Seminar abstracts: SoSe 2025
Minimax learning rates for estimating binary classifiers under margin conditions
Jonathan García Rebellón, 07.04.2025
We bound the minimax error of estimating binary classifiers when the decision boundary of the learning set can be described by horizon functions in a general class and a margin condition is satisfied. Then, we apply these bounds to particular cases of some classes of functions, such as convex functions or Barron-regular functions. In addition, we show that classifiers with a Barron-regular decision boundary can be approximated by ReLU neural networks, with an optimal rate independent of the dimension that is close to $n^{-1}(1+\log n)$ when the margin is large compared to around $n^{-1/3}$ when no margin condition is assumed, here $n$ is the number of training samples. To conclude, we present some numerical experiments that confirm the theoretical results. This is a joint work with Philipp Petersen.
We bound the minimax error of estimating binary classifiers when the decision boundary of the learning set can be described by horizon functions in a general class and a margin condition is satisfied. Then, we apply these bounds to particular cases of some classes of functions, such as convex functions or Barron-regular functions. In addition, we show that classifiers with a Barron-regular decision boundary can be approximated by ReLU neural networks, with an optimal rate independent of the dimension that is close to $n^{-1}(1+\log n)$ when the margin is large compared to around $n^{-1/3}$ when no margin condition is assumed, here $n$ is the number of training samples. To conclude, we present some numerical experiments that confirm the theoretical results. This is a joint work with Philipp Petersen.
Generative AI for the Statistical Computation of Fluids
Samuel Lanthaler, 31.03.2025
In recent years, there has been growing interest in applying neural networks to the data-driven approximation of partial differential equations (PDEs). This talk will discuss a generative AI approach for fast, accurate, and robust statistical computation of three-dimensional turbulent fluid flows. On a set of challenging fluid flows, this approach provides an accurate approximation of relevant statistical quantities of interest while also efficiently generating high-quality realistic samples of turbulent fluid flows. This stands in stark contrast to ensemble forecasts from deterministic machine learning models, which are observed to fail on these challenging tasks. This talk will highlight theoretical results that reveal the underlying mechanisms by which generative AI models can succeed in capturing key physical properties where deterministic ML approaches fall short.
In recent years, there has been growing interest in applying neural networks to the data-driven approximation of partial differential equations (PDEs). This talk will discuss a generative AI approach for fast, accurate, and robust statistical computation of three-dimensional turbulent fluid flows. On a set of challenging fluid flows, this approach provides an accurate approximation of relevant statistical quantities of interest while also efficiently generating high-quality realistic samples of turbulent fluid flows. This stands in stark contrast to ensemble forecasts from deterministic machine learning models, which are observed to fail on these challenging tasks. This talk will highlight theoretical results that reveal the underlying mechanisms by which generative AI models can succeed in capturing key physical properties where deterministic ML approaches fall short.