The discovery of partial differential equations (PDEs) is a challenging task that involves both theoretical and empirical methods. Machine learning approaches have been developed and used to solve this problem; however, it is important to note that existing methods often struggle to identify the underlying equation accurately in the presence of noise. In this study, we present a new approach to discovering PDEs by combining variational Bayes and sparse linear regression. The problem of PDE discovery has been posed as a problem to learn relevant basis from a predefined dictionary of basis functions. To accelerate the overall process, a variational Bayes-based approach for discovering partial differential equations is proposed. To ensure sparsity, we employ a spike and slab prior. We illustrate the efficacy of our strategy in several examples, including Burgers, Korteweg-de Vries, Kuramoto Sivashinsky, wave equation, and heat equation (1D as well as 2D). Our method offers a promising avenue for discovering PDEs from data and has potential applications in fields such as physics, engineering, and biology.

翻译：偏微分方程（PDE）的发现是一项具有挑战性的任务，涉及理论和经验方法。机器学习方法已被开发并用于解决该问题；然而，值得注意的是，现有的方法在存在噪声的情况下往往难以准确识别潜在方程。在本研究中，我们提出了一种结合变分贝叶斯和稀疏线性回归的新方法，用于发现偏微分方程。PDE发现问题被建模为从预定义的基函数字典中学习相关基函数的问题。为了加速整体过程，我们提出了一种基于变分贝叶斯的偏微分方程发现方法。为确保稀疏性，我们采用了尖峰-板先验。通过多个示例（包括Burgers方程、Korteweg-de Vries方程、Kuramoto Sivashinsky方程、波动方程以及热方程（一维和二维））展示了我们策略的有效性。该方法为从数据中发现PDE提供了一条有前景的途径，并在物理、工程和生物学等领域具有潜在应用价值。