We propose a novel framework for discovering Stochastic Partial Differential Equations (SPDEs) from data. The proposed approach combines the concepts of stochastic calculus, variational Bayes theory, and sparse learning. We propose the extended Kramers-Moyal expansion to express the drift and diffusion terms of an SPDE in terms of state responses and use Spike-and-Slab priors with sparse learning techniques to efficiently and accurately discover the underlying SPDEs. The proposed approach has been applied to three canonical SPDEs, (a) stochastic heat equation, (b) stochastic Allen-Cahn equation, and (c) stochastic Nagumo equation. Our results demonstrate that the proposed approach can accurately identify the underlying SPDEs with limited data. This is the first attempt at discovering SPDEs from data, and it has significant implications for various scientific applications, such as climate modeling, financial forecasting, and chemical kinetics.

翻译：我们提出了一种从数据中发现随机偏微分方程（SPDEs）的新框架。该方法融合了随机微积分、变分贝叶斯理论和稀疏学习的概念。我们提出了扩展的Kramers-Moyal展开，以状态响应的形式表达SPDE的漂移项和扩散项，并利用带有稀疏学习技术的Spike-and-Slab先验，高效且准确地发现潜在的SPDEs。该方法已应用于三个典型的SPDEs：(a) 随机热方程、(b) 随机Allen-Cahn方程和(c) 随机Nagumo方程。实验结果表明，所提方法能够在有限数据条件下准确识别潜在的SPDEs。这是首次尝试从数据中发现SPDEs，对气候建模、金融预测和化学动力学等科学应用具有重要意义。