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.

基于核学习和贝叶斯尖峰和板条先验的方程发现方法提供了一种有效的算法来解决数据稀疏性、噪声问题以及目前方法中存在的不确定性量化和训练成本高的限制性问题。

贝叶斯Spike-and-Slab先验与高效核的方程探索