# Divergence-kernel method for linear responses of densities and generative models

arXiv: https://arxiv.org/abs/2509.03992 · HTML: https://mlsanity.com/paper/2509.03992/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 64 references that are in the map.
- First posted on arXiv 4 Sep 2025.

## Builds on (most cited first, 14 of 14)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Pixel Recurrent Neural Networks](https://mlsanity.com/paper/1601.06759/) (2016)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [A Training-Free Conditional Diffusion Model for Learning Stochastic Dynamical Systems](https://mlsanity.com/paper/2410.03108/) (2024)
- [When are dynamical systems learned from time series data statistically accurate?](https://mlsanity.com/paper/2411.06311/) (2024)
- [State-observation augmented diffusion model for nonlinear assimilation with unknown dynamics](https://mlsanity.com/paper/2407.21314/) (2024)
- [Score-based Neural Ordinary Differential Equations for Computing Mean Field Control Problems](https://mlsanity.com/paper/2409.16471/) (2024)
- [Sensitivity analysis on chaotic dynamical system by Non-Intrusive Least Square Shadowing (NILSS)](https://mlsanity.com/paper/1611.00880/) (2016)
- [Malliavin Calculus for Score-based Diffusion Models](https://mlsanity.com/paper/2503.16917/) (2025)
- [Adjoint DSMC for nonlinear Boltzmann equation constrained optimization](https://mlsanity.com/paper/2009.01363/) (2020)
- [Divergence-Kernel method for scores of random systems](https://mlsanity.com/paper/2507.04035/) (2025)
- [Optimal Response for Hyperbolic Systems by the fast adjoint response method](https://mlsanity.com/paper/2501.02395/) (2025)

## Abstract (arXiv)

We derive the divergence-kernel formula for the linear response of random dynamical systems. Specifically, the pathwise expression is for the parameter-derivative of the marginal or stationary density, not an averaged observable. Our formula works for multiplicative and parameterized noise over any period of time; it does not require hyperbolicity. Then we derive a Monte-Carlo algorithm for linear responses. We develop a new framework of generative models, DK-SDE, where the model is a parameterized SDE, that (1) directly uses the KL divergence between the empirical data distribution and the marginal density of the SDE as the training objective, and (2) accommodates parametrizations in both drift and diffusion over a long time span, allowing prior structural knowledge to be incorporated explicitly. The optimization is done by gradient-descent enabled by the divergence-kernel method, which involves only forward processes and therefore substantially reduces memory cost. We demonstrate the new model on a 20-dimensional Lorenz system.