Related papers: New universal operator approximation theorem for e…

We present an approximation theorem for continuous non-decreasing functions on compact preordered spaces, leading to an algebraic characterization of their corresponding function spaces. As an application, we prove that the family of…

The Encoder-Decoder architecture is a main stream deep learning model for biomedical image segmentation. The encoder fully compresses the input and generates encoded features, and the decoder then produces dense predictions using encoded…

The approximation capabilities of Deep Q-Networks (DQNs) are commonly justified by general Universal Approximation Theorems (UATs) that do not leverage the intrinsic structural properties of the optimal Q-function, the solution to a Bellman…

The present paper deals with construction of newly family of Neural Network operators, that is, Steklov Neural Network operators. By using Steklov type integral, we introduce a new version of Neural Network operators and we obtain some…

We prove the converse of the universal approximation theorem, i.e. a neural network (NN) encoding theorem which shows that for every stably converged NN of continuous activation functions, its weight matrix actually encodes a continuous…

We present approximation theories and efficient training methods for derivative-informed Fourier neural operators (DIFNOs) with applications to PDE-constrained optimization. A DIFNO is an FNO trained by minimizing its prediction error…

Operator learning provides methods to approximate mappings between infinite-dimensional function spaces. Deep operator networks (DeepONets) are a notable architecture in this field. Recently, an extension of DeepONet based on model…

In this paper, we present some fixed point theorems for operator systems in the line of Krasnosel'skii's theorem in cones. The cone-compression and cone-expansion type conditions are imposed in a component-wise manner. Unlike related…

Operator splitting techniques have recently gained popularity in convex optimization problems arising in various control fields. Being fixed-point iterations of nonexpansive operators, such methods suffer many well known downsides, which…

Singularly perturbed problems present inherent difficulty due to the presence of a thin boundary layer in its solution. To overcome this difficulty, we propose using deep operator networks (DeepONets), a method previously shown to be…

Operator learning has emerged as a new paradigm for the data-driven approximation of nonlinear operators. Despite its empirical success, the theoretical underpinnings governing the conditions for efficient operator learning remain…

Deep Operator Network (DeepONet), a recently introduced deep learning operator network, approximates linear and nonlinear solution operators by taking parametric functions (infinite-dimensional objects) as inputs and mapping them to…

In this paper we introduce and study a class of structured set-valued operators which we call union averaged nonexpansive. At each point in their domain, the value of such an operator can be expressed as a finite union of single-valued…

We study the closure of the unitary orbit of a given point in the non-commutative Choquet boundary of a unital operator space with respect to the topology of pointwise norm convergence. This may be described more extensively as the…

To a generalized tight continuous frame in a Hilbert space $\H$ indexed by a locally compact space $\Si$ endowed with a Radon measure, one associates a coorbit theory converting spaces of functions on $\Si$ in spaces of vectors comparable…

We introduce AlgoSelect, a principled framework for learning optimal algorithm selection from data, centered around the novel Comb Operator. Given a set of algorithms and a feature representation of problems, AlgoSelect learns to…

This paper explores operators with countable, continuous, and hybrid spectra, focusing on both finite dimensional and infinite dimensional cases, particularly in non-Hermitian systems. For finite dimensional operators, a novel concept of…

Neural oscillators, originating from second-order ordinary differential equations (ODEs), have demonstrated strong performance in stably learning causal mappings between long-term sequences or continuous temporal functions, as well as in…

In this chapter, the Hilbert space framework in the mathematical theory of composite materials is introduced for studying the properties of effective operators. The goal is to introduce some of the key concepts and fundamental theorems in…

The concept of complementability is extended from bounded operators to densely defined operators on Hilbert spaces. By introducing appropriate projections and decomposition techniques, a framework is developed for analyzing…