Abstract:
This thesis comprises a series of publications that contribute to the emerging field of mathematical analysis of
deep learning. The term deep learning refers to machine learning methods that use gradient-based optimization
techniques to fit the parameters of deep neural networks to given data. Over the past decade, such approaches have catalyzed unprecedented advances across a wide range of applications. While a comprehensive
mathematical explanation for their success remains elusive, this work provides fundamental insights that
improve the theoretical understanding of deep learning. To facilitate a rigorous analysis, we focus on learning
problems with known regularity properties, as frequently encountered in the context of differential equations.
Specifically, we analyze deep learning algorithms for the numerical solution of a class of partial differential
equations, known as Kolmogorov equations, employing representations based on stochastic differential
equations. It is demonstrated that empirical risk minimization over deep neural networks efficiently
approximates the solutions of families of Kolmogorov equations, with both the size of the neural networks and
the number of samples scaling only polynomially in the underlying dimension. Furthermore, we introduce
variance-reduced loss functions and identify settings in which local minima of corresponding optimization
problems are nearly optimal. On the other hand, we also address the shortcomings of deep learning and
establish fundamental constraints on learning neural networks from samples. Extensive numerical experiments
corroborate the potential of deep learning to overcome the curse of dimensionality while revealing its inherent
limitations. This comprehensive investigation contributes toward principled and reliable applications of deep
learning in the natural sciences.

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