# Neural Schr"odinger Bridge with Sinkhorn Losses: Application to Data-driven Minimum Effort Control of Colloidal Self-assembly

arXiv: https://arxiv.org/abs/2307.14442 · HTML: https://mlsanity.com/paper/2307.14442/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 74 references that are in the map.
- First posted on arXiv 26 Jul 2023.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Diffusion Models in Vision: A Survey](https://mlsanity.com/paper/2209.04747/) (2022)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Efficient and Accurate Estimation of Lipschitz Constants for Deep Neural Networks](https://mlsanity.com/paper/1906.04893/) (2019)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021)
- [Deep Generative Learning via Schr"odinger Bridge](https://mlsanity.com/paper/2106.10410/) (2021)
- [Safety Verification and Robustness Analysis of Neural Networks via Quadratic Constraints and Semidefinite Programming](https://mlsanity.com/paper/1903.01287/) (2019)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Unpaired Image-to-Image Translation via Neural Schr"odinger Bridge](https://mlsanity.com/paper/2305.15086/) (2023)
- [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)
- [The data-driven Schroedinger bridge](https://mlsanity.com/paper/1806.01364/) (2018)
- [Reflected Schr"odinger Bridge: Density Control with Path Constraints](https://mlsanity.com/paper/2003.13895/) (2020)
- [Stochastic Physics-Informed Neural Ordinary Differential Equations](https://mlsanity.com/paper/2109.01621/) (2021)
- [Finite Horizon Density Steering for Multi-input State Feedback Linearizable Systems](https://mlsanity.com/paper/1909.12511/) (2019)
- [Prediction and Optimal Feedback Steering of Probability Density Functions for Safe Automated Driving](https://mlsanity.com/paper/2009.09055/) (2020)
- [The derivatives of Sinkhorn-Knopp converge](https://mlsanity.com/paper/2207.12717/) (2022)
- [A Physics-informed Deep Learning Approach for Minimum Effort Stochastic Control of Colloidal Self-Assembly](https://mlsanity.com/paper/2208.09182/) (2022)
- [Schr"odinger Meets Kuramoto via Feynman-Kac: Minimum Effort Distribution Steering for Noisy Nonuniform Kuramoto Oscillators](https://mlsanity.com/paper/2202.09734/) (2022)
- [freud: A Software Suite for High Throughput Analysis of Particle Simulation Data](https://mlsanity.com/paper/1906.06317/) (2019)
- [Learning to grow: control of material self-assembly using evolutionary reinforcement learning](https://mlsanity.com/paper/1912.08333/) (2019)
- [Implementation of batched Sinkhorn iterations for entropy-regularized Wasserstein loss](https://mlsanity.com/paper/1907.01729/) (2019)

## Built on by (most cited first, 10 of 10)

- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [On the Hopf-Cole Transform for Control-affine Schr"odinger Bridge](https://mlsanity.com/paper/2503.17640/) (2025)
- [Stochastic Learning of Computational Resource Usage as Graph Structured Multimarginal Schr"odinger Bridge](https://mlsanity.com/paper/2405.12463/) (2024)
- [Weyl Calculus and Exactly Solvable Schr"odinger Bridges with Quadratic State Cost](https://mlsanity.com/paper/2407.15245/) (2024)
- [A Generalized Sinkhorn Algorithm for Mean-Field Schr"odinger Bridge](https://mlsanity.com/paper/2604.06531/) (2026)
- [Schr"odinger Bridge Over A Compact Connected Lie Group](https://mlsanity.com/paper/2603.14049/) (2026)
- [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)
- [Generative Profiling for Soft Real-Time Systems and its Applications to Resource Allocation](https://mlsanity.com/paper/2604.01441/) (2026)
- [Schr"odinger Bridges over Kinetic Swarming Models](https://mlsanity.com/paper/2608.25281/) (2026)
- [On the Equivalence of Stochastic Control and Path Space Formulations for Schr"odinger Bridges over Compact Connected Lie Groups](https://mlsanity.com/paper/2609.13758/) (2026)

## Abstract (arXiv)

We show that the minimum effort control of colloidal self-assembly can be naturally formulated in the order-parameter space as a generalized Schr"odinger bridge problem -- a class of fixed-horizon stochastic optimal control problems that originated in the works of Erwin Schr"odinger in the early 1930s. In recent years, this class of problems has seen a resurgence of research activities in the control and machine learning communities. Different from the existing literature on the theory and computation for such problems, the controlled drift and diffusion coefficients for colloidal self-assembly are typically nonaffine in control, and are difficult to obtain from physics-based modeling. We deduce the conditions of optimality for such generalized problems, and show that the resulting system of equations is structurally very different from the existing results in a way that standard computational approaches no longer apply. Thus motivated, we propose a data-driven learning and control framework, named `neural Schr"odinger bridge', to solve such generalized Schr"odinger bridge problems by innovating on recent advances in neural networks. We illustrate the effectiveness of the proposed framework using a numerical case study of colloidal self-assembly. We learn the controlled drift and diffusion coefficients as two neural networks using molecular dynamics simulation data, and then use these two to train a third network with Sinkhorn losses designed for distributional endpoint constraints, specific for this class of control problems.