3 papers
cs.LG2020
Numerically Solving Parametric Families of High-Dimensional Kolmogorov Partial Differential Equations via Deep Learning
Julius Berner, Markus Dablander, Philipp Grohs
We present a deep learning algorithm for the numerical solution of parametric families of high-dimensional linear Kolmogorov partial differential equations (PDEs). Our method is ba…
cs.LG2019
Towards a regularity theory for ReLU networks -- chain rule and global error estimates
Julius Berner, Dennis Elbrächter, Philipp Grohs +1
Although for neural networks with locally Lipschitz continuous activation functions the classical derivative exists almost everywhere, the standard chain rule is in general not app…
cs.LG2018
Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black-Scholes Partial Differential Equations
Julius Berner, Philipp Grohs, Arnulf Jentzen
The development of new classification and regression algorithms based on empirical risk minimization (ERM) over deep neural network hypothesis classes, coined deep learning, revolu…