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Wasserstein Proximal Algorithms for the Schr\"{o}dinger Bridge Problem: Density Control with Nonlinear Drift
We study the Schr{\"o}dinger bridge problem (SBP) with nonlinear prior dynamics. In control-theoretic language, this is a problem of minimum effort steering of a given joint state probability density function (PDF) to another over a finite time horizon, subject to a controlled stochastic differential evolution of the state vector. For generic nonlinear drift, we reduce the SBP to solving a system of forward and backward Kolmogorov partial differential equations (PDEs) that are coupled through the boundary conditions, with unknowns being the Schr\"{o}dinger factors". We show that if the drift is a gradient vector field, or is of mixed conservative-dissipative nature, then it is possible to transform these PDEs into a pair of initial value problems (IVPs) involving the same forward Kolmogorov operator. We employ a proximal algorithm developed in our prior work to solve these IVPs and compute the Schr\"{o}dinger factors via weighted scattered point cloud evolution in the state space. We provide the algorithmic details and illustrate the proposed framework of solving the SBPs with nonlinear prior dynamics by numerical examples.
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Publication Date:
NSF-PAR ID:
10273797
Journal Name:
IEEE Transactions on Automatic Control
Page Range or eLocation-ID:
1 to 1
ISSN:
0018-9286
Sponsoring Org:
National Science Foundation
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