Resumo:
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
