Related papers: Max-Min Neural Network Operators For Approximation…
Kolmogorov-Arnold Networks (KANs) offer a structured and interpretable framework for multivariate function approximation by composing univariate transformations through additive or multiplicative aggregation. This paper establishes…
The suboptimal performance of wavelets with regard to the approximation of multivariate data gave rise to new representation systems, specifically designed for data with anisotropic features. Some prominent examples of these are given by…
Neural approximate computing gains enormous energy-efficiency at the cost of tolerable quality-loss. A neural approximator can map the input data to output while a classifier determines whether the input data are safe to approximate with…
This paper surveys recent developments at the intersection of operator learning, statistical learning theory, and approximation theory. First, it reviews error bounds for empirical risk minimization with a focus on holomorphic operators and…
Existing value function approximation methods have been successfully used in many applications, but they often lack useful a priori error bounds. We propose a new approximate bilinear programming formulation of value function approximation,…
We introduce the neural tangent kernel (NTK) regime for two-layer neural operators and analyze their generalization properties. For early-stopped gradient descent (GD), we derive fast convergence rates that are known to be minimax optimal…
Although very successfully used in conventional machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of…
In this paper, bivariate Szasz-Mirakjan type operators are introduced along with the estimation of its approximation properties and its rate of convergence. Furthermore, to check the asymptotic behavior of the said bivariate operators, the…
Data-driven modeling techniques have been explored in the spatial-temporal modeling of complex dynamical systems for many engineering applications. However, a systematic approach is still lacking to leverage the information from different…
This work extends the Mond-Pecaric method to functions with multiple operators as arguments by providing arbitrarily close approximations of the original functions. Instead of using linear functions to establish lower and upper bounds for…
Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of…
In this paper, we introduce a Stancu-type generalization of multivariate neural network operators by incorporating two parameters that perturb the sampling nodes. The proposed operators extend the existing neural network operator by…
In training neural networks, it is common practice to use partial gradients computed over batches, mostly very small subsets of the training set. This approach is motivated by the argument that such a partial gradient is close to the true…
We develop a unified operator framework for scalar, multivariate, and functional regression based on integral operators defined with respect to general measures. Within this framework, classical regression models, including…
Operator learning is a recent development in the simulation of Partial Differential Equations (PDEs) by means of neural networks. The idea behind this approach is to learn the behavior of an operator, such that the resulting neural network…
Motivated by the abundance of functional data such as time series and images, there has been a growing interest in integrating such data into neural networks and learning maps from function spaces to R (i.e., functionals). In this paper, we…
This thesis investigates the design of algorithms for solving min-max optimization problems, which form the mathematical foundation of many modern applications in machine learning, game theory, and optimization. This work offers new…
Neural network width and depth are fundamental aspects of network topology. Universal approximation theorems provide that with increasing width or depth, there exists a neural network that approximates a function arbitrarily well. These…
This work focuses on developing methods for approximating the solution operators of a class of parametric partial differential equations via neural operators. Neural operators have several challenges, including the issue of generating…
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…