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

arXiv: https://arxiv.org/abs/2310.05387 · HTML: https://mlsanity.com/paper/2310.05387/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 60 references that are in the map.
- First posted on arXiv 9 Oct 2023.

## Builds on (most cited first, 15 of 15)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [A Simple Baseline for Bayesian Uncertainty in Deep Learning](https://mlsanity.com/paper/1902.02476/) (2019)
- [Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP)](https://mlsanity.com/paper/1503.01057/) (2015)
- [Expectation Propagation for approximate Bayesian inference](https://mlsanity.com/paper/1301.2294/) (2013)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [DeepMoD: Deep learning for Model Discovery in noisy data](https://mlsanity.com/paper/1904.09406/) (2019)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networks](https://mlsanity.com/paper/2205.02902/) (2022)
- [AutoIP: A United Framework to Integrate Physics into Gaussian Processes](https://mlsanity.com/paper/2202.12316/) (2022)
- [Fast Gaussian Process Based Gradient Matching for Parameter Identification in Systems of Nonlinear ODEs](https://mlsanity.com/paper/1804.04378/) (2018)
- [ODIN: ODE-Informed Regression for Parameter and State Inference in Time-Continuous Dynamical Systems](https://mlsanity.com/paper/1902.06278/) (2019)
- [Bayesian Spline Learning for Equation Discovery of Nonlinear Dynamics with Quantified Uncertainty](https://mlsanity.com/paper/2210.08095/) (2022)

## Built on by (most cited first, 5 of 5)

- [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Dynamic Bayesian Optimization Framework for Instruction Tuning in Partial Differential Equation Discovery](https://mlsanity.com/paper/2601.00088/) (2025)
- [Pseudo-Physics-Informed Neural Operators: Enhancing Operator Learning from Limited Data](https://mlsanity.com/paper/2502.02682/) (2025)
- [Discovering Interpretable Ordinary Differential Equations from Noisy Data](https://mlsanity.com/paper/2507.21841/) (2025)

## Abstract (arXiv)

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.