Related papers: Deep Neural Network Structures Solving Variational…

The current understanding of deep neural networks can only partially explain how input structure, network parameters and optimization algorithms jointly contribute to achieve the strong generalization power that is typically observed in…

Deep neural networks are an attractive alternative for simulating complex dynamical systems, as in comparison to traditional scientific computing methods, they offer reduced computational costs during inference and can be trained directly…

Neural networks have become ubiquitous tools for solving signal and image processing problems, and they often outperform standard approaches. Nevertheless, training neural networks is a challenging task in many applications. The prevalent…

In comparison to classical shallow representation learning techniques, deep neural networks have achieved superior performance in nearly every application benchmark. But despite their clear empirical advantages, it is still not well…

Deep neural networks are highly expressive models that have recently achieved state of the art performance on speech and visual recognition tasks. While their expressiveness is the reason they succeed, it also causes them to learn…

As demonstrated in many areas of real-life applications, neural networks have the capability of dealing with high dimensional data. In the fields of optimal control and dynamical systems, the same capability was studied and verified in many…

Deep Neural Networks (DNNs) are generated by sequentially performing linear and non-linear processes. Using a combination of linear and non-linear procedures is critical for generating a sufficiently deep feature space. The majority of…

Standard neural networks can approximate general nonlinear operators, represented either explicitly by a combination of mathematical operators, e.g., in an advection-diffusion-reaction partial differential equation, or simply as a black…

Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…

Recent efforts to understand intermediate representations in deep neural networks have commonly attempted to label individual neurons and combinations of neurons that make up linear directions in the latent space by examining extremal…

In the field of pattern recognition research, the method of using deep neural networks based on improved computing hardware recently attracted attention because of their superior accuracy compared to conventional methods. Deep neural…

We consider neural networks with rational activation functions. The choice of the nonlinear activation function in deep learning architectures is crucial and heavily impacts the performance of a neural network. We establish optimal bounds…

Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies…

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural…

We consider linear and obstacle problems driven by a nonlocal integral operator, for which nonlocal interactions are restricted to a ball of finite radius. These type of operators are used to model anomalous diffusion and, for a special…

In this paper we propose a generalization of deep neural networks called deep function machines (DFMs). DFMs act on vector spaces of arbitrary (possibly infinite) dimension and we show that a family of DFMs are invariant to the dimension of…

We study the complexity of functions computable by deep feedforward neural networks with piecewise linear activations in terms of the symmetries and the number of linear regions that they have. Deep networks are able to sequentially map…

Our ability to interact with the world around us relies on being able to infer what actions objects afford -- often referred to as affordances. The neural mechanisms of object-action associations are realized in the visuomotor pathway where…

Neural networks are famously nonlinear. However, linearity is defined relative to a pair of vector spaces, $f:X \to Y$. Leveraging the algebraic concept of transport of structure, we propose a method to explicitly identify non-standard…

Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit…