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This paper delves into the problem of computing robust controlled invariants for monotone continuous-time systems, with a specific focus on lower-closed specifications. We consider the classes of state monotone (SM) and control-state…

Due to the increase in computational resources and accessibility of data, an increase in large, deep learning models trained on copious amounts of multi-modal data using self-supervised or semi-supervised learning have emerged. These…

The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…

This paper presents a gradient-based reconstruction approach for simulations of compressible single and multi-species Navier-Stokes equations. The novel feature of the proposed algorithm is the efficient reconstruction via derivative…

Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…

We show how to compose robust stability tests for uncertain systems modeled as linear fractional representations and affected by various types of dynamic uncertainties. Our results are formulated in terms of linear matrix inequalities and…

Multiphase poromechanics describes the evolution of multiphase flow in deformable porous media. Mathematical models for such multiphysics system are inheritely nonlinear, potentially degenerate and fully coupled systems of partial…

In the past five years, deep learning methods have become state-of-the-art in solving various inverse problems. Before such approaches can find application in safety-critical fields, a verification of their reliability appears mandatory.…

Machine learning methods can be unreliable when deployed in domains that differ from the domains on which they were trained. There are a wide range of proposals for mitigating this problem by learning representations that are ``invariant''…

This article is concerned with the existence and the long time behavior of weak solutions to certain coupled systems of fourth-order degenerate parabolic equations of gradient flow type. The underlying metric is a Wasserstein-like…

In this paper, we exploit the gradient flow structure of continuous-time formulations of Bayesian inference in terms of their numerical time-stepping. We focus on two particular examples, namely, the continuous-time ensemble Kalman-Bucy…

In this work we study the integrability of a family of nonlinear oscillators. Dynamical systems from this family appear in different applications from mechanics to chemistry. We propose an approach for finding first integrals and…

When studying fluid mechanics in terms of instability, bifurcation and invariant solutions one quickly finds out how little can be done by pen and paper. For flows on sufficiently simple domains and under sufficiently simple boundary…

Incorporating a priori physics knowledge into machine learning leads to more robust and interpretable algorithms. In this work, we combine deep learning techniques and classic numerical methods for differential equations to address two…

Discrete Morse theory emerged as an essential tool for computational geometry and topology. Its core structures are discrete gradient fields, defined as acyclic matchings on a complex $C$, from which topological and geometrical informations…

Multi-fidelity methods are prominently used when cheaply-obtained, but possibly biased and noisy, observations must be effectively combined with limited or expensive true data in order to construct reliable models. This arises in both…

In this work we investigate the existence and uniqueness of Struwe-like solutions for a system of partial differential equations modeling the dynamics of magnetoviscoelastic fluids. The considered system couples a Navier-Stokes type…

In this paper, we introduce and investigate multivariate versions of frequent stability and diam-mean equicontinuity. Given a natural number $m > 1$, we call those notions "frequent $m$-stability" and "diam-mean $m$-equicontinuity". We use…

Deep learning interpretation is essential to explain the reasoning behind model predictions. Understanding the robustness of interpretation methods is important especially in sensitive domains such as medical applications since…

This paper is about statistical properties of quasistatic dynamical systems. These are a class of non-stationary systems that model situations where the dynamics change very slowly over time due to external influence. We focus on the case…