Physics-Informed Neural Network Frameworks for the Analysis of Engineering and Biological Dynamical Systems Governed by Ordinary Differential Equations

TL;DR

This paper demonstrates that Physics-Informed Neural Networks (PINNs) can effectively solve complex ODE-governed systems in engineering and biology, outperforming traditional methods in challenging scenarios by embedding physical laws into neural network training.

Contribution

The study systematically evaluates PINNs for solving complex ODE problems, highlighting the importance of hyperparameter tuning and constraint embedding for improved accuracy and convergence.

Findings

PINNs can handle high stiffness and irregular domains better than traditional methods.

Proper balancing of loss components is crucial for convergence.

Embedding prior knowledge enhances predictive accuracy.

Abstract

In this study, we present and validate the predictive capability of the Physics-Informed Neural Networks (PINNs) methodology for solving a variety of engineering and biological dynamical systems governed by ordinary differential equations (ODEs). While traditional numerical methods a re effective for many ODEs, they often struggle to achieve convergence in problems involving high stiffness, shocks, irregular domains, singular perturbations, high dimensions, or boundary discontinuities. Alternatively, PINNs offer a powerful approach for handling challenging numerical scenarios. In this study, classical ODE problems are employed as controlled testbeds to systematically evaluate the accuracy, training efficiency, and generalization capability under controlled conditions of the PINNs framework. Although not a universal solution, PINNs can achieve superior results by embedding physical laws directly into the learning process. We first analyze the existence and uniqueness properties of several benchmark problems and subsequently validate the PINNs methodology on these model systems. Our results demonstrate that for complex problems to converge to correct solutions, the loss function components data loss, initial condition loss, and residual loss must be appropriately balanced through careful weighting. We further establish that systematic tuning of hyperparameters, including network depth, layer width, activation functions, learning rate, optimization algorithms, w eight initialization schemes, and collocation point sampling, plays a crucial role in achieving accurate solutions. Additionally, embedding prior knowledge and imposing hard constraints on the network architecture, without loss the generality of the ODE system, significantly enhances the predictive capability of PINNs.