Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernels

TL;DR

This paper introduces KBASS, a novel equation discovery method combining kernel regression with Bayesian spike-and-slab priors, improving robustness to data issues and enabling uncertainty quantification through efficient inference techniques.

Contribution

The paper proposes KBASS, a new approach that integrates kernel learning with Bayesian priors for robust, efficient equation discovery with uncertainty quantification.

Findings

Outperforms existing methods on benchmark ODE and PDE discovery tasks.

Robust to data sparsity and noise due to kernel-based function estimation.

Provides effective uncertainty quantification through Bayesian inference.

Abstract

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.