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

Area of Science:

• Computational Physics
• Machine Learning

Background:

• Physics-aware machine learning, particularly physics-informed neural networks (PINNs), integrates physical laws into machine learning models.
• PINNs enable interpretable and physically consistent solutions but face practical implementation challenges.

Purpose of the Study:

• To demonstrate the implementation of PINNs for systems of ordinary differential equations (ODEs).
• To address the forward problem (solving ODEs) and the inverse problem (parameter estimation) for ODEs using PINNs.
• To provide practical insights and identify future research directions for PINN frameworks.

Main Methods:

• Implementation of PINNs for systems of ODEs.
• Utilized a Python-based framework, DeepXDE, for practical case studies.
• Investigated both forward and inverse problem formulations within the ODE context.

Main Results:

• Successfully demonstrated PINN implementation for ODE systems, covering both forward and inverse problems.
• Presented two case studies offering practical insights into the application of PINNs for ODEs.
• Highlighted key challenges and potential future research avenues in PINN frameworks.

Conclusions:

• PINNs offer a viable and powerful tool for addressing ODEs, an area less explored than PDEs in the physics community.
• Practical implementation through frameworks like DeepXDE facilitates the application of PINNs.
• Further research is needed to overcome current challenges and expand the utility of PINNs for ODEs.