Generalızed Kuramoto-Sıvashınsky equatıon solvıng by B-splınes based neural network

This study presents a comprehensive case study investigating the application of Variational Physics-Informed Neural Networks (VPINNs) combined with B-spline test functions for solving the generalized Kuramoto-Sivashinsky equation in one space dimension. The Kuramoto-Sivashinsky equation, a nonlinear fourth-order PDE, serves as a benchmark for modeling instabilities in fluid dynamics, combustion, and pattern formation. Our methodology leverages variational formulations to enhance solution accuracy while ensuring robust convergence. Through systematic numerical experiments, we demonstrate that VPINNs effectively capture complex dynamical behaviors in high-order nonlinear systems. The framework’s robustness is evaluated across different network architectures and multi-stage optimization strategies using Adam and L-BFGS optimizers. Results establish VPINNs as a powerful tool for addressing complex nonlinear PDEs while maintaining computational efficiency. This study provides insights into practical VPINNs implementation for scientific computing, with future directions focusing on computational optimization and adaptive collocation strategies.