Lipschitz Operator

Lipschitz operators, functions with bounded output sensitivity to input changes, are central to various machine learning and scientific computing problems, with research focusing on efficiently approximating and learning them. Current efforts concentrate on analyzing the approximation capabilities of neural networks, particularly deep operator networks and shallow ReLU networks, and developing algorithms for estimating Lipschitz constants for different architectures, including convolutional networks. Understanding the limitations and capabilities of these approaches is crucial for improving the robustness, generalization, and efficiency of machine learning models and numerical methods across diverse applications.