# Learning the Infinitesimal Generator of Stochastic Diffusion Processes

arXiv: https://arxiv.org/abs/2405.12940 · HTML: https://mlsanity.com/paper/2405.12940/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 57 references that are in the map.
- First posted on arXiv 21 May 2024.

## Builds on (most cited first, 17 of 17)

- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1708.06850/) (2017)
- [Sobolev Norm Learning Rates for Regularized Least-Squares Algorithm](https://mlsanity.com/paper/1702.07254/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Variational approach for learning Markov processes from time series data](https://mlsanity.com/paper/1707.04659/) (2017)
- [Optimal Rates for Regularized Conditional Mean Embedding Learning](https://mlsanity.com/paper/2208.01711/) (2022)
- [Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/1712.01572/) (2017)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Learning Stable Koopman Embeddings](https://mlsanity.com/paper/2110.06509/) (2021)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks](https://mlsanity.com/paper/2110.14523/) (2021)
- [Consistent Long-Term Forecasting of Ergodic Dynamical Systems](https://mlsanity.com/paper/2312.13426/) (2023)
- [The Galerkin method beats Graph-Based Approaches for Spectral Algorithms](https://mlsanity.com/paper/2306.00742/) (2023)
- [Kernelized Diffusion maps](https://mlsanity.com/paper/2302.06757/) (2023)

## Built on by (most cited first, 9 of 9)

- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2406.09028/) (2024)
- [A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systems](https://mlsanity.com/paper/2509.24920/) (2025)
- [Kernel-Based Optimal Control: An Infinitesimal Generator Approach](https://mlsanity.com/paper/2412.01591/) (2024)
- [kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learning](https://mlsanity.com/paper/2512.21409/) (2025)
- [Toeplitz Based Spectral Methods for Data-driven Dynamical Systems](https://mlsanity.com/paper/2602.09791/) (2026)
- [Koopman operator theory: fundamentals, control, and applications](https://mlsanity.com/paper/2607.01819/) (2026)
- [Adaptive Kernel Selection for Kernelized Diffusion Maps](https://mlsanity.com/paper/2604.18402/) (2026)
- [Langevin-Informed Transfer Learning: Replacing Target Samples by Black-Box Feedback](https://mlsanity.com/paper/2610.01522/) (2026)

## Abstract (arXiv)

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems. The unbounded nature of the generator poses significant challenges, rendering conventional analysis techniques for Hilbert-Schmidt operators ineffective. To overcome this, we introduce a novel framework based on the energy functional for these stochastic processes. Our approach integrates physical priors through an energy-based risk metric in both full and partial knowledge settings. We evaluate the statistical performance of a reduced-rank estimator in reproducing kernel Hilbert spaces (RKHS) in the partial knowledge setting. Notably, our approach provides learning bounds independent of the state space dimension and ensures non-spurious spectral estimation. Additionally, we elucidate how the distortion between the intrinsic energy-induced metric of the stochastic diffusion and the RKHS metric used for generator estimation impacts the spectral learning bounds.