Physics-Informed Machine Learning for the Numerical Solution of Linear Oscillatory Systems

This study explores the application of Physics-Informed Neural Networks (PINNs) for solving systems of ordinary differential equations (ODES), with a particular focus on capturing oscillatory solutions. Initial experiments with a simple neural network (SNN) without physics-informed constraints revealed significant limitations in purely data-driven approaches, emphasizing the necessity of embedding physical laws into the learning process. By incorporating governing equations into the loss function, PINNS offer a robust framework for approximating solutions beyond the training domain. However, early implementations of PINNS with curriculum learning faced challenges, including phase shifts in predicted oscillations and sensitivity to activation functions. To enhance accuracy, we introduced the activation function f(x) = x + sin(x), which improved the network's capacity to learn oscillatory behaviors. Additionally, systematic tuning of curriculum learning parameters and loss function balancing proved critical in refining performance. The optimized PINN approach successfully captured the oscillatory dynamics of the ODE system, closely matching numerical solutions. These findings highlight the importance of activation function design, curriculum learning strategies, and loss function optimization in training PINNS effectively. Future work will focus on adaptive curriculum learning, theoretical analysis of activation function properties, and application to more complex differential equation problems.

کلیدواژه ها:

Oscillatory ، Physics-Informed Neural Networks ، Solutions ، Curriculum Learning ، Differential Equations