EXPLICIT ESTIMATION OF DERIVATIVES FROM DATA AND DIFFERENTIAL EQUATIONS BY GAUSSIAN PROCESS REGRESSION

In this work, we employ the Bayesian inference framework to robustly estimate the derivatives of a function from noisy observations of only the function values at given location points, under the assumption of a physical model in the form of a differential equation governing the function and its derivatives. To overcome the instability of numerical differentiation of the fitted function solely from the data or the prohibitive costs of solving the differential equation on the whole domain, we use the Gaussian processes to jointly model the solution, the derivatives, and the differential equation, by utilizing the fact that differentiation is a linear operator. By regarding the linear differential equation as a linear constraint, we develop the Gaussian process regression with the constraint method (GPRC) at the Bayesian perspective to improve the prediction accuracy of derivatives. For nonlinear equations, we propose a Picard-iteration approximation of linearization around the Gaussian process obtained only from data to iteratively apply our GPRC. Besides, a product of experts method is applied if the initial or boundary condition is also available. We present several numerical results to illustrate the advantages of our new method and show the new estimation of the derivatives from GPRC improves the parameter identification with fewer data samples.