# STAR-seminars: Luca Galimberti

The webinars will take place on Zoom and a link to the virtual room will be sent out to all those who registered at the registration page.

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Speaker: Luca Galimberti (NTNU)

Title: Neural Networks in Fréchet spaces

Abstract: In this talk we present some novel results obtained by Fred Espen Benth (UiO), Nils Detering (University of California Santa Barbara) and myself on abstract neural networks and deep learning. More precisely, we derive an approximation result for continuous functions from a Fréchet space $$\mathfrak X$$ into its field $$\mathbb{F}, (\mathbb{F}\in\{\mathbb{R},\mathbb{C} \})$$. The approximation is similar to the well known universal approximation theorems for continuous functions from $$\mathbb{R}^n$$ to $$\mathbb{R}$$ with (multilayer) neural networks [1, 2, 3, 4]. Similar to classical neural networks, the approximating function is easy to implement and allows for fast computation and fitting.

Few applications geared toward derivative pricing and numerical solutions of parabolic partial differential equations will be outlined.

[1] G. Cybenko. Approximation by superpositions of a sigmoidal function. Mathematics of Control, Signals and Systems, 2(4):303–314, 1989.
[2] K. Hornik, M. Stinchcombe, and H. White. Multilayer feedforward networks are universal approximators. Neural Networks, 2(5):359–366, 1989.
[3] K.-I. Funahashi. On the approximate realization of continuous mappings by neural networks. NeuralNetworks, 2(3):183–192, 1989.
[4] M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks, 6(6):861–867, 1993.

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This series of webinars addresses all interested people in probability, stochastic analysis, control, risk evaluation, statistics, with a view towards applications, in particular to renewable energy markets and production. This series brings together the major research themes of the projects STORM, SCROLLER, and SPATUS

## Organizer

Fred Espen Benth, Kristina Rognlien Dahl, Giulia Di Nunno, Salvador Ortiz-Latorre and Anton Yurchenko-Tytarenko
Published Sep. 16, 2021 10:36 AM - Last modified Sep. 16, 2021 10:38 AM