# Tangent Schr"odinger Bridge Matching: Learning Stochastic Transport with Mechanistic Sensitivities

arXiv: https://arxiv.org/abs/2610.02906 · HTML: https://mlsanity.com/paper/2610.02906/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 23 of its 34 references that are in the map.
- First posted on arXiv 2 Oct 2026.

## Builds on (most cited first, 23 of 23)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Neural SDEs as Infinite-Dimensional GANs](https://mlsanity.com/paper/2102.03657/) (2021)
- [Sobolev Training for Neural Networks](https://mlsanity.com/paper/1706.04859/) (2017)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Knowledge Transfer with Jacobian Matching](https://mlsanity.com/paper/1803.00443/) (2018)
- [Conditional Simulation Using Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2202.13460/) (2022)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators](https://mlsanity.com/paper/2305.20053/) (2023)
- [Derivative-Informed Fourier Neural Operator: Universal Approximation and Applications to PDE-Constrained Optimization](https://mlsanity.com/paper/2512.14086/) (2025)
- [SPDEBench: An Extensive Benchmark for Learning Stochastic PDEs](https://mlsanity.com/paper/2505.18511/) (2025)
- [Tilted Schr"odinger Bridge Matching](https://mlsanity.com/paper/2609.34642/) (2026)

## Abstract (arXiv)

Predicting how stochastic systems respond to changes in viscosity, reaction rates, or external forces requires costly simulations, motivating reusable learned models. Yet matching observed outcome distributions does not ensure accurate intervention responses. We introduce Tangent Schr"odinger Bridge Matching (Tangent-SBM), which learns stochastic transports from endpoint observations and mechanistic sensitivities. It propagates parameter derivatives alongside trajectories and supervises them against supplied targets. For average-response targets, single-rollout squared error also penalizes response variability; our objective uses two independent rollouts to match the mean without this additional penalty. We establish conditions under which sensitivity accuracy bounds finite-change prediction error and decision regret. Across Gaussian, stochastic double-well, PDEBench reaction--diffusion, and stochastic Navier--Stokes systems, Tangent-SBM improves sensitivity and finite-change prediction over matched conditional-bridge baselines while maintaining comparable endpoint and distributional accuracy. Controls examine target correctness, response objectives, and simulator-budget allocation. To test decision usefulness, we evaluate calibrated viscosity selection in Navier--Stokes: Tangent-SBM reduces tracking error relative to taking no action on every evaluated task.