VARIATIONAL METHODS FOR NEURAL NETWORK TRAINING: APPLICATIONS OF STURM-LIOUVILLE ENERGY ESTIMATES

This paper establishes a novel connection between local minimization principles for Sturm-Liouville equations and optimization techniques used in training neural networks. By interpreting the training of neural networks as a variational problem, we demonstrate how recent results on energy estimates for mixed boundary value problems in Sturm-Liouville theory can be adapted to analyze and improve neural network convergence. We present two main theorems: the first establishes conditions for guaranteed convergence to non-zero local minima in neural network training, and the second demonstrates the existence of multiple critical points with energy estimates. Our theoretical results are supported by experimental validation on benchmark datasets, showing improved performance in avoiding trivial solutions during training. This work bridges the gap between classical differential equation theory and modern machine learning optimization.