Related papers: A Priori Estimation of the Approximation, Optimiza…

Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the…

This work compares the advantages and limitations of the Finite Difference Method with Physics-Informed Neural Networks, showing where each can best be applied for different problem scenarios. Analysis on the L2 relative error based on…

Physics-Informed Neural Networks (PINNs) are a class of deep neural networks that are trained, using automatic differentiation, to compute the response of systems governed by partial differential equations (PDEs). The training of PINNs is…

Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep…

Deep neural nets have caused a revolution in many classification tasks. A related ongoing revolution -- also theoretically not understood -- concerns their ability to serve as generative models for complicated types of data such as images…

Optimal a priori estimates are derived for the population risk, also known as the generalization error, of a regularized residual network model. An important part of the regularized model is the usage of a new path norm, called the weighted…

The numerical solution of partial differential equations (PDEs) is challenging because of the need to resolve spatiotemporal features over wide length and timescales. Often, it is computationally intractable to resolve the finest features…

While much attention of neural network methods is devoted to high-dimensional PDE problems, in this work we consider methods designed to work for elliptic problems on domains $\Omega \subset \mathbb{R} ^d, $ $d=1,2,3$ in association with…

Recent research has used margin theory to analyze the generalization performance for deep neural networks (DNNs). The existed results are almost based on the spectrally-normalized minimum margin. However, optimizing the minimum margin…

While traditional Deep Learning (DL) optimization methods treat all training samples equally, Distributionally Robust Optimization (DRO) adaptively assigns importance weights to different samples. However, a significant gap exists between…

Physics-Informed Neural Networks (PINNs) have emerged as powerful tools for solving partial differential equations (PDEs). However, training PINNs from scratch is often computationally intensive and time-consuming. To address this problem,…

The theoretical and empirical performance of Empirical Risk Minimization (ERM) often suffers when loss functions are poorly behaved with large Lipschitz moduli and spurious sharp minimizers. We propose and analyze a counterpart to ERM…

We study the problem of learning classification functions from noiseless training samples, under the assumption that the decision boundary is of a certain regularity. We establish universal lower bounds for this estimation problem, for…

Physics-informed neural networks (PINNs) are an increasingly powerful way to solve partial differential equations, generate digital twins, and create neural surrogates of physical models. In this manuscript we detail the inner workings of…

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based…

Randomized neural networks (RNN) are a variation of neural networks in which the hidden-layer parameters are fixed to randomly assigned values and the output-layer parameters are obtained by solving a linear system by least squares. This…

How to develop slim and accurate deep neural networks has become crucial for real- world applications, especially for those employed in embedded systems. Though previous work along this research line has shown some promising results, most…

Deep neural networks (DNNs) have brought significant advancements in various applications in recent years, such as image recognition, speech recognition, and natural language processing. In particular, Vision Transformers (ViTs) have…

Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods…

In this paper, neural network approximation methods are developed for elliptic partial differential equations with multi-frequency solutions. Neural network work approximation methods have advantages over classical approaches in that they…