Neural Networks for Non-convex Lattice Geometry: Extending Stoka’s Theory with Machine Learning Applications to Financial Risk and Economic Forecasting

We establish a comprehensive theoretical and computational framework connecting Marius Stoka’s classical geometric probability theory for non-convex lattices with modern neural network approximation methods. Our contribution consists of two fundamental theorems: first, we prove universal approximation capabilities of deep neural networks for geometric intersection probabilities with explicit convergence guarantees; second, we develop an optimization framework using reinforcement learning for multi-dimensional generalized Buffon problems. The framework demonstrates significant applications in financial risk assessment, economic forecasting, and quantitative modeling where geometric constraints naturally arise. Experimental validation on both synthetic and real financial datasets confirms the practical utility of our approach for contemporary risk management and portfolio optimization problems.