This lecture explores the mathematical foundations underlying neural network approximation, focusing on the development of rigorous theories that explain how and why neural networks approximate functions effectively. We talk about key topics such as error estimation, convergence analysis, and the role of activation functions in enhancing network performance. Additionally, the lecture will demonstrate convergence analysis in the context of scientific machine learning, further bridging the gap between empirical success and theoretical understanding. Our goal is to provide deeper insights into the mechanisms driving neural network efficiency, reliability, and their applications in scientific computing.
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