Related papers: Computing Lyapunov functions using deep neural net…
This paper proves an abstract theorem addressing in a unified manner two important problems in function approximation: avoiding curse of dimensionality and estimating the degree of approximation for out-of-sample extension in manifold…
In this paper, we develop a framework for showing that neural networks can overcome the curse of dimensionality in different high-dimensional approximation problems. Our approach is based on the notion of a catalog network, which is a…
We study the training of deep neural networks by gradient descent where floating-point arithmetic is used to compute the gradients. In this framework and under realistic assumptions, we demonstrate that it is highly unlikely to find ReLU…
A wide range of numerical methods exists for computing polynomial approximations of solutions of ordinary differential equations based on Chebyshev series expansions or Chebyshev interpolation polynomials. We consider the application of…
We propose a technique for the design and analysis of adaptation algorithms in dynamical systems. The technique applies both to systems with conventional Lyapunov-stable target dynamics and to ones of which the desired dynamics around the…
We provide general methods for explicitly constructing strict Lyapunov functions for fully nonlinear slowly time-varying systems. Our results apply to cases where the given dynamics and corresponding frozen dynamics are not necessarily…
In this article, we investigate the existence of a deep neural network (DNN) capable of approximating solutions to partial integro-differential equations while circumventing the curse of dimensionality. Using the Feynman-Kac theorem, we…
Recently, it has been proposed in the literature to employ deep neural networks (DNNs) together with stochastic gradient descent methods to approximate solutions of PDEs. There are also a few results in the literature which prove that DNNs…
Deep Reinforcement Learning (RL) powered by neural net approximation of the Q function has had enormous empirical success. While the theory of RL has traditionally focused on linear function approximation (or eluder dimension) approaches,…
We present a stability analysis framework for the general class of discrete-time linear switching systems for which the switching sequences belong to a regular language. They admit arbitrary switching systems as special cases. Using recent…
We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for…
In this paper, we establish universal approximation theorems for neural networks applied to general nonlinear ill-posed operator equations. In addition to the approximation error, the measurement error is also taken into account in our…
It is well-known that neural networks are universal approximators, but that deeper networks tend in practice to be more powerful than shallower ones. We shed light on this by proving that the total number of neurons $m$ required to…
We derive upper bounds on the complexity of ReLU neural networks approximating the solution maps of parametric partial differential equations. In particular, without any knowledge of its concrete shape, we use the inherent…
Accurate approximation of scalar-valued functions from sample points is a key task in computational science. Recently, machine learning with Deep Neural Networks (DNNs) has emerged as a promising tool for scientific computing, with…
Deep neural network with rectified linear units (ReLU) is getting more and more popular recently. However, the derivatives of the function represented by a ReLU network are not continuous, which limit the usage of ReLU network to situations…
A Lyapunov design method is used to analyze the nonlinear stability of a generic reservoir computer for both the cases of continuous-time and discrete-time dynamics. Using this method, for a given nonlinear reservoir computer, a radial…
Based on the tree architecture, the objective of this paper is to design deep neural networks with two or more hidden layers (called deep nets) for realization of radial functions so as to enable rotational invariance for near-optimal…
Projection-based model reduction has become a popular approach to reduce the cost associated with integrating large-scale dynamical systems so they can be used in many-query settings such as optimization and uncertainty quantification. For…
A method for approximating continuous functions $\mathbb{Z}_{p}^{n}\rightarrow\mathbb{Z}_{p}$ by a linear superposition of continuous functions $\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{p}$ is presented and a polynomial regression model is…