arXiv subjects

Hannes Stuke

Publications and source records attributed to Hannes Stuke.

7 recordsLinked to original sources

arXiv2024

Real eternal PDE solutions are not complex entire: a quadratic parabolic example

In parabolic or hyperbolic PDEs, solutions which remain uniformly bounded for all real times $t=r\in\mathbb{R}$ are often called PDE entire or eternal. For example, consider the quadratic parabolic PDE \begin{equation*} \label{*} w_t=w_{xx}+6w^2-\lambda, \tag{*} \end{equation*} for $0 0$, and are currently limited to unstable dimensions $n\leq22$, or to fast unstable manifolds of dimensions $d<1+\tfrac{1}{\sqrt{2}}n$.

math.AP↗

arXiv2023

Hybrid Bifurcations: Periodicity from Eliminating a Line of Equilibria

We describe a new mechanism that triggers periodic orbits in smooth dynamical systems. To this end, we introduce the concept of hybrid bifurcations: Such bifurcations occur when a line of equilibria with an exchange point of normal stability vanishes. Our main result is the existence and stability criteria of periodic orbits that bifurcate from breaking a line of equilibria. As an application, we obtain stable periodic coexistent solutions in an ecosystem for two competing predators with Holling's type II functional response.

math.DS↗

arXiv2019

Robustness Against Outliers For Deep Neural Networks By Gradient Conjugate Priors

We analyze a new robust method for the reconstruction of probability distributions of observed data in the presence of output outliers. It is based on a so-called gradient conjugate prior (GCP) network which outputs the parameters of a prior. By rigorously studying the dynamics of the GCP learning process, we derive an explicit formula for correcting the obtained variance of the marginal distribution and removing the bias caused by outliers in the training set. Assuming a Gaussian (input-dependent) ground truth distribution contaminated with a proportion $\varepsilon$ of outliers, we show that the fitted mean is in a $c e^{-1/\varepsilon}$-neighborhood of the ground truth mean and the corrected variance is in a $b\varepsilon$-neighborhood of the ground truth variance, whereas the uncorrected variance of the marginal distribution can even be infinite. We explicitly find $b$ as a function of the output of the GCP network, without a priori knowledge of the outliers (possibly input-dependent) distribution. Experiments with synthetic and real-world data sets indicate that the GCP network fitted with a standard optimizer outperforms other robust methods for regression.

stat.ML↗

arXiv2019

Gradient conjugate priors and multi-layer neural networks

The paper deals with learning probability distributions of observed data by artificial neural networks. We suggest a so-called gradient conjugate prior (GCP) update appropriate for neural networks, which is a modification of the classical Bayesian update for conjugate priors. We establish a connection between the gradient conjugate prior update and the maximization of the log-likelihood of the predictive distribution. Unlike for the Bayesian neural networks, we use deterministic weights of neural networks, but rather assume that the ground truth distribution is normal with unknown mean and variance and learn by the neural networks the parameters of a prior (normal-gamma distribution) for these unknown mean and variance. The update of the parameters is done, using the gradient that, at each step, directs towards minimizing the Kullback--Leibler divergence from the prior to the posterior distribution (both being normal-gamma). We obtain a corresponding dynamical system for the prior's parameters and analyze its properties. In particular, we study the limiting behavior of all the prior's parameters and show how it differs from the case of the classical full Bayesian update. The results are validated on synthetic and real world data sets.

math.ST↗

arXiv2018

Complex time blow-up of the nonlinear heat equation

This paper investigates the connection between blow-up solutions of scalar reaction-diffusion equations, in particular of $u_t = u_{xx} + u^2, $ and its counterpart - eternally existing solutions like heteroclinic orbits - by complex time. We prove that heteroclinic orbits in one-dimensional unstable manifolds are accompanied by blow-up solutions. Furthermore we show, that we can continue blow-up solutions into a slit complex time and eventually back to the real axis. The solution picks up an imaginary factor after continuation which is related to the eigenvalue relations of the linearizations at the source and the sink of the heteroclinic orbit.

math.DS↗

arXiv2018

Pairing an arbitrary regressor with an artificial neural network estimating aleatoric uncertainty

We suggest a general approach to quantification of different forms of aleatoric uncertainty in regression tasks performed by artificial neural networks. It is based on the simultaneous training of two neural networks with a joint loss function and a specific hyperparameter $λ>0$ that allows for automatically detecting noisy and clean regions in the input space and controlling their {\em relative contribution} to the loss and its gradients. After the model has been trained, one of the networks performs predictions and the other quantifies the uncertainty of these predictions by estimating the locally averaged loss of the first one. Unlike in many classical uncertainty quantification methods, we do not assume any a priori knowledge of the ground truth probability distribution, neither do we, in general, maximize the likelihood of a chosen parametric family of distributions. We analyze the learning process and the influence of clean and noisy regions of the input space on the loss surface, depending on $λ$. In particular, we show that small values of $λ$ increase the relative contribution of clean regions to the loss and its gradients. This explains why choosing small $λ$ allows for better predictions compared with neural networks without uncertainty counterparts and those based on classical likelihood maximization. Finally, we demonstrate that one can naturally form ensembles of pairs of our networks and thus capture both aleatoric and epistemic uncertainty and avoid overfitting.

stat.ML↗

arXiv2016

Global Dynamics, Blow-Up, and Bianchi Cosmology

Many central problems in geometry, topology, and mathematical physics lead to questions concerning the long-time dynamics of solutions to ordinary and partial differential equations. Examples range from the Einstein field equations of general relativity to quasilinear reaction-advection-diffusion equations of parabolic type. Specific questions concern the convergence to equilibria, the existence of periodic, homoclinic, and heteroclinic solutions, and the existence and geometric structure of global attractors. On the other hand, many solutions develop singularities in finite time. The singularities have to be analyzed in detail before attempting to extend solutions beyond their singularities, or to understand their geometry in conjunction with globally bounded solutions. In this context we have also aimed at global qualitative descriptions of blow-up and grow-up phenomena.

math.DS↗