Error Analysis of Physics-Informed Neural Networks (PINNs) in Typical Dynamical Systems

Neural network (NN) related research has grown significantly due to its accuracy in estimating solutions after learning the behavior of the dataset. Throughout the learning process, NN uses backpropagation feedback to identify hidden patterns and correlations in raw data, classify them, and continuously improve over time. However, models constructed by NN often lack model reasoning. To say, NN could quickly build a predictive or generative model from a dataset but not be able to explain why the model works. Hence, to overcome this, the Physics-Informed Neural Networks (PINNs) method is introduced, which respects the laws of physics and embeds prior scientific knowledge into machine learning models. We use PINNs to compare the NN solutions with the numerical solution of some typical dynamical systems. Specifically, differential equations are used to formulate dynamical processes, which are used as a representation of modeling done with reasoning. We will demonstrate the use of PINNs in solving the system of first-order ODE and stiff system. The resulting PINNs solutions are expected to reproduce the character of the trajectory. We will illustrate the differences between numerical and PINNs methods to the ODE system and comment on the ability of the PINNs model to reproduce the unique solution to the initial value problem.