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Every definable neural network can learn from finite data

A new preprint in collaboration with Anastasis Kratsios, Greg Cousins, Haitz Sáez de Ocáriz Borde, and Bum Jun Kim is out!

https://arxiv.org/abs/2605.07097

We show that  every definable neural network can successfully perform classification and regression from a finite dataset, in the sense of Probably Approximately Correct (PAC) learnability. Definability is formalized using o-minimality, a notion borrowed from logic and model theory. Notably, virtually all the architectures and individual components considered in practice are definable in this sense (...incredible, but true!). This inlcudes MLPs, CNNs, GNNs, and transformers with fixed sequence length. A key implication of our results is that finite-sample PAC learnability should be considered a baseline rather than a differentiator between neural networks architectures, since it holds for practically every model.

A short tour of operator learning

I am happy to announce a new paper just uploaded to the arXiv with Nicola Rares Franco and Nick Nelsen:

https://arxiv.org/abs/2603.00819

We provide a concise (12-page) survey of recent theoretical developments in operator learning, a fast-evolving research area at the intersection of numerical analysis, statistical learning and approximation theory. The goal is to learn operators between function (e.g., Hilbert) spaces, motivated by applications such as Partial Differential Equations (PDEs) with paramteric coefficients. In our paper, we review recently established converence rates for holomorphic operators and fundamental learnability limits for different operator classes according to the mimimax framework. We conclude by identifying promising research avenues for future work.

This "mini survey paper" is my entry point in the field and writing it with Nick and Nicola has been a great learning experience. I hope this paper will be useful to researchers who want to get started in operator learning theory!