Related papers: Data-based stochastic model reduction for the Kura…

Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features…

Many physical systems are described by nonlinear differential equations that are too complicated to solve in full. A natural way to proceed is to divide the variables into those that are of direct interest and those that are not, formulate…

We examine the problem of predicting the evolution of solutions of the Kuramoto-Sivashinsky equation when initial data are missing. We use the optimal prediction method to construct equations for the reduced system. The resulting equations…

Leveraging recent work on data-driven methods for constructing a finite state space Markov process from dynamical systems, we address two problems for obtaining further reduced statistical representations. The first problem is to extract…

This paper investigates how models of spatiotemporal dynamics in the form of nonlinear partial differential equations can be identified directly from noisy data using a combination of sparse regression and weak formulation. Using the…

Simulations of chaotic systems can only produce high-fidelity trajectories if the initial and boundary conditions are well specified. When these conditions are unknown but measurements are available, variational state estimation can…

In many real-world applications, optimization problems evolve continuously over time and are often subject to stochastic noise. We consider a stochastic time-varying optimization problem in which the objective function $f(x;t)$ changes…

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…

Estimating the likelihood, timing, and nature of events is a major goal of modeling stochastic dynamical systems. When the event is rare in comparison with the timescales of simulation and/or measurement needed to resolve the elemental…

The polynomial NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous input) is a model that represents the dynamics of physical systems. This polynomial contains information from the past of the inputs and outputs of the…

We show the skills of a data-driven low-dimensional linear model in predicting the spatio-temporal evolution of turbulent Rayleigh-B\'enard convection. The model is based on dynamic mode decomposition with delay-embedding, which provides a…

This study evaluates data-driven models from a dynamical system perspective, such as unstable fixed points, periodic orbits, chaotic saddle, Lyapunov exponents, manifold structures, and statistical values. We find that these dynamical…

Sparse regression has recently emerged as an attractive approach for discovering models of spatiotemporally complex dynamics directly from data. In many instances, such models are in the form of nonlinear partial differential equations…

Systematic discovery of reduced-order closure models for multi-scale processes remains an important open problem in complex dynamical systems. Even when an effective lower-dimensional representation exists, reduced models are difficult to…

We propose a novel functional approach to surrogate modeling of dynamical systems with exogenous inputs. This approach, named Functional Nonlinear AutoRegressive with eXogenous inputs (F-NARX), approximates the system response based on…

We present a class of efficient parametric closure models for 1D stochastic Burgers equations. Casting it as statistical learning of the flow map, we derive the parametric form by representing the unresolved high wavenumber Fourier modes as…

Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time…

While data-driven model reduction techniques are well-established for linearizable mechanical systems, general approaches to reducing non-linearizable systems with multiple coexisting steady states have been unavailable. In this paper, we…

We present a variational principle for the extraction of a time-dependent orthonormal basis from random realizations of transient systems. The optimality condition of the variational principle leads to a closed-form evolution equation for…

Data-driven modeling techniques have been explored in the spatial-temporal modeling of complex dynamical systems for many engineering applications. However, a systematic approach is still lacking to leverage the information from different…