Discovering governing equations from data is important to many scientific and engineering applications. Despite promising successes, existing methods are still challenged by data sparsity as well as 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 methods to enable efficient computation and optimization. We show the significant 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许多科学和工程应用至关重要。尽管已取得可喜的成功，现有方法仍面临数据稀疏性和噪声问题的挑战，这两种问题在实践中普遍存在。此外，最先进的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）。我们采用核回归估计目标函数，该方法灵活、富有表现力且对数据稀疏性和噪声具有更强鲁棒性。结合贝叶斯尖峰-平板先验——一种理想的贝叶斯稀疏分布——实现有效的算子选择与不确定性量化。我们开发了一种期望传播期望最大化（EP-EM）算法，用于高效的后验推理与函数估计。为克服核回归的计算挑战，我们将函数值置于网格上并诱导克罗内克积构造，利用张量代数方法实现高效计算与优化。我们在基准ODE和PDE发现任务列表上展示了KBASS的显著优势。