Thoughts on Machine Learning
Dienstag, 20.1.26, 14:15-15:15, Seminarraum 226, HH10
Techniques of machine learning (ML) find a rapidly increasing range of applications touching upon social, economic, and technological aspects of everyday life. They are also being used with great enthusiasm to fill in gaps in our scientific knowledge by data-based modelling approaches. I have followed these developments for a while with interest, concern, and mounting disappointment. When these technologies are employed to take over decisive functionality in safety-critical applications, we would like to exactly know how to guarantee their compliance with pre-defined guardrails and limitations. Moreover, when they are utilized as building blocks in scientific research, it would violate scientific standards -in my opinion- if these building blocks were used without a throrough understanding of their functionality, including inaccuracies, uncertainties, and other pitfalls. In this context, I will juxtapose (a subset of) deep neural network methods with the family of entropy-optimal Sparse Probabilistic Approximation (eSPA) and entropy-optimal network (EON) techniques developed recently by Illia Horenko (RPTU Kaiserslautern-Landau) and colleagues.
On some results for Threshold Diffusions
Dienstag, 13.1.26, 14:15-15:15, Seminarraum 226, HH10
Threshold diffusions are solutions to stochastic differential equations whose coefficients change discontinuously depending on the position of the process relative to certain barriers (thresholds). They are used to model a variety of phenomena in finance/economics, engineering, and other sciences. Many questions remain about existence and uniqueness, numerical approximation, and statistical inference for these processes. This seminar will primarily offer a mathematical introduction to some of these processes. We will also consider parameter inference problems, presenting asymptotic results under different assumptions and highlighting technical challenges arising, in particular, from the interplay between discontinuities and local times.
Geometric Optimization in Scientific Machine Learning
Dienstag, 16.12.25, 14:15-15:15, Seminarraum 226, HH10
We discuss an “optimize-then-project” approach for applications in scientific machine learning. The key idea is to design algorithms at the infinite-dimensional level and subsequently discretize them in the tangent space of the neural network ansatz. We illustrate this approach in the context of the variational Monte Carlo method for quantum many-body problems, where neural quantum states have recently emerged as powerful representations of high-dimensional wavefunctions. In this setting, we recover the celebrated stochastic reconfiguration algorithm, interpreting it as a projected Riemannian L2 gradient descent method. We further explore extensions to Riemannian Newton methods, and conclude with considerations related to the scalability of these schemes.
Space-time least squares approximation for Schrödinger equation and efficient solver
Dienstag, 2.12.25, 14:15-15:15, Seminarraum 226, HH10
We propose a space-time least-squares Galerkin formulation for the numerical solution of the Schrödinger equation, which overcomes the numerical instability of the plain Galerkin formulation in space-time and provides a well-posed method with quasi-optimal convergence. When discretized on a tensor product grid, the arising linear system is a sum of Kronecker product matrices. We propose a preconditioning strategy following a variant of the Fast Diagonalization approach. We also derive an explicit bound for the eigenvalues of the preconditioned system when a tensor product spline space is considered for the discretization. This bound is independent of the mesh size and depends on the quasi-uniformity parameter of the mesh and the spline degree p. Numerical results validate the theoretical convergence and demonstrate the computational efficiency of the approach.
A greedy reconstruction algorithm for minimal neural network architectures
Dienstag, 18.11.25, 14:15-15:15, Seminarraum 226, HH10
n many machine learning applications the choice of an appropriate/optimal neural network architecture is based purely on heuristic experience, or determined by trial and error. Moreover, even when a “good” network is found, it is a common issue that the training data is not distributed evenly, leading to bias in the networks.
To address these issues, we introduce a new greedy algorithm that selects simultaneously a subset of optimal training data points and the smallest neural network that is able to learn the selected data, while also representing well the non-selected data. By this approach, we are able to keep a perfect balance between under- and overfitting. Additionally, the non-selected training data is turned into validation data, which is especially useful in settings where only limited data is available.
We demonstrate the effectiveness of our new method by numerical experiments for function approximation and classification problems. This talk is based on a joint work with Gabriele Ciaramella and Marco Verani.
Remarks to exact Poincaré Constants in n-dimensional Annuli and Balls
Dienstag, 11.11.25, 14:15-15:15, Seminarraum 226, HH10
We study n-dimensional annuli and n-dimensional balls, where we suppose n ∈ {2,..,N} with N < ∞. We investigate in our non-dimensional setting each annulus ΩA- defined via two concentrical balls with radii A/2 and A/2 + 1 in Rn - and n-dimensional open unit balls as ”limits” of ΩAfor A → 0. We provide calculated (precise) Poincar´ e constants for scalar functions (with vanishing Dirichlet traces on the boundary) in dependence of the inner diameter A and the dimension nof the space Rn for these geometries. Addi- tionally we lay open the direct match of the Poincar´ e constants for solenoidal vector fields and the Poincaré constants for scalar functions (both with vanishing Dirichlet traces on the boundary) for solenoidal vector in space R2 resp. R3 with the Poincar´ e constants for scalar functions in R4 resp. R5. Generally we use the first eigenvalues of the scalar Laplacian (or the first eigenvalues the Stokes operator) for the calculation of the Poincar´ e constants. Supplementary, corresponding problems in domains Ω∗ σ (cf. e.g. the 3d-annuli from [12]) are investigated - for comparison but also to provide the limits for A → 0. These domains Ω∗ σ enable us to use the Green’s function of the Laplacian on Ω∗ σ with vanishing Dirichlet traces on ∂Ω∗ σ to show that for σ → 0 the first eigenvalue here tends to the first eigenvalue of the corresponding problem on the open unit ball in Rn . On the other hand, we take advantage of the so-called small-gap limit for A → ∞ like in our papers to Poincar´ e constants in annuli (cf. [10] and [11]).