Solving Oscillator Ordinary Differential Equations in the Time Domain with High Performance via Soft-constrained Physics-informed Neural Network with Small Data

Abstract

In many scientific and engineering (e.g., physical, biochemical, medical) practices, data generated through expensive experiments or large-scale simulations, are often sparse and noisy. Physics-informed neural network (PINN) incorporates physical information and knowledge into network topology or computational processes as model priors, with the unique advantage of achieving strong generalization with small data. This study aims to investigate the performance characteristics of the soft-constrained PINN method to solving typical linear and nonlinear ordinary differential equations (ODEs) such as primer, Van der Pol and Duffing oscillators, especially the effectiveness, efficiency, and robustness to noise with minimal data. It is verified that the soft-constrained PINN significantly reduces the need for labeled data. With the aid of appropriate collocation points no need to be labeled, it can predict and also extrapolate with minimal data. First-order and second-order ODEs, no matter linear or nonlinear oscillators, require only one and two training data (containing initial values) respectively, just like classical analytic or Runge-Kutta methods, and with equivalent precision and comparable efficiency (fast training in seconds for scalar ODEs). Furthermore, it can conveniently impose a physical law (e.g., conservation of energy) constraint by adding a regularization term to the total loss function, improving the performance to deal with various complexities such as nonlinearity like Duffing. The DeepXDE-based PINN implementation is light code and can be efficiently trained on both GPU and CPU platforms. The mathematical and computational framework of this alternative and feasible PINN method to ODEs, can be easily extended to PDEs, etc., and is becoming a favorable catalyst for the era of Digital Twins.