Implementing physics-informed neural networks with deep learning for differential equations.

Physics-aware machine learning integrates domain-specific physical knowledge into machine learning models, leading to the development of physics-informed neural networks (PINNs). PINNs embed physical laws directly into the learning process, enabling interpretable and physically consistent solutions to complex problems. However, the practical use of PINNs presents challenges and their applications are complex. Therefore, in this paper, we demonstrate the implementation of PINNs for systems of ordinary differential equations (ODEs), an area that is often overlooked by the physics community, which typically focuses on partial differential equations. We discuss two key challenges: the inverse problem, which involves estimating unknown parameters of ODEs, and the forward problem, which provides an approximate solution to ODEs. To provide practical insights into PINNs, we present two case studies based on a Python implementation using DeepXDE. Drawing on these studies, we discuss key challenges and identify promising directions for future research in PINN-based implementation frameworks.