Related papers: A generalized nonstandard finite difference method…
The theory of slow invariant manifolds (SIMs) is the foundation of various model-order reduction techniques for dissipative dynamical systems with multiple time-scales, e.g. in chemical kinetic models. The construction of SIMs and many…
The Finite Elements with Switch Detection (FESD) method is a highly accurate direct transcription method for optimal control of several classes of nonsmooth dynamical systems. This paper extends the FESD method to Projected Dynamical…
An implicit finite difference method with non-uniform timesteps for solving the fractional diffusion equation in the Caputo form is proposed. The method allows one to build adaptive methods where the size of the timesteps is adjusted to the…
We describe a family of descent algorithms which generalizes common existing schemes used in applications such as neural network training and more broadly for optimization of smooth functions--potentially for global optimization, or as a…
We introduce a methodology for seeking conservation laws within a Hamiltonian dynamical system, which we term ``neural deflation''. Inspired by deflation methods for steady states of dynamical systems, we propose to {iteratively} train a…
We develop a new method which extends Dynamic Mode Decomposition (DMD) to incorporate the effect of control to extract low-order models from high-dimensional, complex systems. DMD finds spatial-temporal coherent modes, connects local-linear…
This article establishes the foundation for a new theory of invariant/integral manifolds for non-autonomous dynamical systems. Current rigorous support for dimensional reduction modelling of slow-fast systems is limited by the rare events…
We describe a method to model nonlinear dynamical systems using periodic solutions of delay-differential equations. We show that any finite-time trajectory of a nonlinear dynamical system can be loaded approximately into the initial…
We present an explicit second order staggered finite difference (FD) discretization scheme for forward simulation of natural gas transport in pipeline networks. By construction, this discretization approach guarantees that the conservation…
The method of distributions is developed for systems that are governed by hyperbolic conservation laws with stochastic forcing. The method yields a deterministic equation for the cumulative density distribution (CDF) of a system state,…
Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to…
We propose a dynamic factor model (DFM) where the latent factors are linked to observed variables with unknown and potentially nonlinear functions. The key novelty and source of flexibility of our approach is a nonparametric observation…
Recently we proposed a new constant-pressure molecular dynamics method for finite systems. In this paper, we discuss the current understanding of this method and its technique details. We also review the recent theoretical advances of…
Computational fluid dynamics (CFD) simulations are broadly applied in engineering and physics. A standard description of fluid dynamics requires solving the Navier-Stokes (N-S) equations in different flow regimes. However, applications of…
In this work, we demonstrate how physical principles -- such as symmetries, invariances, and conservation laws -- can be integrated into the dynamic mode decomposition (DMD). DMD is a widely-used data analysis technique that extracts…
Many successful methods to learn dynamical systems from data have recently been introduced. However, ensuring that the inferred dynamics preserve known constraints, such as conservation laws or restrictions on the allowed system states,…
In this thesis we introduce the concept of a guided dynamical system, and exploit this idea to solve various problems in functional equations and PDE's. Our main results are 1) a necessary and sufficient condition for unique-solvability of…
We consider structure-preserving methods for conservative systems, which rigorously replicate the conservation property yielding better numerical solutions. There, corresponding to the skew-symmetry of the differential operator, that of…
The problem of finite-dimensional asymptotics of infinite-dimensional dynamic systems is studied. A non-linear kinetic system with conservation of supports for distributions has generically finite-dimensional asymptotics. Such systems are…
The DMD (Dynamic Mode Decomposition) method has attracted widespread attention as a representative modal-decomposition method and can build a predictive model. However, the DMD may give predicted results that deviate from physical reality…