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：概率求解器为动力系统的模拟、不确定性量化与推断提供了灵活高效的框架。然而，与标准求解器类似，其在特定刚性系统中面临性能损失：步长选择需服从稳定性而非数值精度要求。本文开发的概率指数积分器显著缓解了半线性问题中的这一困境。通过将快速线性动力学纳入先验，我们构建了一类具有优良特性的概率积分器，其被证明具备L稳定性，且在特定情形下可退化为经典指数积分器——同时额外提供数值误差的概率化描述。该方法进一步推广至任意非线性系统：通过利用向量场在先前估计点处的雅可比矩阵，对先验施加分段半线性约束，由此形成概率指数罗森布罗克方法。我们在多个刚性微分方程上评估了所提方法，验证其相较于现有概率求解器在稳定性与效率上的改进。本文贡献拓展了概率数值方法能够有效处理的问题范畴。