相关论文: Globally injective and bijective neural operators

In practice, multi-task learning (through learning features shared among tasks) is an essential property of deep neural networks (NNs). While infinite-width limits of NNs can provide good intuition for their generalization behavior, the…

With the increased prevalence of neural operators being used to provide rapid solutions to partial differential equations (PDEs), understanding the accuracy of model predictions and the associated error levels is necessary for deploying…

Recently, graph neural networks have attracted great attention and achieved prominent performance in various research fields. Most of those algorithms have assumed pairwise relationships of objects of interest. However, in many real…

Smooth Equations of the form G[z]=0 are investigated in Banach spaces with the aim of continuing the basic solution G[0]=0 to a solution curve of G[z]=0 with the implicit function theorem. If the linearization is surjective, then the…

Leveraging the infinite dimensional neural network architecture we proposed in arXiv:2109.13512v4 and which can process inputs from Fr\'echet spaces, and using the universal approximation property shown therein, we now largely extend the…

Understanding the loss surface of neural networks is essential for the design of models with predictable performance and their success in applications. Experimental results suggest that sufficiently deep and wide neural networks are not…

This article derives and validates three principles for initialization and architecture selection in finite width graph neural networks (GNNs) with ReLU activations. First, we theoretically derive what is essentially the unique…

The classical Universal Approximation Theorem holds for neural networks of arbitrary width and bounded depth. Here we consider the natural `dual' scenario for networks of bounded width and arbitrary depth. Precisely, let $n$ be the number…

In recent years, there has been considerable innovation in the world of predictive methodologies. This is evident by the relative domination of machine learning approaches in various classification competitions. While these algorithms have…

The input space of a neural network with ReLU-like activations is partitioned into multiple linear regions, each corresponding to a specific activation pattern of the included ReLU-like activations. We demonstrate that this partition…

Residual neural networks are state-of-the-art deep learning models. Their continuous-depth analog, neural ordinary differential equations (ODEs), are also widely used. Despite their success, the link between the discrete and continuous…

In this paper, we prove that in the overparametrized regime, deep neural network provide universal approximations and can interpolate any data set, as long as the activation function is locally in $L^1(\RR)$ and not an affine function.…

Solving ill-posed inverse problems necessitates effective regularization strategies to stabilize the inversion process against measurement noise. While classical methods like Tikhonov regularization require heuristic parameter tuning, and…

Large neural network models have been successful in learning functions of importance in many branches of science, including physics, chemistry and biology. Recent theoretical work has shown explicit learning bounds for wide networks and…

Learning embeddings from large-scale networks is an open challenge. Despite the overwhelming number of existing methods, is is unclear how to exploit network structure in a way that generalizes easily to unseen nodes, edges or graphs. In…

Non-injective functions are not globally invertible. However, they can often be restricted to locally injective subdomains where the inversion is well-defined. In many settings a preferred solution can be selected even when multiple valid…

Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous…

The number of linear regions is one of the distinct properties of the neural networks using piecewise linear activation functions such as ReLU, comparing with those conventional ones using other activation functions. Previous studies showed…

The great advances of learning-based approaches in image processing and computer vision are largely based on deeply nested networks that compose linear transfer functions with suitable non-linearities. Interestingly, the most frequently…

The Sinkhorn operator has recently experienced a surge of popularity in computer vision and related fields. One major reason is its ease of integration into deep learning frameworks. To allow for an efficient training of respective neural…