approximators

An approximator, in mathematics and computer science, is a model, algorithm, or function class designed to estimate a target function from input data. It aims to produce outputs that closely match the true values of the function being approximated, often when the exact function is unknown or too costly to evaluate directly.

Common analytic approximators include polynomials, Fourier series, splines, radial basis functions, and wavelets. In data-driven contexts, practitioners use linear models, polynomial regression, kernel methods, neural networks, decision trees, and ensemble methods as function approximators.

Approximation theory studies how well a function can be approximated by elements of a chosen class, with error measured in norms such as Lp or the sup norm. The universal approximation theorem states that certain architectures, notably feedforward neural networks with a non-polynomial activation, can approximate any continuous function on a compact set to arbitrary accuracy, given sufficient capacity.

In practice, performance is judged by approximation error on held-out data and by generalization ability. Trade-offs between bias and variance, regularization, model complexity, and the quality and quantity of training data affect results. Training involves selecting parameters to minimize a loss that reflects the discrepancy between the approximator’s outputs and observed values.

Common applications include numerical analysis, statistics, signal processing, control engineering, economics, and simulation. Approximators serve as surrogate models, interpolation tools, or predictors when the true function is unknown, expensive to evaluate, or defined only empirically.