Related papers: Projection Methods for Operator Learning and Unive…
We present an algorithm for learning operators between Banach spaces, based on the use of Leray-Schauder mappings to learn a finite-dimensional approximation of compact subspaces. We show that the resulting method is a universal…
Operator learning refers to the application of ideas from machine learning to approximate (typically nonlinear) operators mapping between Banach spaces of functions. Such operators often arise from physical models expressed in terms of…
Operator learning is the approximation of operators between infinite dimensional Banach spaces using machine learning approaches. While most progress in this area has been driven by variants of deep neural networks such as the Deep Operator…
Previously unknown estimates of uniform continuity of projection operators in Banach space have been obtained. They can be used in the investigations of approximation methods, in particular, the method of quasisolutions, methods of…
We study the universal approximation properties of transformers and neural integral operators for operators in Banach spaces. In particular, we show that the transformer architecture is a universal approximator of integral operators between…
Operator learning based on neural operators has emerged as a promising paradigm for the data-driven approximation of operators, mapping between infinite-dimensional Banach spaces. Despite significant empirical progress, our theoretical…
This paper studies approximation properties of linear sampling operators in general Banach lattices $X$. We obtain matching direct and inverse approximation estimates, convergence criteria, equivalence results involving special…
In this paper we study the convergence of an iterative algorithm for finding zeros with constraints for not necessarily monotone set-valued operators in a reflexive Banach space. This algorithm, which we call the proximal-projection method…
Several non-linear operators in stochastic analysis, such as solution maps to stochastic differential equations, depend on a temporal structure which is not leveraged by contemporary neural operators designed to approximate general maps…
Metric projection operators can be defined in similar wayin Hilbert and Banach spaces. At the same time, they differ signifitiantly in their properties. Metric projection operator in Hilbert space is a monotone and nonexpansive operator. It…
We introduce a Banach space-valued extension of random feature learning, a data-driven supervised machine learning technique for large-scale kernel approximation. By randomly initializing the feature maps, only the linear readout needs to…
While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning…
As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all…
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear…
The standard Universal Approximation Theorem for operator neural networks (NNs) holds for arbitrary width and bounded depth. Here, we prove that operator NNs of bounded width and arbitrary depth are universal approximators for continuous…
We study universal approximation of continuous functionals on compact subsets of products of Hilbert spaces. We prove that any such functional can be uniformly approximated by models that first take finitely many continuous linear…
DeepONets have recently been proposed as a framework for learning nonlinear operators mapping between infinite dimensional Banach spaces. We analyze DeepONets and prove estimates on the resulting approximation and generalization errors. In…
Motivated by the rapidly growing field of mathematics for operator approximation with neural networks, we present a novel universal operator approximation theorem for a broad class of encoder-decoder architectures. In this study, we focus…
Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions. They are gaining increased attention in computational science and engineering, due to their potential both to accelerate…
Derivative-Informed Operator Learning (DIOL), i.e. learning a (nonlinear) operator and its derivatives, is an open research frontier at the foundations of the influential field of Operator Learning (OL). In particular, Universal…