# Score-based Generative Modeling Through Backward Stochastic Differential Equations: Inversion and Generation

arXiv: https://arxiv.org/abs/2304.13224 · HTML: https://mlsanity.com/paper/2304.13224/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 22 of its 59 references that are in the map.
- First posted on arXiv 26 Apr 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 22 of 22)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding](https://mlsanity.com/paper/2205.11487/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Elucidating the Design Space of Diffusion-Based Generative Models](https://mlsanity.com/paper/2206.00364/) (2022)
- [SDEdit: Guided Image Synthesis and Editing with Stochastic Differential Equations](https://mlsanity.com/paper/2108.01073/) (2021)
- [Imagen Video: High Definition Video Generation with Diffusion Models](https://mlsanity.com/paper/2210.02303/) (2022)
- [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018)
- [CSDI: Conditional Score-based Diffusion Models for Probabilistic Time Series Imputation](https://mlsanity.com/paper/2107.03502/) (2021)
- [Diffusion Probabilistic Modeling for Video Generation](https://mlsanity.com/paper/2203.09481/) (2022)
- [Gotta Go Fast When Generating Data with Score-Based Models](https://mlsanity.com/paper/2105.14080/) (2021)
- [Riemannian Score-Based Generative Modelling](https://mlsanity.com/paper/2202.02763/) (2022)
- [Diffusion-based Time Series Imputation and Forecasting with Structured State Space Models](https://mlsanity.com/paper/2208.09399/) (2022)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [A Variational Perspective on Diffusion-Based Generative Models and Score Matching](https://mlsanity.com/paper/2106.02808/) (2021)
- [Leapfrog Diffusion Model for Stochastic Trajectory Prediction](https://mlsanity.com/paper/2303.10895/) (2023)
- [Convergence of the Deep BSDE Method for Coupled FBSDEs](https://mlsanity.com/paper/1811.01165/) (2018)
- [Zero-shot-Learning Cross-Modality Data Translation Through Mutual Information Guided Stochastic Diffusion](https://mlsanity.com/paper/2301.13743/) (2023)
- [Introduction to Solving Quant Finance Problems with Time-Stepped FBSDE and Deep Learning](https://mlsanity.com/paper/1911.12231/) (2019)
- [Value Function Estimators for Feynman-Kac Forward-Backward SDEs in Stochastic Optimal Control](https://mlsanity.com/paper/2103.14246/) (2021)

## Built on by (most cited first, 5 of 5)

- [Free-Form Motion Control: Controlling the 6D Poses of Camera and Objects in Video Generation](https://mlsanity.com/paper/2501.01425/) (2025)
- [A Diffusion-Based Framework for Terrain-Aware Remote Sensing Image Reconstruction](https://mlsanity.com/paper/2504.12112/) (2025)
- [Diffusion Models for Generative Artificial Intelligence: An Introduction for Applied Mathematicians](https://mlsanity.com/paper/2312.14977/) (2023)
- [Vision-Driven Prompt Optimization for Large Language Models in Multimodal Generative Tasks](https://mlsanity.com/paper/2501.02527/) (2025)
- [GaiaFlow: Semantic-Guided Diffusion Tuning for Carbon-Frugal Search](https://mlsanity.com/paper/2602.15423/) (2026)

## Abstract (arXiv)

The proposed BSDE-based diffusion model represents a novel approach to diffusion modeling, which extends the application of stochastic differential equations (SDEs) in machine learning. Unlike traditional SDE-based diffusion models, our model can determine the initial conditions necessary to reach a desired terminal distribution by adapting an existing score function. We demonstrate the theoretical guarantees of the model, the benefits of using Lipschitz networks for score matching, and its potential applications in various areas such as diffusion inversion, conditional diffusion, and uncertainty quantification. Our work represents a contribution to the field of score-based generative learning and offers a promising direction for solving real-world problems.