Related papers: A comparative study of two stochastic mode reducti…

Model reduction techniques have emerged as a powerful paradigm across different fronts of scientific computing. Despite their success, the provided tools and methodologies remain limited if high-dimensional dynamical systems subject to…

The well-known Mori-Zwanzig theory tells us that model reduction leads to memory effect. For a long time, modeling the memory effect accurately and efficiently has been an important but nearly impossible task in developing a good reduced…

We present a scalable methodology to verify stochastic hybrid systems. Using the Mori-Zwanzig reduction method, we construct a finite state Markov chain reduction of a given stochastic hybrid system and prove that this reduced Markov chain…

We examine the challenging problem of constructing reduced models for the long time prediction of systems where there is no timescale separation between the resolved and unresolved variables. In previous work we focused on the case where…

A theoretical framework which unifies the conventional Mori-Zwanzig formalism and the approximate Koopman learning is presented. In this framework, the Mori-Zwanzig formalism, developed in statistical mechanics to tackle the hard problem of…

In this chapter we review stochastic modelling methods in climate science. First we provide a conceptual framework for stochastic modelling of deterministic dynamical systems based on the Mori-Zwanzig formalism. The Mori-Zwanzig equations…

We present a general numerical approach for constructing governing equations for unknown dynamical systems when only data on a subset of the state variables are available. The unknown equations for these observed variables are thus a…

Reduced models for the (defocusing) nonlinear Schr\"odinger equation are developed. In particular, we develop reduced models that only involve the low-frequency modes given noisy observations of these modes. The ansatz of the reduced…

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…

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…

We consider a composite convex minimization problem associated with regularized empirical risk minimization, which often arises in machine learning. We propose two new stochastic gradient methods that are based on stochastic dual averaging…

We propose stochastic variance reduced algorithms for solving convex-concave saddle point problems, monotone variational inequalities, and monotone inclusions. Our framework applies to extragradient, forward-backward-forward, and…

This work proposes novel techniques for the efficient numerical simulation of parameterized, unsteady partial differential equations. Projection-based reduced order models (ROMs) such as the reduced basis method employ a (Petrov-)Galerkin…

We develop two novel stochastic variance-reduction methods to approximate solutions of a class of nonmonotone [generalized] equations. Our algorithms leverage a new combination of ideas from the forward-reflected-backward splitting method…

We introduce a predictor-corrector discretisation scheme for the numerical integration of a class of stochastic differential equations and prove that it converges with weak order 1.0. The key feature of the new scheme is that it builds up…

We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori--Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any…

In this paper we study the problem of model reduction by moment matching for stochastic systems. We characterize the mathematical object which generalizes the notion of moment to stochastic differential equations and we find a class of…

We present a novel way of constructing reduced models for systems of ordinary differential equations. The reduced models we construct depend on coefficients which measure the importance of the different terms appearing in the model and need…

We propose a stochastic model reduction strategy for deterministic and stochastic slow-fast systems with finite time-scale separation. The stochastic model reduction relaxes the assumption of infinite time-scale separation of classical…

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…