150 citations · 323 across the 10 of their papers we have counts for
10 papers
Towards accurate extreme event likelihoods from diffusion model climate emulators
Peter Manshausen, Noah Brenowitz, Julius Berner +2
ML climate model emulators are useful for scenario planning and adaptation, allowing for cost-efficient experimentation. Recently, the diffusion model Climate in a Bottle (cBottle)…
Improved sampling via learned diffusions
Lorenz Richter, Julius Berner
Recently, a series of papers proposed deep learning-based approaches to sample from target distributions using controlled diffusion processes, being trained only on the unnormalize…
An optimal control perspective on diffusion-based generative modeling
Julius Berner, Lorenz Richter, Karen Ullrich
We establish a connection between stochastic optimal control and generative models based on stochastic differential equations (SDEs), such as recently developed diffusion probabili…
Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning
Lorenz Richter, Julius Berner
The combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related lear…
Learning ReLU networks to high uniform accuracy is intractable
Julius Berner, Philipp Grohs, Felix Voigtlaender
Statistical learning theory provides bounds on the necessary number of training samples needed to reach a prescribed accuracy in a learning problem formulated over a given target c…
The Modern Mathematics of Deep Learning
Julius Berner, Philipp Grohs, Gitta Kutyniok +1
We describe the new field of mathematical analysis of deep learning. This field emerged around a list of research questions that were not answered within the classical framework of…
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…
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…
How degenerate is the parametrization of neural networks with the ReLU activation function?
Julius Berner, Dennis Elbrächter, Philipp Grohs
Neural network training is usually accomplished by solving a non-convex optimization problem using stochastic gradient descent. Although one optimizes over the networks parameters,…
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…