Aprendizado de máquina aplicado em equações diferenciais com soluções do tipo sóliton com variação de parâmetros.

Abstract

This work introduces an extended version of the Physics-Informed Neural Networks (PINNs) framework to solve parametrized partial differential equations (PDEs) by incorporating an additional input dimension for a variable parameter 𝑚. This enhancement enables the model to generalize solutions across a specified parameter range without retraining for each value. Notably, the extended model maintained training times comparable to the standard approach (≈280 seconds), demonstrating computational efficiency despite the added dimension. Initially, the model was validated on a second-order ordinary differential equation (ODE) with variable coefficients, effectively generalizing solutions over parameter ranges. The investigation then progressed to the nonlinear Korteweg-de Vries (KdV) equation, demonstrating the model’s ability to simulate complex soliton interactions using limited initial data points. Optimal network architectures were identified for different scenarios, emphasizing the importance of hyperparameter selection. Although physical constraints were easily implemented using DeepXDE, no significant accuracy improvements were observed, suggesting that the equation and initial conditions provided sufficient restrictions. The primary focus was on the Sine-Gordon equation, a nonlinear PDE with soliton solutions, to evaluate the model’s ability to generalize for variable parameter 𝑚. Accurate solutions were achieved with minimal data, highlighting the model’s efficiency. Challenges arose near 𝑚 = 0, where the equation approaches a linear wave form, reducing the model’s generalization capacity. Despite this, the approach consistently delivered reliable solutions across most of the parameter range, demonstrating its potential for efficiently solving parametrized PDEs