提出可逆微分方程求解器Rex，实现高精度反向积分。
Rex: A Family of Reversible Exponential (Stochastic) Runge-Kutta Solvers
- 通过龙格-库塔法构造代数可逆求解器，支持扩散微分方程与随机微分方程。
- 实验表明Rex可实现接近机器精度的重建，且在图像生成中提升采样质量。
- 适合需要精确反演的生成模型应用，如流模型与扩散模型。
基于神经微分方程的深度生成模型已在多种生成任务中达到领先水平。这些模型依赖于从先验分布到数据分布的ODE/SDE求解器；在许多应用中，反向积分也至关重要。然而，标准求解器累积离散化误差，导致无法精确反演，这对高精度场景不可接受。现有反演方法稳定性差、收敛阶低，且仅限于ODE情形。本文提出Rex，一类通过Lawson方法将任意显式(随机)龙格-库塔格式转化为代数可逆形式的可逆指数(随机)龙格-库塔求解器，适用于扩散ODE与SDE。除严格理论分析（证明任意阶收敛性及非零线性稳定性区域）外，实验证明Rex实现了近机器精度重建，显著提升流模型的Boltzmann采样性能，以及扩散模型在图像生成与编辑中的表现。
原文摘要 · Abstract (English)
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.
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