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Partial differential equations, and their chaotic solutions, are pervasive in the modelling of complex systems in engineering, science, and beyond. Data-driven methods can find solutions to partial differential equations with a…

Deep generative models for anomaly detection in multivariate time-series are typically trained by maximizing data likelihood. However, likelihood in observation space measures marginal density rather than conformity to structured temporal…

Modeling the evolution of system with time-series data is a challenging and critical task in a wide range of fields, especially when the time-series data is regularly sampled and partially observable. Some methods have been proposed to…

The prediction of periodical time-series remains challenging due to various types of data distortions and misalignments. Here, we propose a novel model called Temporal embedding-enhanced convolutional neural Network (TeNet) to learn…

Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, we rigorously establish the convergence of this…

In this paper, the performance of three deep learning methods for predicting short-term evolution and for reproducing the long-term statistics of a multi-scale spatio-temporal Lorenz 96 system is examined. The methods are: echo state…

Transformer models have consistently achieved remarkable results in various domains such as natural language processing and computer vision. However, despite ongoing research efforts to better understand these models, the field still lacks…

We study the problem of graph structure identification, i.e., of recovering the graph of dependencies among time series. We model these time series data as components of the state of linear stochastic networked dynamical systems. We assume…

In nonlinear time series analysis and dynamical systems theory, Takens' embedding theorem states that the sliding window embedding of a generic observation along trajectories in a state space, recovers the region traversed by the dynamics.…

Deep learning of the Artificial Neural Networks (ANN) can be treated as a particular class of interpolation problems. The goal is to find a neural network whose input-output map approximates well the desired map on a finite or an infinite…

This paper shows that the celebrated Embedding Theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible…

We introduce a self-consistent deep-learning framework which, for a noisy deterministic time series, provides unsupervised filtering, state-space reconstruction, identification of the underlying differential equations and forecasting.…

We investigate the ability to discover data assimilation (DA) schemes meant for chaotic dynamics with deep learning. The focus is on learning the analysis step of sequential DA, from state trajectories and their observations, using a simple…

Deep neural networks (DNNs) are widely used in pattern-recognition tasks for which a human comprehensible, quantitative description of the data-generating process, e.g., in the form of equations, cannot be achieved. While doing so, DNNs…

This paper addresses the data-driven identification of latent dynamical representations of partially-observed systems, i.e., dynamical systems for which some components are never observed, with an emphasis on forecasting applications,…

Learning long-term behaviors in chaotic dynamical systems, such as turbulent flows and climate modelling, is challenging due to their inherent instability and unpredictability. These systems exhibit positive Lyapunov exponents, which…

Nudging is an empirical data assimilation technique that incorporates an observation-driven control term into the model dynamics. The trajectory of the nudged system approaches the true system trajectory over time, even when the initial…

In this article we propose a new deep learning approach to approximate operators related to parametric partial differential equations (PDEs). In particular, we introduce a new strategy to design specific artificial neural network (ANN)…

We present an algorithm to learn the relevant latent variables of a large-scale discretized physical system and predict its time evolution using thermodynamically-consistent deep neural networks. Our method relies on sparse autoencoders,…

Complex chaotic dynamics, seen in natural and industrial systems like turbulent flows and weather patterns, often span vast spatial domains with interactions across scales. Accurately capturing these features requires a high-dimensional…