相关论文: Constraint-defined Manifolds: a Legacy Code Approa…

Many interesting functions arising in applications map into Riemannian manifolds. We present an algorithm, using the manifold exponential and logarithm, for approximating such functions. Our approach extends approximation techniques for…

Input-affine dynamical systems often arise in control and modeling scenarios, such as the data-driven case when state-derivative observations are recorded under bounded noise. Common tasks in system analysis and control include optimal…

Accelerated gradient descent iterations are widely used in optimization. It is known that, in the continuous-time limit, these iterations converge to a second-order differential equation which we refer to as the accelerated gradient flow.…

We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main…

The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous encoder network cannot embed it in a one-to-one…

Discrete translational symmetry plays a fundamental role in condensed matter physics and lattice gauge theories, enabling the analysis of systems that would otherwise be intractable. Despite this, many open problems remain. Quantum…

Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the "curse of dimensionality," becoming ineffective as the dimension of the parameter space grows. One feature of a…

The discrete class algorithm presented in this paper is an efficient simulation tool for stochastic processes governed by a reasonably small set of transition rates. The algorithm is presented, its performance compared to prevailing methods…

An efficient representation of observed data has many benefits in various domains of engineering and science. Representing static data sets, such as images, is a living branch in machine learning and eases downstream tasks, such as…

We derive conditions under which a general nonlinear mechanical system can be exactly reduced to a lower-dimensional model that involves only the most flexible degrees of freedom. This Slow-Fast Decomposition (SFD) enslaves exponentially…

Many systems in physics, engineering, and biology exhibit multiscale stochastic dynamics, where low-dimensional slow variables evolve under the influence of high-dimensional fast processes. In practice, observations are often limited to a…

High-index saddle dynamics (HiSD) is an effective approach for computing saddle points of a prescribed Morse index and constructing solution landscapes for complex nonlinear systems. However, for problems with ill-conditioned Hessians…

Multiscale dynamics are ubiquitous in applications of modern science. Because of time scale separation between relatively small set of slowly evolving variables and (typically) much larger set of rapidly changing variables, direct numerical…

Numerical solutions to fractional differential equations can be extremely computationally intensive due to the effect of non-local derivatives in which all previous time points contribute to the current iteration. In finite difference…

Shape optimization models with one or more shapes are considered in this chapter. Of particular interest for applications are problems in which where a so-called shape functional is constrained by a partial differential equation (PDE)…

We develop a general compactification framework to facilitate analysis of nonlinear nonautonomous ODEs where nonautonomous terms decay asymptotically. The strategy is to compactify the problem: the phase space is augmented with a bounded…

Random invariant manifolds are geometric objects useful for understanding complex dynamics under stochastic influences. Under a nonuniform hyperbolicity or a nonuniform exponential dichotomy condition, the existence of random pseudo-stable…

Accurate and efficient prediction of multi-scale flows remains a formidable challenge. Constructing theoretical models and numerical methods often involves the design and optimization of parameters. While gradient descent methods have been…

Manifold learning techniques seek to discover structure-preserving mappings of high-dimensional data into low-dimensional spaces. While the new sets of coordinates specified by these mappings can closely parameterize the data, they are…

While much work on deep latent variable models of text uses continuous latent variables, discrete latent variables are interesting because they are more interpretable and typically more space efficient. We consider several approaches to…