Related papers: Towards Size-Independent Generalization Bounds for…

Deep Neural Networks can generalize despite being significantly overparametrized. Recent research has tried to examine this phenomenon from various view points and to provide bounds on the generalization error or measures predictive of the…

Statistical learning theory has largely focused on learning and generalization given independent and identically distributed (i.i.d.) samples. Motivated by applications involving time-series data, there has been a growing literature on…

This paper studies the approximation capacity of neural networks with an arbitrary activation function and with norm constraint on the weights. Upper and lower bounds on the approximation error of these networks are computed for smooth…

The traditional limitations of neural networks in reliably generalizing beyond the convex hulls of their training data present a significant problem for computational physics, in which one often wishes to solve PDEs in regimes far beyond…

Operator learning has become a powerful tool in machine learning for modeling complex physical systems governed by partial differential equations (PDEs). Although Deep Operator Networks (DeepONet) show promise, they require extensive data…

Understanding and certifying the generalization performance of machine learning algorithms -- i.e. obtaining theoretical estimates of the test error from the training error -- is a central theme of statistical learning theory. Among the…

Despite existing work on ensuring generalization of neural networks in terms of scale sensitive complexity measures, such as norms, margin and sharpness, these complexity measures do not offer an explanation of why neural networks…

Finding tight bounds on the optimal solution is a critical element of practical solution methods for discrete optimization problems. In the last decade, decision diagrams (DDs) have brought a new perspective on obtaining upper and lower…

Despite the tremendous empirical success of deep learning models to solve various learning tasks, our theoretical understanding of their generalization ability is very limited. Classical generalization bounds based on tools such as the VC…

Neural networks are universal approximators and are studied for their use in solving differential equations. However, a major criticism is the lack of error bounds for obtained solutions. This paper proposes a technique to rigorously…

We show that the Rademacher complexity-based framework can establish non-vacuous generalization bounds for Convolutional Neural Networks (CNNs) in the context of classifying a small set of image classes. A key technical advancement is the…

This paper presents novel generalization bounds for vector-valued neural networks and deep kernel methods, focusing on multi-task learning through an operator-theoretic framework. Our key development lies in strategically combining a…

This paper presents a general framework for norm-based capacity control for $L_{p,q}$ weight normalized deep neural networks. We establish the upper bound on the Rademacher complexities of this family. With an $L_{p,q}$ normalization where…

Operator regression provides a powerful means of constructing discretization-invariant emulators for partial-differential equations (PDEs) describing physical systems. Neural operators specifically employ deep neural networks to approximate…

Deep Operator Networks (DeepOnets) have revolutionized the domain of scientific machine learning for the solution of the inverse problem for dynamical systems. However, their implementation necessitates optimizing a high-dimensional space…

While deep learning models and techniques have achieved great empirical success, our understanding of the source of success in many aspects remains very limited. In an attempt to bridge the gap, we investigate the decision boundary of a…

There has been considerable effort to better understand the generalization capabilities of deep neural networks both as a means to unlock a theoretical understanding of their success as well as providing directions for further improvements.…

Machine learning, especially deep learning is gaining much attention due to the breakthrough performance in various cognitive applications. Recently, neural networks (NN) have been intensively explored to model partial differential…

In this paper, we evaluate the effectiveness of deep operator networks (DeepONets) in solving both forward and inverse problems of partial differential equations (PDEs) on unknown manifolds. By unknown manifolds, we identify the manifold by…

The paper establishes generalization bounds for multitask deep neural networks using operator-theoretic techniques. The authors propose a tighter bound than those derived from conventional norm based methods by leveraging small condition…