Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernels
This addresses data sparsity and noise issues in scientific and engineering applications, offering uncertainty quantification and efficiency, though it appears incremental as an enhancement to existing kernel-based methods.
The authors tackled the problem of discovering governing equations from sparse and noisy data by proposing KBASS, a method combining kernel regression with Bayesian spike-and-slab priors, which achieved improved performance on benchmark ODE and PDE tasks.
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.