arXiv subjects

Sachin Saini

Publications and source records attributed to Sachin Saini.

5 recordsLinked to original sources

arXiv2026

Stancu-Type Generalizations of Neural Network Operators with Perturbed Sampling Nodes

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 allowing greater flexibility in the placement of sampling nodes. We establish the well-definedness and boundedness of the operators and prove uniform convergence on compact domains. Furthermore, quantitative error estimates are derived in terms of the modulus of continuity, leading to convergence rate results. Numerical experiments are presented to illustrate the approximation behavior of the proposed operators and to demonstrate the effect of the Stancu parameters on the sampling nodes and the approximation accuracy. Finally, the application of signal denoising is demonstrated using a synthetic ECG signal, showing that the proposed operators effectively suppress noise while preserving the signal's main characteristics.

math.NA↗

arXiv2026

A Universal Approximation Theorem for Neural Networks with Outputs in Locally Convex Spaces

In this paper, a universal approximation theorem (UAT) for shallow neural networks whose inputs belong to a topological vector space (TVS) and whose outputs take values in a Hausdorff locally convex TVS is established. The networks are constructed using scalar activation functions applied to continuous linear functionals of the input, while the output coefficients lie in the target space. It is shown that this class of neural networks is dense in the space of continuous mappings from a compact subset of the input space into the target space with respect to the topology of uniform convergence induced by the defining seminorms. The result extends existing scalar-valued approximation theorems on TVS and includes Banach and Hilbert-valued approximation as special cases. Several corollaries and examples are provided, illustrating applications to operator approximation between function spaces.

math.FA↗

arXiv2026

Taylor-Accelerated Neural Network Interpolation Operators on Irregular Grids with Higher Order Approximation

In this paper, a new class of \emph{Taylor-accelerated neural network interpolation operators} is introduced on quasi-uniform irregular grids. These operators improve existing neural network interpolation operators by incorporating Taylor polynomials at the sampling nodes, thereby exploiting higher smoothness of the target function. The proposed operators are shown to be well defined, uniformly bounded, and to satisfy an exact interpolation property at the grid points. In addition, polynomial reproduction up to a prescribed degree is established. Jackson-type approximation estimates are derived in terms of higher-order moduli of smoothness, yielding enhanced convergence rates for sufficiently smooth functions. Numerical experiments are presented to support the theoretical analysis and to demonstrate the significant accuracy improvement achieved through the Taylor-accelerated construction. In particular, higher-order convergence on irregular grids is obtained, and the proposed approach outperforms existing neural network interpolation operators on irregular grids, including Lagrange-based schemes.

math.NA↗

arXiv2026

Kantorovich-Type Stochastic Neural Network Operators for the Mean-Square Approximation of Certain Second-Order Stochastic Processes

Artificial neural network operators (ANNOs) have been widely used for approximating deterministic input-output functions; however, their extension to random dynamics remains comparatively unexplored. In this paper, we construct a new class of \textbf{Kantorovich-type Stochastic Neural Network Operators (K-SNNOs)} in which randomness is incorporated not at the coefficient level, but through \textbf{stochastic neurons} driven by stochastic integrators. This framework enables the operator to inherit the probabilistic structure of the underlying process, making it suitable for modeling and approximating stochastic signals. We establish mean-square convergence of K-SNNOs to the target stochastic process and derive quantitative error estimates expressing the rate of approximation in terms of the modulus of continuity. Numerical simulations further validate the theoretical results by demonstrating accurate reconstruction of sample paths and rapid decay of the mean square error (MSE). Graphical results, including sample-wise approximations and empirical MSE behaviour, illustrate the robustness and effectiveness of the proposed stochastic-neuron-based operator.

cs.LG↗

arXiv2025

Constructive Approximation of Random Process via Stochastic Interpolation Neural Network Operators

In this paper, we construct a class of stochastic interpolation neural network operators (SINNOs) with random coefficients activated by sigmoidal functions. We establish their boundedness, interpolation accuracy, and approximation capabilities in the mean square sense, in probability, as well as path-wise within the space of second-order stochastic (random) processes \( L^2(Ω, \mathcal{F},\mathbb{P}) \). Additionally, we provide quantitative error estimates using the modulus of continuity of the processes. These results highlight the effectiveness of SINNOs for approximating stochastic processes with potential applications in COVID-19 case prediction.

stat.ML↗