Inferring the unknown parameters in Differential Equation by Gaussian Process Regression with Constraint

Differential Equation (DE) is a commonly used modeling method in various scientific subjects such as finance and biology. The parameters in DE models often have interesting scientific interpretations, but their values are often unknown and need to be estimated from the measurements of the DE. In this work, we propose a Bayesian inference framework to solve the problem of estimating the parameters of the DE model, from the given noisy and scarce observations of the solution only. A key issue in this problem is 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 differential equation governing the function and its derivatives. To address the key issue, we use the Gaussian Process Regression with Constraint (GPRC) method which jointly model the solution, the derivatives, and the parametric differential equation, to estimate the solution and its derivatives. For nonlinear differential equations, a Picard-iteration-like approximation of linearization method is used so that the GPRC can be still iteratively applicable. A new potential which combines the data and equation information, is proposed and used in the likelihood for our inference. With numerical examples, we illustrate that the proposed method has competitive performance against existing approaches for estimating the unknown parameters in DEs.