# Weyl Calculus and Exactly Solvable Schr"odinger Bridges with Quadratic State Cost

arXiv: https://arxiv.org/abs/2407.15245 · HTML: https://mlsanity.com/paper/2407.15245/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 36 references that are in the map.
- First posted on arXiv 21 Jul 2024.

## Builds on (most cited first, 12 of 12)

- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [I2SB: Image-to-Image Schr"odinger Bridge](https://mlsanity.com/paper/2302.05872/) (2023)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [Bayesian Learning via Neural Schr"odinger-F"ollmer Flows](https://mlsanity.com/paper/2111.10510/) (2021)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Entropic and displacement interpolation: a computational approach using the Hilbert metric](https://mlsanity.com/paper/1506.04255/) (2015)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly](https://mlsanity.com/paper/2307.14442/) (2023)
- [Prediction and Optimal Feedback Steering of Probability Density Functions for Safe Automated Driving](https://mlsanity.com/paper/2009.09055/) (2020)
- [Solution of the Probabilistic Lambert Problem: Connections with Optimal Mass Transport, Schr"odinger Bridge and Reaction-Diffusion PDEs](https://mlsanity.com/paper/2401.07961/) (2024)

## Built on by (most cited first, 3 of 3)

- [Harmonic Path Integral Diffusion](https://mlsanity.com/paper/2409.15166/) (2024)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Nonlinear Non-Gaussian Density Steering with Input and Noise Channel Mismatch: Sinkhorn with Memory for Solving the Control-affine Schr"odinger Bridge Problem](https://mlsanity.com/paper/2604.23370/) (2026)

## Abstract (arXiv)

Schr"odinger bridge--a stochastic dynamical generalization of optimal mass transport--exhibits a learning-control duality. Viewed as a stochastic control problem, the Schr"odinger 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 Schr"odinger 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 Schr"odinger 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.