Physics-informed Lane-Emden solvers using Lynx-Net: Implementing radial basis functions in Kolmogorov representation

This paper introduces a novel approach for solving Lane-Emden equations using Lynx-Net, a Physics-Informed Neural Network that integrates Radial Basis Functions (RBFs) within the Kolmogorov representation framework. Lynx-Net addresses these challenges by combining RBF-enhanced function approximation with physics-informed constraints: differential-equation residuals are enforced during training, ensuring stability and rapid convergence. Across a spectrum of polytropic indices, our experiments show that Lynx-Net consistently outperforms prior machine-learning approaches, achieving lower errors without incurring excessive computational cost. The proposed model leverages the function approximation capabilities of RBFs and physics-informed constraints to enhance solution stability and convergence. By incorporating differential equation residuals into the learning process, Lynx-Net minimizes errors while maintaining computational efficiency. Experimental results across multiple test cases demonstrate its superiority over conventional solvers and existing machine learning-based approaches. This research highlights the potential of integrating RBFs with PINNs for solving nonlinear differential equations, providing a scalable and efficient framework applicable to broader problems in astrophysics and engineering.