Adaptive Learning of Periodic Solutions for Nonlinear Systems by Physics-informed Gaussian Process
Abstract
Abstract This paper presents a novel physics-informed Gaussian process (PIGP) for the adaptive computation of periodic solutions to nonlinear dynamical systems. The proposed approach is built upon two key steps. First, the Gaussian process (GP), as a Bayesian non-parametric machine learning framework, is employed to approximate the periodic solution without requiring a predetermined set of basis functions. To this end, tailored physics-informed periodic kernel and mean functions are constructed for GP regression. The nonlinear governing equation is then collocated to generate observational data, and in conjunction with an iterative linearization scheme, the periodic solution is obtained with both mean and covariance information. Second, recognizing that the performance of GP regression critically depends on the selection of collocation points, an adaptive collocation strategy based on variance-or residual-based criterion is developed to enable accurate and efficient solving of nonlinear systems. Numerical examples are conducted to validate the accuracy, robustness, and efficiency of the proposed PIGP. Notably, under the same level of accuracy, the PIGP demonstrates superior computational efficiency compared to the widely-recognized harmonic balance method (HBM).
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