Discovering governing equations from data is important to many scientific and engineering applications. Despite promising successes, existing methods are still challenged by data sparsity and noise issues, both of which are ubiquitous in practice. Moreover, state-of-the-art methods lack uncertainty quantification and/or are costly in training. To overcome these limitations, we propose a novel equation discovery method based on Kernel learning and BAyesian Spike-and-Slab priors (KBASS). We use kernel regression to estimate the target function, which is flexible, expressive, and more robust to data sparsity and noises. We combine it with a Bayesian spike-and-slab prior -- an ideal Bayesian sparse distribution -- for effective operator selection and uncertainty quantification. We develop an expectation-propagation expectation-maximization (EP-EM) algorithm for efficient posterior inference and function estimation. To overcome the computational challenge of kernel regression, we place the function values on a mesh and induce a Kronecker product construction, and we use tensor algebra to enable efficient computation and optimization. We show the advantages of KBASS on a list of benchmark ODE and PDE discovery tasks.

翻译：从数据中发现控制方程对众多科学与工程应用至关重要。尽管已有方法取得显著成功，但在实践中普遍存在的数据稀疏性和噪声问题仍对其构成挑战。此外，现有方法缺乏不确定性量化，且/或训练成本高昂。为克服这些局限，我们提出了一种基于核学习与贝叶斯尖峰-平板先验的新型方程发现方法（KBASS）。该方法采用核回归估计目标函数，具有灵活性强、表达力丰富且对数据稀疏性和噪声更具鲁棒性的特点。我们将其与贝叶斯尖峰-平板先验（一种理想的贝叶斯稀疏分布）相结合，以实现有效的算子选择与不确定性量化。我们开发了期望传播-期望最大化（EP-EM）算法，用于高效的后验推断与函数估计。为克服核回归的计算挑战，我们将函数值置于网格上并引入克罗内克积构造，利用张量代数实现高效计算与优化。我们在多个基准常微分方程和偏微分方程发现任务中展示了KBASS的优势。