Upcoming Events
- Wednesday, 05.02.2025, 10:00 (WIAS-HVP-3.13)
- Forschungsseminar Mathematische Statistik
Dr. Sophie Langer, University of Twente, Niederlande:
Deep learning theory - what's next?
more ... Location
Weierstraß-Institut, Hausvogteiplatz 11A, 10117 Berlin, 3. Etage, Raum: 3.13
Abstract
Since several years, deep learning has emerged as a transformative field, with its theory involving several disciplines such as approximation theory, statistics and optimization. Despite remarkable advances, the rapid evolution of AI-driven methods continually outpaces our theoretical understanding. New challenges, from overparametrization and diffusion models to Transformer learning arise almost yearly, underscoring the gap between theory and practice. In this talk, we delve into key theoretical breakthroughs, with a particular focus on statistical results. We critically question the prevailing frameworks and introduce a novel statistical approach to image analysis. Rather than treating images as high-dimensional data entities, our framework reconceptualized them as structured objects shaped by geometric deformations like shifts, scales, and orientations. The goal of the classification rule is then to learn the uninformative deformations, resulting in convergence rates with more favorable tradeoffs between input dimension and sample size. This fresh perspective not only provides new guarantees for approximation and convergence in deep learning-based image classification but also redefines how we approach image analysis with the potential of broader applications to other learning tasks. We conclude by discussing emerging research directions and riflecting on the role of theory in the field. This talk is based on joint work with Johannes Schmidt-Hieber and Juntong Chen.
Further Informations
Dieser Vortrag findet hybrid statt. Die Teilnahme per Zoom ist über den (neuen!) Link: https://hu-berlin.zoom-x.de/j/62476510180?pwd=1bws9DORlDM2Iub3ANrb7zzDNANvsJ.1
Host
Humboldt-Universität zu Berlin
Universität Potsdam
WIAS Berlin
- Wednesday, 05.02.2025, 15:15 (WIAS-405-406)
- Berliner Oberseminar „Nichtlineare partielle Differentialgleichungen” (Langenbach-Seminar)
Ass. Prof. Sebastian Schwarzacher, Uppsala University, Schweden:
Time-periodic solutions for fluid-solid interactions
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406
Abstract
I will discuss several recent analytic discoveries and numerical experiments about the appearance of time-periodic motions when fluids interact with solids. On one hand, I will present abstract results on the existence and uniqueness of solutions when a parabolic PDE interacts with a hyperbolic PDE. In this setting, geometric conditions will be explored that allow for unique solutions and, as such, exclude hyperbolic resonances. Additionally, I will discuss some results for deformable shells interacting with fluids. On the other hand, I will show numerical experiments related to the appearance of bifurcations in the Navier-Stokes equations, known as the von Karman vortex street. The work presented was established in collaboration with J. Cach, C. Midnrila, S. Mosny, B. Muha, K. Tuma, and J. Webster.
Further Informations
Oberseminar “Nichtlineare partielle Differentialgleichungen” (Langenbach-Seminar)
Host
Humboldt-Universität zu Berlin
WIAS Berlin
- Thursday, 06.02.2025, 10:00 (WIAS-405-406)
- Software and Data Seminar
Jan Philipp Thiele, WIAS Berlin:
Easy to use tools for software quality
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406
Further Informations
Software and Data Seminar
Host
WIAS Berlin
- Tuesday, 11.02.2025, 11:30 (WIAS-405-406)
- Seminar Interacting Random Systems
Benedikt Jahnel, WIAS Berlin and TU Braunschweig:
Large and moderate deviations in Poisson navigations
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406
Abstract
We derive large- and moderate-deviation results in random networks given as planar directed navigations on homogeneous Poisson point processes. In this non-Markovian routing scheme, starting from the origin, at each consecutive step a Poisson point is joined by an edge to its nearest Poisson point to the right within a cone. We establish precise exponential rates of decay for the probability that the vertical displacement of the random path is unexpectedly large. The proofs rest on controlling the dependencies of the individual steps and the randomness in the horizonal displacement as well as renewal-process arguments. This is joint work with Partha Ghosh and Sanjoy Jhawar.
Host
WIAS Berlin
- Wednesday, 12.02.2025, 10:00 (WIAS-HVP-3.13)
- Forschungsseminar Mathematische Statistik
Prof. Dr. Judith Rousseau, University of Oxford, GB:
Convergence of diffusion models under the manifold hypothesis in high-dimensions
more ... Location
Weierstraß-Institut, Hausvogteiplatz 11A, 10117 Berlin, 3. Etage, Raum: 3.13
Abstract
Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio and video generation as well as many more applications in science and beyond. The manifold hypothesis states that high-dimensional data often lie on lower-dimensional manifolds within an ambient space of large dimension D , and is widely believed to hold in provided examples. While recent results have provided invaluable insight into how diffusion models adapt to the manifold hypothesis, they do not capture the great empirical success of these models. In this work, we study DDPMs under the manifold hypothesis and prove that they achieve rates independent of the ambient dimension in terms of learning the score. In terms of sampling, we obtain rates independent of the ambient dimension w.r.t. the Kullback-Leibler divergence, and $O(sqrtD)$ w.r.t. the Wasserstein distance. We do this by developing a new framework connecting diffusion models to the well-studied theory of extrema of Gaussian Processes. This is a joint work with I. Azangulov and G. Deligliannidis (University of Oxford)
Further Informations
Dieser Vortrag findet hybrid statt. Die Teilnahme per Zoom ist über den (neuen!) Link: https://hu-berlin.zoom-x.de/j/62476510180?pwd=1bws9DORlDM2Iub3ANrb7zzDNANvsJ.1
Host
Humboldt-Universität zu Berlin
Universität Potsdam
WIAS Berlin
- February 12 – 13, 2025 (WIAS-405-406)
- Workshop/Konferenz: Joint RICAM WIAS Workshop
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406
Host
WIAS Berlin
Johann Radon Institute for Computational and Applied Mathematics
- Wednesday, 12.02.2025, 15:15 (WIAS-Library)
- Berliner Oberseminar „Nichtlineare partielle Differentialgleichungen” (Langenbach-Seminar)
Prof. Dr. André Schlichting, Universität Ulm:
Breakdown of the mean-field description of interacting systems: Phase transitions, metastability and coarsening
more ... Location
Weierstraß-Institut, Hausvogteiplatz 5-7, 10117 Berlin, R411
Abstract
We present results concerning the qualitative and quantitative description of interacting systems, with particular emphasis on those possessing a phase transition under the change of relevant system parameters. For this, we first discuss and identify continuous and discontinuous phase for mean-field limits of interacting particle systems on the torus and spheres. Since phase transitions are intimately related to the metastability of the stochastic particle system, we show how a suitable mountain pass theorem in the space of probability measures can describe the metastable behaviour of the underlying finite particle system. We also argue that the mean-field description of the particle system in the regime of strong local interaction has to break down. In this regime, coarsening is observed, where smaller clusters grow through coagulation events. We provide numerical experiments with a positivity preserving numerical scheme for a SPDE of Dean-Kawasaki type, consisting of the McKean-Vlasov equation and conservative noise. Joint works with Nicolai Gerber (U Ulm), Rishabh Gvalani (ETH Zürich), Greg Pavliotis (Imperial London) and Anna Shalova (TU Eindhoven).
Further Informations
Oberseminar “Nichtlineare partielle Differentialgleichungen” (Langenbach-Seminar)
Host
Humboldt-Universität zu Berlin
WIAS Berlin
- Thursday, 20.02.2025, 10:15 (WIAS-405-406)
- Seminar Nichtlineare Optimierung und Inverse Probleme
Prof. Dr. Giulio Schimperna, Università degli Studi di Pavia, Italien:
Some results on a modified Cahn-Hilliard model with chemotaxis
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406
Abstract
We will present some mathematical results for a new model coupling the Cahn-Hilllard system with an evolutionary equation describing the active (chemotactic) transport of a chemical species influencing the phase separation process. Specifically, the model may arise in connection with tumor growth processes; mathematically speaking, it may be interesting in itself as it provides a new coupling between a Keller-Segel-like relation (the equation describing the evolution of the concentration of the chemical substance) and a fourth order (rather than a second order as in most models for chemotaxis) evolutionary system. Our main result will be devoted to proving existence of weak solutions in the case when the chemotaxis sensitivity function has a controlled growth at infinity; a particular emphasis will be given to discussing the occurrence of critical exponents and to presenting a regularization scheme compatible with the a-priori estimates. Moreover, we will discuss an extension of the model where the effects of a macroscopic velocity flow of Brinkman type are taken into account and analyze the Darcy limit regime. Finally, referring to the (more difficult) case of linear chemotactic sensitivity we will shortly present some work in progress, in collaboration with Elisabetta Rocca (Pavia) and Robert Lasarzik (WIAS), related to the existence of very weak solutions as well as weak-strong uniqueness.
Host
WIAS Berlin
- Wednesday, 26.02.2025, 11:30 (WIAS-405-406)
- Seminar Interacting Random Systems
Prof. Dr. Renato Soares dos Santos, Universidade Federal de Minas Gerais (UFMG), Brasilien:
tba
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406
Further Informations
Seminar Interacting Random Systems
Host
WIAS Berlin
- Thursday, 27.02.2025, 14:00 (WIAS-405-406)
- Seminar Numerische Mathematik
Dr. El-Houssaine Quenjel, La Rochelle Université:
Stable finite volume methods for transient convection-diffusion systems with anisotropy
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406
Host
WIAS Berlin
- Thursday, 27.03.2025, 14:00 (WIAS-405-406)
- Seminar Numerische Mathematik
Prof. Cornelis Vuik, Delft University of Technology:
Resolving divergence: the first multigrid scheme for the highly indefinite Helmholtz equation using classical components
more ... Location
Weierstraß-Institut, Mohrenstr. 39, 10117 Berlin, 4. Etage, Raum: 405/406
Abstract
In this talk, we (V. Dwarka and C. Vuik) present the first stand-alone classical multigrid solver for the highly in- definite 2D Helmholtz equation with constant costs per iteration, addressing a longstanding open problem in numerical analysis [1]. Our work covers both large constant and non- constant wavenumbers up to k = 500 in 2D. We obtain a full V - and W -cycle without any level-dependent restrictions. Another powerful feature is that it can be combined with the computationally cheap weighted Jacobi smoother. The key novelty lies in the use of higher-order inter-grid transfer operators [2]. When combined with coarsening on the Complex Shifted Laplacian, rather than the original Helmholtz operator, our solver is h-independent and scales linearly with the wavenumber k. If we use GMRES(3) smoothing we obtain k- independent convergence, and can coarsen on the original Helmholtz operator, as long as the higher-order transfer operators are used. This work opens doors to study robustness of contemporary solvers, such as machine learning solvers inspired by multigrid components, without adding to the black-box complexity.
Host
WIAS Berlin