Active learning with Gaussian Process Regression for solving non-linear time-dependent partial differential equations

Abstract

The efficient and accurate solution of nonlinear time-dependent partial differential equations (PDEs) remains a challenging problem in various scientific and engineering domains. Traditional numerical methods often require fine-grained discretizations and become computationally expensive as the complexity of the PDE increases. This paper aims to present a novel approach for solving nonlinear time-dependent PDEs using active learning with Gaussian Process Regression (GPR). GPR provides a probabilistic prediction of the solution, which includes a mean (the predicted value) and uncertainty (variance). Active learning with GPR in solving PDEs is a data-efficient approach that strategically selects the most informative points for model training, thus reducing the overall computational burden. Nonlinearity in PDEs is addressed by using kernel functions, such as the Matern and Radial Basis kernels. We predict numerical solutions for four distinct PDEs, namely the Burger’s equation, the Allen–Cahn equation, the parabolic interface PDE (Stefan equation) and the Korteweg–de Vries equation. By integrating active learning with GPR, the proposed method intelligently chooses the most informative data points for training, optimizing the model’s accuracy and efficiency. Our numerical results reveal that active learning with GPR significantly reduces the number of data points required for accurate predictions, offering a substantial reduction in computational costs compared to conventional numerical methods. Furthermore, the accuracy of the predictions is maintained, even with a limited number of labeled samples.