Probabilistic solvers provide a flexible and efficient framework for simulation, uncertainty quantification, and inference in dynamical systems. However, like standard solvers, they suffer performance penalties for certain stiff systems, where small steps are required not for reasons of numerical accuracy but for the sake of stability. This issue is greatly alleviated in semi-linear problems by the probabilistic exponential integrators developed in this paper. By including the fast, linear dynamics in the prior, we arrive at a class of probabilistic integrators with favorable properties. Namely, they are proven to be L-stable, and in a certain case reduce to a classic exponential integrator -- with the added benefit of providing a probabilistic account of the numerical error. The method is also generalized to arbitrary non-linear systems by imposing piece-wise semi-linearity on the prior via Jacobians of the vector field at the previous estimates, resulting in probabilistic exponential Rosenbrock methods. We evaluate the proposed methods on multiple stiff differential equations and demonstrate their improved stability and efficiency over established probabilistic solvers. The present contribution thus expands the range of problems that can be effectively tackled within probabilistic numerics.

翻译：概率求解器为动态系统的模拟、不确定性量化和推断提供了灵活高效的框架。然而，与标准求解器类似，它们在应对某些刚性系统时面临性能损失——在步长限制非源于数值精度要求、而是出于稳定性考量时尤其明显。本文开发的概率指数积分器通过在先验中融入快速线性动力学，有效缓解了半线性问题中的这一困境。由此诞生的概率积分器被证明具有L-稳定性等优良特性，且在特定情形下可简化为经典指数积分器，同时额外提供了数值误差的概率化描述。通过利用向量场雅可比矩阵对前一估计值施加分段半线性先验，该方法进一步推广至任意非线性系统，衍生出概率指数罗斯布洛克法。我们在多个刚性微分方程上验证了所提方法，证明其相较于现有概率求解器具有更优的稳定性与计算效率。本研究由此拓展了概率数值方法能够有效处理的问题范畴。