Rex: A Family of Reversible Exponential (Stochastic) Runge-Kutta Solvers

Zander W. Blasingame, Chen Liu
Proceedings of the 43rd International Conference on Machine Learning, PMLR 306:8373-8445, 2026.

Abstract

Deep generative models based on neural differential equations have become state-of-the-art for many generation tasks. These models rely on ODE/SDE solvers that integrate from a prior distribution to the data distribution; in many applications it is also highly desirable to integrate in the inverse direction. Standard solvers, however, accumulate discretization errors that prohibit exact inversion, an inaccuracy that is unacceptable in precision-critical applications. Existing inversion methods suffer from poor stability and low order of convergence, and are strictly limited to the ODE setting. In this work, we propose Rex, a family of reversible exponential (stochastic) Runge-Kutta solvers obtained by applying Lawson methods to convert any explicit (stochastic) Runge-Kutta scheme into an algebraically reversible one for both diffusion ODEs and SDEs. Beyond a rigorous theoretical analysis—establishing arbitrary-order convergence and a non-zero region of linear stability—we empirically demonstrate that Rex achieves near-machine-precision reconstruction and improves Boltzmann sampling with flow models as well as image generation and editing with diffusion models.

Cite this Paper

BibTeX
@InProceedings{pmlr-v306-blasingame26a, title = {Rex: A Family of Reversible Exponential ({S}tochastic) Runge-Kutta Solvers}, author = {Blasingame, Zander W. and Liu, Chen}, booktitle = {Proceedings of the 43rd International Conference on Machine Learning}, pages = {8373--8445}, year = {2026}, editor = {Zhang, Tong and Dudik, Miroslav and Jaggi, Martin and Agarwal, Alekh and Li, Sharon and Schuurmans, Dale and Zhu, Jerry and Berkenkamp, Felix and Dong, Hanze and Bietti, Alberto}, volume = {306}, series = {Proceedings of Machine Learning Research}, month = {06--11 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v306/main/assets/blasingame26a/blasingame26a.pdf}, url = {https://proceedings.mlr.press/v306/blasingame26a.html}, abstract = {Deep generative models based on neural differential equations have become state-of-the-art for many generation tasks. These models rely on ODE/SDE solvers that integrate from a prior distribution to the data distribution; in many applications it is also highly desirable to integrate in the inverse direction. Standard solvers, however, accumulate discretization errors that prohibit exact inversion, an inaccuracy that is unacceptable in precision-critical applications. Existing inversion methods suffer from poor stability and low order of convergence, and are strictly limited to the ODE setting. In this work, we propose Rex, a family of reversible exponential (stochastic) Runge-Kutta solvers obtained by applying Lawson methods to convert any explicit (stochastic) Runge-Kutta scheme into an algebraically reversible one for both diffusion ODEs and SDEs. Beyond a rigorous theoretical analysis—establishing arbitrary-order convergence and a non-zero region of linear stability—we empirically demonstrate that Rex achieves near-machine-precision reconstruction and improves Boltzmann sampling with flow models as well as image generation and editing with diffusion models.} }
Endnote
%0 Conference Paper %T Rex: A Family of Reversible Exponential (Stochastic) Runge-Kutta Solvers %A Zander W. Blasingame %A Chen Liu %B Proceedings of the 43rd International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2026 %E Tong Zhang %E Miroslav Dudik %E Martin Jaggi %E Alekh Agarwal %E Sharon Li %E Dale Schuurmans %E Jerry Zhu %E Felix Berkenkamp %E Hanze Dong %E Alberto Bietti %F pmlr-v306-blasingame26a %I PMLR %P 8373--8445 %U https://proceedings.mlr.press/v306/blasingame26a.html %V 306 %X Deep generative models based on neural differential equations have become state-of-the-art for many generation tasks. These models rely on ODE/SDE solvers that integrate from a prior distribution to the data distribution; in many applications it is also highly desirable to integrate in the inverse direction. Standard solvers, however, accumulate discretization errors that prohibit exact inversion, an inaccuracy that is unacceptable in precision-critical applications. Existing inversion methods suffer from poor stability and low order of convergence, and are strictly limited to the ODE setting. In this work, we propose Rex, a family of reversible exponential (stochastic) Runge-Kutta solvers obtained by applying Lawson methods to convert any explicit (stochastic) Runge-Kutta scheme into an algebraically reversible one for both diffusion ODEs and SDEs. Beyond a rigorous theoretical analysis—establishing arbitrary-order convergence and a non-zero region of linear stability—we empirically demonstrate that Rex achieves near-machine-precision reconstruction and improves Boltzmann sampling with flow models as well as image generation and editing with diffusion models.
APA
Blasingame, Z.W. & Liu, C.. (2026). Rex: A Family of Reversible Exponential (Stochastic) Runge-Kutta Solvers. Proceedings of the 43rd International Conference on Machine Learning, in Proceedings of Machine Learning Research 306:8373-8445 Available from https://proceedings.mlr.press/v306/blasingame26a.html.

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