The Mathematics of Artificial Intelligence

Overview

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Explore the mathematics behind artificial intelligence in this 45-minute lecture by Gitta Kutyniok. Delve into the impact of AI on public life, sciences, and mathematics, focusing on inverse problems and numerical analysis of partial differential equations. Gain insights into the development of a mathematical foundation for AI and novel approaches to solving complex mathematical problems. Examine fundamental limitations of AI methodologies, particularly in terms of computability. Learn about neural networks, generalization capabilities, explainability, and the application of AI to mathematical challenges such as image processing and computed tomography. Discover the potential and limitations of deep neural networks in solving partial differential equations, and consider the implications of computability on digital machines for AI development.

Syllabus

Intro
The Dawn of Artificial Intelligence in Public Life
Spectacular Success in Science
Impact on Mathematical Problem Settings
Artificial Intelligence = Alchemy?
Problem with Reliability
Role of Mathematics Two Key Challenges for Mathematics
First Appearance of Neural Networks
Artificial Neurons
Affine Linear Maps and Weights
Definition of a Deep Neural Network
Training of Deep Neural Networks
Mathematics for Artificial Intelligence
Glimpse into Generalization
Graph Neural Networks
A Special Form of Generalization Capability
Generalization Result
Glimpse into Explainability
Artificial Intelligence for Mathematics
Anisotropic Structures as Model for Images
(Cone-adapted) Discrete Shearlet Systems
Optimally Sparse Approximation Theorem (K. Lin, 2011)
Solving Inverse Problems
(Limited Angle) Computed Tomography
Zooming in on the Limited-Angle CT Problem
Numerical Results
Deep Network Shearlet Edge Extractor (DeNSE)
Numerical Deep Learning Approaches to PDES
What can Deep Neural Networks do?
Theoretical Results
A Serious Problem Computability on Digital Machines (informal)
Some Thoughts on the Result
Conclusions