Data-Driven Deep Learning Algorithms for Dynamical Systems

Abstract

Artificial neural networks have revolutionized scientific problem-solving, offering reliable techniques for modeling and understanding complex processes. This trend is most apparent in dynamical system simulation, where experts continuously seek methods to increase accuracy and efficiency. Learning the time-varying parameters of a dynamical system is essential, especially when trying to understand how a system reacts to different situations over time. We present three deep learning algorithm approaches to address the learning of time-varying parameters from a dynamical system. The first algorithm, a logistic-informed neural network, is motivated by using physics-informed neural networks on logistic differential equations to predict the number of individuals infected by the COVID-19 Omicron variant. This algorithm learns the time-varying parameters of four mathematical models to predict individuals infected with the COVID-19 Omicron in a country with strict and partial mitigation measures. The second algorithm, optimized physics-informed neural networks, allows us to understand nonlinear dynamics in various fields, such as biochemistry, ecology, and epidemiology. By optimizing the model, a constant parameter of a system of ordinary differential equations can be learned as a time-varying parameter, revealing hidden patterns in complex systems. Finally, we tackle the difficulties of learning the time-varying parameter and solving stiff dynamical systems by introducing a novel approach called physics-informed transfer learning neural network. This model is developed by transferring prior knowledge of optimized neural network parameters and pre-training using physics-informed neural networks with a simple adaptive scheme to learn the time-varying parameters of stiff dynamical systems. These algorithms improve neural network's capacities for analyzing dynamic systems, learning time-varying parameters, and predicting system behavior. We are confident that our research significantly contributes to the journey toward more complex and accurate modeling of dynamic processes.