Sobolev Norm

Sobolev norms measure the smoothness of functions by incorporating both function values and derivatives, finding applications in diverse fields like machine learning and partial differential equations. Current research focuses on understanding how Sobolev norms impact the approximation capabilities of various models, including shallow and deep neural networks, kernel methods, and generative adversarial networks (GANs), often within the context of minimizing approximation error or optimizing learning algorithms. This work is significant because it provides theoretical foundations for improving the efficiency and accuracy of these models, leading to advancements in areas such as image generation, solving inverse problems, and approximating solutions to PDEs.