigid body from a given statistics to another over a prescribed time horizon. This is of interest, for example, in controlling the spin of a spacecraft in the presence of stochastic uncertainties. The second case study involves controlled colloidal self-assembly for the purpose of advanced manufactur- ing of materials with desirable properties. In this case, first principle physics-based control-oriented models are difficult to obtain due to complex molecular dynamics, quantum effects and thermal fluctuations. We show how such colloidal self-assembly problems are amenable to generalized Schr ̈odinger bridge formulation, and solve the data-driven distribution steering problems for such systems using our proposed framework. We provide detailed numerical results and discuss the implementation details for the proposed computational architecture and algorithms.