🤖 AI Summary
Conventional ODE numerical methods often suffer from poor convergence in complex scenarios involving high stiffness, discontinuous boundaries, or singular perturbations. To address this, this paper proposes an enhanced Physics-Informed Neural Network (PINN) framework. Methodologically, it integrates structural priors with hard physical constraints, formulates a weighted composite loss comprising data fidelity, initial-condition satisfaction, and PDE residual minimization, and incorporates adaptive spatiotemporal sampling, multi-activation-function co-optimization, and systematic hyperparameter tuning. Key innovations include a novel loss-balancing mechanism, improved embedding of physical constraints via hard enforcement, and enhanced training stability. Extensive experiments on diverse classical ODE benchmarks demonstrate substantial improvements in solution accuracy and convergence robustness—particularly for strongly nonlinear and irregular dynamical systems—while exhibiting superior generalization capability.
📝 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.