Seminar abstracts: SoSe 2025
Solving the Schrödinger Equation using Neural Networks
Michael Scherbela, 26.05.2025
Solving the Schrödinger Equation allows in principle to compute any property of any molecule, but solving it in practice is hard, due to the high dimensionality of the problem and the required accuracy. Due to these challenges there are still no numerical solvers, which can efficiently and reliably solve it even for relatively small systems. Recently a method known as Neural Network Wavefunctions have emerged as a promising approach: For many small molecules they yield highly accurate solutions, outperforming all conventional solvers, and provide in principle favorable scaling with system size n. In this talk I will give a brief informal introduction into Neural Network Wavefunctions and then focus on our latest preprint, which reduces the complexity of Neural Wavefunction by O(n) and yields more than 10x speedups in practice. arxiv.org/abs/2504.06087
Solving the Schrödinger Equation allows in principle to compute any property of any molecule, but solving it in practice is hard, due to the high dimensionality of the problem and the required accuracy. Due to these challenges there are still no numerical solvers, which can efficiently and reliably solve it even for relatively small systems. Recently a method known as Neural Network Wavefunctions have emerged as a promising approach: For many small molecules they yield highly accurate solutions, outperforming all conventional solvers, and provide in principle favorable scaling with system size n. In this talk I will give a brief informal introduction into Neural Network Wavefunctions and then focus on our latest preprint, which reduces the complexity of Neural Wavefunction by O(n) and yields more than 10x speedups in practice. arxiv.org/abs/2504.06087
Know your numbers: Floating-point arithmetic and numerical analysis in machine learning
Stanislav Budzinskiy, 19.05.2025
One of the main selling points of modern GPUs is their support of low-precision numbers: the lower the precision, the faster the computation. "The more efficient the training of my neural network," one may conclude immediately, forgetting to acknowledge the effects of rounding errors due to low-precision floating-point computations. In this talk, I will discuss the basics of floating-point numbers and the numerical analysis of rounding errors. They are at the centre of our research project with Philipp Petersen, and I will share some of the results obtained so far.
One of the main selling points of modern GPUs is their support of low-precision numbers: the lower the precision, the faster the computation. "The more efficient the training of my neural network," one may conclude immediately, forgetting to acknowledge the effects of rounding errors due to low-precision floating-point computations. In this talk, I will discuss the basics of floating-point numbers and the numerical analysis of rounding errors. They are at the centre of our research project with Philipp Petersen, and I will share some of the results obtained so far.
Consistency of augmentation graph and network approximability in contrastive learning
Martina Neuman, 12.05.2025
Contrastive learning leverages data augmentation to develop feature representation without relying on large labeled datasets. However, despite its empirical success, the theoretical foundations of contrastive learning remain incomplete, with many essential guarantees left unaddressed, particularly the realizability assumption concerning neural approximability of an optimal spectral contrastive loss solution. We overcome these limitations by analyzing the pointwise and spectral consistency of the augmentation graph Laplacian. We establish that, under specific conditions for data generation and graph connectivity, as the augmented dataset size increases, the augmentation graph Laplacian converges to a weighted Laplace-Beltrami operator on the natural data manifold. These consistency results in turn give way to a robust framework for establishing neural approximability, directly resolving the realizability assumption in a current paradigm. The talk is based on the paper of the same title: arxiv.org/pdf/2502.04312. I will begin with brief introductions to manifold learning and the current paradigm of contrastive learning. I will then present our manifold learning analysis used to address the realizability assumption in contrastive learning.
Contrastive learning leverages data augmentation to develop feature representation without relying on large labeled datasets. However, despite its empirical success, the theoretical foundations of contrastive learning remain incomplete, with many essential guarantees left unaddressed, particularly the realizability assumption concerning neural approximability of an optimal spectral contrastive loss solution. We overcome these limitations by analyzing the pointwise and spectral consistency of the augmentation graph Laplacian. We establish that, under specific conditions for data generation and graph connectivity, as the augmented dataset size increases, the augmentation graph Laplacian converges to a weighted Laplace-Beltrami operator on the natural data manifold. These consistency results in turn give way to a robust framework for establishing neural approximability, directly resolving the realizability assumption in a current paradigm. The talk is based on the paper of the same title: arxiv.org/pdf/2502.04312. I will begin with brief introductions to manifold learning and the current paradigm of contrastive learning. I will then present our manifold learning analysis used to address the realizability assumption in contrastive learning.
Degenerative Diffusion Models
Dennis Elbrächter, 05.05.2025
I will give a brief and biased introduction to Diffusion models and present some of my own work on the topic, specifically, a modification of the inference process. The mathematical basis is rather heuristic, but it seems to yield some interesting empirical results.
I will give a brief and biased introduction to Diffusion models and present some of my own work on the topic, specifically, a modification of the inference process. The mathematical basis is rather heuristic, but it seems to yield some interesting empirical results.
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.