Weyl Calculus and Exactly Solvable Schrödinger Bridges with Quadratic State Cost
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| Format: | Preprint |
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2024
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| _version_ | 1866916355015442432 |
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| author | Teter, Alexis M. H. Wang, Wenqing Halder, Abhishek |
| author_facet | Teter, Alexis M. H. Wang, Wenqing Halder, Abhishek |
| contents | Schrödinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schrödinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schrödinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schrödinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_15245 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weyl Calculus and Exactly Solvable Schrödinger Bridges with Quadratic State Cost Teter, Alexis M. H. Wang, Wenqing Halder, Abhishek Optimization and Control Machine Learning Systems and Control Mathematical Physics Schrödinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schrödinger bridge finds an optimal control policy that steers a given joint state statistics to another while minimizing the total control effort subject to controlled diffusion and deadline constraints. Viewed as a stochastic learning problem, the Schrödinger bridge finds the most-likely distribution-valued trajectory connecting endpoint distributional observations, i.e., solves the two point boundary-constrained maximum likelihood problem over the manifold of probability distributions. Recent works have shown that solving the Schrödinger bridge problem with state cost requires finding the Markov kernel associated with a reaction-diffusion PDE where the state cost appears as a state-dependent reaction rate. We explain how ideas from Weyl calculus in quantum mechanics, specifically the Weyl operator and the Weyl symbol, can help determine such Markov kernels. We illustrate these ideas by explicitly finding the Markov kernel for the case of quadratic state cost via Weyl calculus, recovering our earlier results but avoiding tedious computation with Hermite polynomials. |
| title | Weyl Calculus and Exactly Solvable Schrödinger Bridges with Quadratic State Cost |
| topic | Optimization and Control Machine Learning Systems and Control Mathematical Physics |
| url | https://arxiv.org/abs/2407.15245 |