Related papers: Deep Learning Models for Global Coordinate Transfo…

Being symmetric positive-definite (SPD), covariance matrix has traditionally been used to represent a set of local descriptors in visual recognition. Recent study shows that kernel matrix can give considerably better representation by…

The complexity of real-world geophysical systems is often compounded by the fact that the observed measurements depend on hidden variables. These latent variables include unresolved small scales and/or rapidly evolving processes, partially…

There have been growing interests in leveraging experimental measurements to discover the underlying partial differential equations (PDEs) that govern complex physical phenomena. Although past research attempts have achieved great success…

We extend the methodology in [Yang et al., 2023] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a…

Recently, there has been a growing interest in the problem of learning rich implicit models - those from which we can sample, but can not evaluate their density. These models apply some parametric function, such as a deep network, to a base…

The accuracy of solving partial differential equations (PDEs) on coarse grids is greatly affected by the choice of discretization schemes. In this work, we propose to learn time integration schemes based on neural networks which satisfy…

In recent years, deep learning techniques have shown great success in various tasks related to inverse problems, where a target quantity of interest can only be observed through indirect measurements by a forward operator. Common approaches…

In a recent article, we presented a framework to control nonlinear partial differential equations (PDEs) by means of Koopman operator based reduced models and concepts from switched systems. The main idea was to transform a control system…

In data-driven modeling of spatiotemporal phenomena careful consideration often needs to be made in capturing the dynamics of the high wavenumbers. This problem becomes especially challenging when the system of interest exhibits shocks or…

Operator learning has emerged as a promising paradigm for developing efficient surrogate models to solve partial differential equations (PDEs). However, existing approaches often overlook the domain knowledge inherent in the underlying PDEs…

In this paper we present a continuation method which transforms spatially distributed ODE systems into continuous PDE. We show that this continuation can be performed both for linear and nonlinear systems, including multidimensional, space-…

The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of…

Dynamical systems theory has long provided a foundation for understanding evolving phenomena across scientific domains. Yet, the application of this theory to complex real-world systems remains challenging due to issues in mathematical…

Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy.…

While many phenomena in physics and engineering are formally high-dimensional, their long-time dynamics often live on a lower-dimensional manifold. The present work introduces an autoencoder framework that combines implicit regularization…

Machine learning has made important headway in helping to improve the treatment of quantum many-body systems. A domain of particular relevance are correlated inhomogeneous systems. What has been missing so far is a general, scalable…

We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard…

Data-driven model predictive control based on Willems' fundamental lemma has proven effective for linear systems, but extending stability guarantees to nonlinear systems remains an open challenge. In this paper, we establish conditions…

Neural ordinary differential equations (Neural ODEs) propose the idea that a sequence of layers in a neural network is just a discretisation of an ODE, and thus can instead be directly modelled by a parameterised ODE. This idea has had…

We present a new framework for optimal and feedback control of PDEs using Koopman operator-based reduced order models (K-ROMs). The Koopman operator is a linear but infinite-dimensional operator which describes the dynamics of observables.…