Explicit Estimation of Derivatives from Data and Differential Equations by Gaussian Process Regression

2020/04/13 by Hongqiao Wang, Xiang Zhou, Wang, Hongqiao +1 · 3 citations
Computer Science · Decision Sciences · Physics and Astronomy · #Computation (stat.CO) #FOS: Computer and information sciences #Gaussian Processes and Bayesian Inference #Model Reduction and Neural Networks #Other Statistics (stat.OT) #Probabilistic and Robust Engineering Design

paper · pdf · doi:10.48550/arxiv.2004.05796

openalex publication_date 2020/04/13 · openalex created_date 2020/04/17 · openalex updated_date 2026/07/28

Abstract

In this work, we employ the Bayesian inference framework to solve the problem of estimating the solution and particularly, its derivatives, which satisfy a known differential equation, from the given noisy and scarce observations of the solution data only. To address the key issue of accuracy and robustness of derivative estimation, we use the Gaussian processes to jointly model the solution, the derivatives, and the differential equation. By regarding the linear differential equation as a linear constraint, a Gaussian process regression with constraint method (GPRC) is developed to improve the accuracy of prediction of derivatives. For nonlinear differential equations, we propose a Picard-iteration-like approximation of linearization around the Gaussian process obtained only from data so that our GPRC can be still iteratively applicable. Besides, a product of experts method is applied to ensure the initial or boundary condition is considered to further enhance the prediction accuracy of the derivatives. We present several numerical results to illustrate the advantages of our new method in comparison to the standard data-driven Gaussian process regression.

Citations

• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

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