We show how to efficiently compute asymptotically sharp estimates of extreme event probabilities in stochastic differential equations (SDEs) with small multiplicative Brownian noise. The underlying approximation is known as sharp large deviation theory or precise Laplace asymptotics in mathematics, the second-order reliability method (SORM) in reliability engineering, and the instanton or optimal fluctuation method with 1-loop corrections in physics. It is based on approximating the tail probability in question with the most probable realization of the stochastic process, and local perturbations around this realization. We first recall and contextualize the relevant classical theoretical result on precise Laplace asymptotics of diffusion processes [Ben Arous (1988), Stochastics, 25(3), 125-153], and then show how to compute the involved infinite-dimensional quantities - operator traces and Carleman-Fredholm determinants - numerically in a way that is scalable with respect to the time discretization and remains feasible in high spatial dimensions. Using tools from automatic differentiation, we achieve a straightforward black-box numerical computation of the SORM estimates in JAX. The method is illustrated in examples of SDEs and stochastic partial differential equations, including a two-dimensional random advection-diffusion model of a passive scalar. We thereby demonstrate that it is possible to obtain efficient and accurate SORM estimates for very high-dimensional problems, as long as the infinite-dimensional structure of the problem is correctly taken into account. Our JAX implementation of the method is made publicly available.

翻译：我们展示了如何高效计算带小乘性布朗噪声的随机微分方程(SDEs)中极端事件概率的渐近精确估计。该近似方法在数学领域称为尖锐大偏差理论或精确拉普拉斯渐近，在可靠性工程中称为二阶可靠性方法(SORM)，在物理学中称为带有1-loop修正的瞬子或最优涨落方法。其核心思想是通过随机过程最可能实现路径及其局部扰动来近似尾部概率。我们首先回顾并阐明关于扩散过程精确拉普拉斯渐近的相关经典理论结果[Ben Arous (1988), Stochastics, 25(3), 125-153]，继而展示如何对涉及的无穷维量——算子迹和Carleman-Fredholm行列式——进行数值计算，使其在时间离散化方面具有可扩展性且在空间高维情形下依然可行。利用自动微分工具，我们在JAX框架中实现了黑箱化的SORM估计数值计算。该方法通过SDE和随机偏微分方程实例进行验证，包括二维无源标量的随机平流-扩散模型。实验证明，只要正确考虑问题的无穷维结构，即可对极高维问题获得高效精确的SORM估计。我们的JAX实现代码已公开发布。