Solving Oscillator ODEs via Soft-constrained Physics-informed Neural Network with Small Data

This paper compared physics-informed neural network (PINN), conventional neural network (NN) and numerical discretization methods on solving differential equations through literature research. We formalized the mathematical framework and computational flow of the soft-constrained PINN method for solving differential equations (e.g., ODEs/PDEs). Its working mechanism and its accuracy and efficiency were experimentally verified by solving typical linear and non-linear oscillator ODEs. The implementation of the PINN method based on DeepXDE is not only light code and efficient in training, but also flexible across platforms. PINN greatly reduces the need for labeled data: when the nonlinearity of the ODE is weak, a very small amount of supervised training data plus a small amount of collocation points are sufficient to predict the solution; in the minimalist case, only one or two training points (with initial values) are needed for first- or second-order ODEs, respectively. Strongly nonlinear ODE also require only an appropriate increase in the number of training and collocation points, which still has significant advantages over conventional NN. With the aid of collocation points and the use of physical information, PINN has the ability to extrapolate data outside the time domain covered by the training set, and is robust to noisy data, thus with enhanced generalization capabilities. Training is accelerated when the gains obtained along with the reduction in the amount of data outweigh the delay caused by the increase in the loss function terms. The soft-constrained PINN method can easily impose a physical law (e.g., energy conservation) constraint by adding a regularization term to the total loss function, thus improving the solution performance of ODEs that obey this physical law.