Efficient Bayesian Kernel Learning for Discovering Governing Equations from Sparse and Noisy Data

Khái niệm cốt lõi

A novel equation discovery method based on Kernel learning and Bayesian Spike-and-Slab priors (KBASS) that is flexible, expressive, and robust to data sparsity and noise, with efficient posterior inference and function estimation.

Tóm tắt

The authors propose a novel equation discovery method called KBASS that combines kernel regression and Bayesian spike-and-slab priors. The key highlights are:

Model: KBASS uses kernel regression to estimate the target function, which is flexible, expressive, and more robust to data sparsity and noise. It combines this with a Bayesian spike-and-slab prior to select equation operators and estimate the operator weights, enabling effective operator selection and uncertainty quantification.

Algorithm: To overcome the computational challenge of kernel learning, KBASS places the function values on a mesh and induces a Kronecker product structure in the kernel matrices. It then develops an expectation-propagation expectation-maximization (EP-EM) algorithm for efficient posterior inference and function estimation.

Results: KBASS is evaluated on a range of benchmark ODE and PDE discovery tasks. It is shown to outperform state-of-the-art methods in terms of sample efficiency, robustness to noise, accuracy of weight estimation, and computational efficiency. KBASS can recover the equations from much sparser and noisier data compared to the competing methods.

The authors demonstrate the advantages of KBASS through extensive experiments on various ODE and PDE systems, showcasing its superior performance in terms of sample efficiency, robustness to noise, and computational efficiency compared to existing data-driven equation discovery methods.

Thống kê

The target function u(t, x1, x2) is modeled as a kernel interpolation from the function values U on a mesh M.
The derivatives of u, such as ∂x1x2u, can be efficiently computed using the product structure of the kernel.

Trích dẫn

"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."
"To overcome these limitations, we propose a novel equation discovery method based on Kernel learning and BAyesian Spike-and-Slab priors (KBASS)."

Thông tin chi tiết chính được chắt lọc từ

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

by Da Long,Wei ... lúc arxiv.org 04-23-2024

https://arxiv.org/pdf/2310.05387.pdf

Yêu cầu sâu hơn

How can the KBASS method be extended to handle dynamically expanding operator dictionaries to further improve the accuracy and robustness of equation discovery

To extend the KBASS method to handle dynamically expanding operator dictionaries, we can introduce a mechanism that automatically adds new operators to the dictionary based on the data and the complexity of the system being modeled. This can be achieved by incorporating a mechanism that evaluates the relevance of potential new operators based on their contribution to the model's performance metrics.
One approach could involve monitoring the performance of the model with the current dictionary and periodically evaluating the need for additional operators. When the model's performance starts to plateau or deteriorate, new operators can be introduced and tested to see if they improve the model's accuracy. This dynamic expansion of the operator dictionary would allow the model to adapt to the complexity of the underlying system and potentially uncover more intricate relationships present in the data.

What are the potential limitations of the Bayesian spike-and-slab prior approach, and how could alternative sparse Bayesian priors be explored to address them

The Bayesian spike-and-slab prior approach, while powerful for sparse Bayesian learning, may have limitations in terms of computational complexity and scalability, especially when dealing with high-dimensional data or large-scale problems. The spike-and-slab prior requires sampling binary selection indicators and continuous weights, which can be computationally intensive, particularly when the number of operators or dimensions is large.
To address these limitations, alternative sparse Bayesian priors could be explored. For example, hierarchical priors such as horseshoe priors or automatic relevance determination (ARD) priors could be considered. These priors offer a more structured approach to sparsity by automatically shrinking irrelevant weights towards zero while allowing relevant weights to be estimated accurately. By incorporating such priors, the model may achieve better scalability and computational efficiency while maintaining the benefits of sparsity in the Bayesian framework.

Given the success of KBASS in PDE and ODE discovery, how could the method be adapted to tackle other types of differential equations, such as stochastic differential equations or partial integro-differential equations

To adapt the KBASS method for other types of differential equations, such as stochastic differential equations (SDEs) or partial integro-differential equations (PIDEs), several modifications and extensions can be considered:

Stochastic Differential Equations (SDEs): For SDEs, the KBASS method can be extended to incorporate stochastic processes and noise terms in the model. This would involve modifying the kernel regression to account for the stochastic nature of the data and incorporating Bayesian priors that capture the uncertainty in the system dynamics.

Partial Integro-Differential Equations (PIDEs): Handling PIDEs would require extending the operator dictionary to include integral operators and their combinations. The KBASS method can be adapted to estimate the solution of PIDEs by incorporating integral kernels and operators in the model. Additionally, the algorithm may need to be adjusted to handle the unique characteristics of PIDEs, such as memory effects and non-local interactions.

By customizing the KBASS method to suit the specific requirements of SDEs and PIDEs, it can be effectively applied to a broader range of differential equations, enabling accurate and robust equation discovery in diverse scientific and engineering applications.