Research Projects

The study of constructive approximation using neural networks has seen significant developments, especially in areas where traditional methods face limitations. The core challenge lies in constructing neural network operators (NNOs) that can effectively interpolate and approximate given functions, leveraging the powerful computational capabilities of neural networks. Neural networks, with their highly flexible structure and learning abilities, have proven to be a robust tool for function approximation, especially when dealing with complex and high-dimensional data. However, constructing these operators in a systematic, efficient, and mathematically sound manner remains a crucial research problem. At the heart of a neural network lies the concept of neurons, which are simple yet highly interconnected processing units. Each neuron stores certain weight w and a threshold θ in the memory and evaluates certain activation function σ(w.x+θ) on receiving an input x. These neurons are arranged in layers. The first layer is the input layer, last layer is the output layer and the intermediate layers are called the hidden layers. A single layered neural network with activation function σ:R→R, d input nodes, one output node and one hidden layer of n neurons can be mathematically expressed as follows: N_(n,σ) (x)=∑_(j=1)^n▒〖c_j σ(w_j.x+θ_j)〗,x∈R^d,d∈N, where c_j∈R are the coefficients, w_j∈R^d are the weights and θ_j∈R (j=1,2,…,n) are the thresholds of the neural network operator. In the context of function approximation, the simplest form of a neural network operator (NNO) associated with bounded measurable functions given by Anastassiou [1] can be put in the following form: Definition: Let f:[a,b]→R be a bounded and measurable function. The neural network operators (NNOs), activated by σ and acting on f, are defined by N_n (f,x)≔(∑_(k=⌈na⌉)^⌊nb⌋▒〖f(k/n) ϕ_σ (nx-k) 〗)/(∑_(k=⌈na⌉)^⌊nb⌋▒〖ϕ_σ (nx-k) 〗),x∈[a,b],n∈N, (1) where ϕ_σ≔1/2 [σ(x+1)-σ(x-1)], ⌊ .⌋ and ⌈ .⌉ denote, respectively, the greatest integer function and ceiling function. One can note the sample points k/n are equally spaced, and the operator N_n is linear. Various researchers have estimated upper bounds for the deviation N_n (f,x)-f(x) in appropriate norms. The overarching goal of this project is to investigate and construct various forms of NNOs for the approximation and interpolation of functions, focusing on the following key objectives: Nonlinear NNOs: Neural network operators with nonlinear activation functions, which can better model complex functions that linear methods cannot. Irregular Grid NNOs: NNOs that operate on irregularly spaced data points, which is common in real-world datasets where measurements are not uniformly distributed. Stochastic NNOs: Operators that approximate stochastic processes, such as time series or random fields, capturing uncertainty and variability in the data.