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Distance metric learning (DML) has been studied extensively in the past decades for its superior performance with distance-based algorithms. Most of the existing methods propose to learn a distance metric with pairwise or triplet…

Linear time-invariant systems are very popular models in system theory and applications. A fundamental problem in system identification that remains rather unaddressed in extant literature is to leverage commonalities amongst related linear…

Discovering the governing equations of a dynamical system from observed trajectories provides deeper insight into its structure than mere prediction of future states. We present a data-driven approach to model discovery based on…

In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic)…

The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems…

Critical transitions are the abrupt shifts between qualitatively different states of a system, and they are crucial to understanding tipping points in complex dynamical systems across ecology, climate science, and biology. Detecting these…

While linear systems have been useful in solving problems across different fields, the need for improved performance and efficiency has prompted them to operate in nonlinear modes. As a result, nonlinear models are now essential for the…

There have been extensive studies on solving differential equations using physics-informed neural networks. While this method has proven advantageous in many cases, a major criticism lies in its lack of analytical error bounds. Therefore,…

The Hidden Markov Model (HMM) is one of the most widely used statistical models for sequential data analysis. One of the key reasons for this versatility is the ability of HMM to deal with missing data. However, standard HMM learning…

The vast majority of systems of practical interest are characterised by nonlinear dynamics. This renders the control and optimization of such systems a complex task due to their nonlinear behaviour. Additionally, standard methods such as…

Port-Hamiltonian neural networks (pHNNs) are emerging as a powerful modeling tool that integrates physical laws with deep learning techniques. While most research has focused on modeling the entire dynamics of interconnected systems, the…

Deep Metric Learning (DML) approaches learn to represent inputs to a lower-dimensional latent space such that the distance between representations in this space corresponds with a predefined notion of similarity. This paper investigates how…

Distance metric learning (DML) approaches learn a transformation to a representation space where distance is in correspondence with a predefined notion of similarity. While such models offer a number of compelling benefits, it has been…

This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank…

The limits of molecular dynamics (MD) simulations of macromolecules are steadily pushed forward by the relentless developments of computer architectures and algorithms. This explosion in the number and extent (in size and time) of MD…

Lifted linear predictor (LLP) is an artificial linear dynamical system designed to predict trajectories of a generally nonlinear dynamical system based on the current state (or measurements) and the input. The main benefit of the LLP is its…

To quantify the fundamental evolution of time-varying networks, and detect abnormal behavior, one needs a notion of temporal difference that captures significant organizational changes between two successive instants. In this work, we…

We analyze multi-layer neural networks in the asymptotic regime of simultaneously (A) large network sizes and (B) large numbers of stochastic gradient descent training iterations. We rigorously establish the limiting behavior of the…

The number of units of a network dynamical system, its size, arguably constitutes its most fundamental property. Many units of a network, however, are typically experimentally inaccessible such that the network size is often unknown. Here…

Deep networks are commonly used to model dynamical systems, predicting how the state of a system will evolve over time (either autonomously or in response to control inputs). Despite the predictive power of these systems, it has been…