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Rex introduces a family of reversible exponential Runge-Kutta solvers that address the critical limitation of discretization errors in standard ODE/SDE integrators used for deep generative models. By leveraging Lawson methods to achieve algebraic reversibility, the approach enables near-machine-precision reconstruction and improved sampling for both diffusion and flow models. This advancement is particularly significant for precision-critical applications requiring exact inversion in inverse direction integration.
Rex: A Family of Reversible Exponential (Stochastic) Runge-Kutta Solvers
arXiv:2502.08834v4 Announce Type: replace-cross 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.
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.
Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Machine Learning (stat.ML)
arXiv:2502.08834 [cs.LG]
(or arXiv:2502.08834v4 [cs.LG] for this version)
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