Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernels
Equation Discovery with Bayesian Spike-and-Slab Priors and Efficient Kernels
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DOI：
10.48550/arxiv.2310.05387
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发表时间：
2023-10
期刊：
影响因子：
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通讯作者：
Da Long;Wei W. Xing;Aditi S. Krishnapriyan;R. Kirby;Shandian Zhe;Michael W. Mahoney
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作者：
Da Long;Wei W. Xing;Aditi S. Krishnapriyan;R. Kirby;Shandian Zhe;Michael W. Mahoney
从数据中发现控制方程对许多科学和工程应用都很重要。尽管取得了成功，但现有的方法仍然受到数据稀疏性和噪声问题的挑战，这两个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和/或培训费用高昂。为了克服这些限制，我们提出了一种新的基于核学习和贝叶斯尖峰- slab先验（KBASS）的方程发现方法。我们使用核回归来估计目标函数，这是灵活的，富有表现力的，并且对数据稀疏性和噪声具有更强的鲁棒性。我们将其与贝叶斯尖峰-板先验（一种理想的贝叶斯稀疏分布）相结合，用于有效的算子选择和不确定性量化。我们开发了一种期望-传播期望-最大化（EP-EM）算法，用于有效的后验推理和函数估计。为了克服核回归的计算挑战，我们将函数值放在网格上并诱导Kronecker积构造，并且我们使用张量代数来实现高效的计算和优化。我们在一系列基准ODE和PDE发现任务上展示了KBASS的优势。
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.