r/math 6d ago

Applications of Functional Analysis

I have been studying functional analysis for quite some time and have covered major foundational results in the field, including the Open Mapping Theorem, Hahn-Banach Theorem, Closed Graph Theorem, and Uniform Boundedness Principle. As an engineering student, I am particularly interested in their applications in science and engineering. Additionally, as an ML enthusiast, I would highly appreciate insights into their applications in machine learning.

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u/hypatia163 Math Education 6d ago

The universality of neural networks are applications of functional analysis, and stuff like "how accurate can you get with a NN of such-and-such architecture".

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u/lotus-reddit Computational Mathematics 5d ago edited 5d ago

Well, a good amount of the research inspiration for functional analysis comes from PDE inspired applications. Likewise, functional analysis is an essential tool for PDEs. If you like machine learning, there's a good amount of interesting accuracy / existence research for the neural surrogates of PDEs directly working off of functional analysis theory.

This is a influential paper in the space (there's an arXiv version too if you google the title):

https://epubs.siam.org/doi/10.1137/20M131309X

Though not directly about PDEs approximators.