13.04.2026 14:00 Dario Trevisan: Large Deviations, Kernels, and Feature Learning in Bayesian Wide Neural Networks
Wide neural networks are often described by Gaussian process limits, which capture their typical behavior at initialization and under lazy training regimes. However, these approximations fail to describe the rare but structurally significant fluctuations that govern finite-width effects, posterior concentration, and feature learning. In this talk, we present a large deviation perspective on deep neural networks that goes beyond Gaussian limits.
We first discuss recent results establishing a functional large deviation principle for fully connected networks with Gaussian initialization and Lipschitz activations, including ReLU. This provides a probabilistic description of the entire network output as a random function on compact input sets, with a rate function characterized by a recursive variational structure across layers.
We then turn to Bayesian neural networks and show how large deviation theory leads to an explicit variational characterization of the posterior over predictors. In contrast with Gaussian process theory, where the kernel is fixed, the large deviation rate function involves a joint optimization over both outputs and internal kernels, yielding a natural notion of feature learning at the functional level. Numerical experiments illustrate how this framework captures non-Gaussian effects, posterior deformation, and data-dependent kernel selection in moderately wide networks.
These results suggest that large deviations provide a principled framework to understand representation learning in wide neural networks, bridging probabilistic asymptotics with practical behavior beyond Gaussian approximations.
Based on arXiv:2601.18276 and arXiv:2602.22925.
Quelle
13.04.2026 15:00 Jan Friedrich: An introduction to nonlocal conservation laws: Theory and numerics
In recent years, conservation laws with space-dependent nonlocal fluxes have attracted growing attention due to their wide range of applications in fluid mechanics, including granular flow, sedimentation, aggregation phenomena, crowd dynamics, and, in particular, traffic flow, which serves as the main motivating example of this talk.
We provide an introduction to nonlocal conservation laws, where nonlocality is modeled through a spatial convolution operator. We discuss key analytical properties of these models, including the uniqueness of weak solutions and the singular limit as the convolution kernel converges to a Dirac delta.
Furthermore, we present a general framework for the numerical approximation of such equations and highlight possible extensions of both the theory and the numerical methods, as well as remaining open problems in the literature.
Quelle
15.04.2026 12:15 Daniel Rademacher (Universität Heidelberg): Mercer Expansions in Sobolev Spaces and Applications to Stochastic Processes
Mercer's celebrated theorem is refined and extended by introducing a novel class of higher-order kernel operators that includes the common integral operator only as a special case.
\[ \]
These operators genuinely take into account information encoded in the (weak) derivatives of a kernel, and their natural domains are Sobolev spaces of order k over some bounded d-dimensional space. domain, where k depends on the order of (weak) differentiability.
\[ \]
The spectral decomposition of such higher-order kernel operators leads to Mercer-type expansions, which are optimal in terms of the Sobolev norm and, if k>d, also converge uniformly without requiring the kernel to be positive definite.
\[ \]
Nuclearity of higher order kernel operators is confirmed for positive definite kernels, and a major refinement of Mercer's theorem is obtained that implies trace formulas and a simple rate for the uniform convergence (including derivatives) in terms of the eigenvalues.
A further immediate consequence is novel spectral representations of RKHS's.
\[ \]
Finally, applied to the covariance kernel of a (weakly) differentiable stochastic process, these refinements also yield novel Karhunen-Loève-type expansions allowing for simultaneous approximations of the process and its (weak) derivatives in a mean-square-optimal sense.
Quelle
20.04.2026 15:00 Aliaksei Kuzmenka: Dynamics of the Caputo FODEs
In recent years, Fractional Ordinary differential equations, FODEs, became an essential tool for modelling of viscoelasticity, neuron behaviour, fluid dynamics, electrical circuits and more. The distinguishing feature of the FODEs is the use of a fractional derivative, which generalises the classical derivative to a non-integer order.
The fractional derivative is a non-local operator, meaning the whole history of the function affects the value of the derivative at a given point. The non-locality introduces analytical difficulties when extending the standard method from the classical dynamical systems to the FODE framework, similar to the challenges faced with time-delay systems. This is particularly evident in the theory of invariant manifolds. For example, the classical notion of invariance is no longer well-posed for the fractional dynamical systems. The literature presents conflicting results on this topic, some studies claim that stable and invariant centre manifolds exist, and one work disputes that claim.
The aim is to resolve this contradiction and provide a concrete framework for the analysis of the
fractional dynamical systems along the way.
Quelle
20.04.2026 16:30 Piotr Dyszewski: TBA
TBA
Quelle