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We establish a continuous-time framework for analyzing Deep Q-Networks (DQNs) via stochastic control and Forward-Backward Stochastic Differential Equations (FBSDEs). Considering a continuous-time Markov Decision Process (MDP) driven by a…

Deep neural networks (NN) have achieved great success in many applications. However, why do deep neural networks obtain good generalization at an over-parameterization regime is still unclear. To better understand deep NN, we establish the…

We introduce Neural Radiosity, an algorithm to solve the rendering equation by minimizing the norm of its residual similar as in traditional radiosity techniques. Traditional basis functions used in radiosity techniques, such as piecewise…

This paper establishes rates of universal approximation for the shallow neural tangent kernel (NTK): network weights are only allowed microscopic changes from random initialization, which entails that activations are mostly unchanged, and…

We study the approximation of multivariate functions with tensor networks (TNs), providing some answers to the following two questions: ``what are the approximation capabilities of TNs for functions from classical smoothness classes?'' and…

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…

Due to the over-parameterization nature, neural networks are a powerful tool for nonlinear function approximation. In order to achieve good generalization on unseen data, a suitable inductive bias is of great importance for neural networks.…

The density functional renormalization group (density-fRG) is proposed to investigate the density fluctuations within the functional renormalization group approach, which allows us to quantify the medium effect and study physics of high…

We consider approximations of general continuous functions on finite-dimensional cubes by general deep ReLU neural networks and study the approximation rates with respect to the modulus of continuity of the function and the total number of…

In this paper we present a generalized Deep Learning-based approach for solving ill-posed large-scale inverse problems occuring in medical image reconstruction. Recently, Deep Learning methods using iterative neural networks and cascaded…

Quantum neural networks (QNNs) are an analog of classical neural networks in the world of quantum computing, which are represented by a unitary matrix with trainable parameters. Inspired by the universal approximation property of classical…

Classical neural network approximation results take the form: for every function $f$ and every error tolerance $\epsilon > 0$, one constructs a neural network whose architecture and weights depend on $\epsilon$. This paper introduces a…

We propose a systematic procedure for the approximation of density functionals in density functional theory that consists of two parts. First, for the efficient approximation of a general density functional, we introduce an efficient ansatz…

Overparameterized neural networks enjoy great representation power on complex data, and more importantly yield sufficiently smooth output, which is crucial to their generalization and robustness. Most existing function approximation…

We provide here a universal approximation theorem with precise quantitative error bounds for noisy quantum neural networks. We focus on applications to Quantitative Finance, where target functions are often given as expectations. We further…

The paper introduces the weighted convolution, a novel approach to the convolution for signals defined on regular grids (e.g., 2D images) through the application of an optimal density function to scale the contribution of neighbouring…

The universal approximation theorem asserts that a single hidden layer neural network approximates continuous functions with any desired precision on compact sets. As an existential result, the universal approximation theorem supports the…

In the study of condensed matter physics, spectral information plays an important role for understand the mechanism of materials. However, it is difficult to obtain the spectrum directly through experiments or simulation. For example, the…

We propose a new molecular simulation framework that combines the transferability, robustness and chemical flexibility of an ab initio method with the accuracy and efficiency of a machine learned force field. The key to achieve this mix is…

Machine learning techniques have received growing attention as an alternative strategy for developing general-purpose density functional approximations, augmenting the historically successful approach of human designed functionals derived…