A Comprehensive Survey on Functional Approximation

The theory of functional approximation has numerous applications in sciences and industry. This thesis focuses on the possible approaches to approximate a continuous function on a compact subset of R2 using a variety of constructions. The results are presented from the following four general topics: polynomials, Fourier series, wavelets, and neural networks. Approximation with polynomials on subsets of R leads to the discussion of the Stone-Weierstrass theorem. Convergence of Fourier series is characterized on the unit circle. Wavelets are introduced following the Fourier transform, and their construction as well as ability to approximate functions in L2(R) is discussed. At the end, the universal approximation theorem for artificial neural networks is presented, and the function representation and approximation with single- and multilayer neural networks on R2 is constructed

Compressive Fourier collocation methods for high-dimensional diffusion equations with periodic boundary conditions

High-dimensional Partial Differential Equations (PDEs) are a popular mathematical modelling tool, with applications ranging from finance to computational chemistry. However, standard numerical techniques for solving these PDEs are typically affected by the curse of dimensionality. In this work, we tackle this challenge while focusing on stationary diffusion equations defined over a high-dimensional domain with periodic boundary conditions. Inspired by recent progress in sparse function approximation in high dimensions, we propose a new method called compressive Fourier collocation. Combining ideas from compressive sensing and spectral collocation, our method replaces the use of structured collocation grids with Monte Carlo sampling and employs sparse recovery techniques, such as orthogonal matching pursuit and $\ell^1$ minimization, to approximate the Fourier coefficients of the PDE solution. We conduct a rigorous theoretical analysis showing that the approximation error of the proposed method is comparable with the best $s$-term approximation (with respect to the Fourier basis) to the solution. Using the recently introduced framework of random sampling in bounded Riesz systems, our analysis shows that the compressive Fourier collocation method mitigates the curse of dimensionality with respect to the number of collocation points under sufficient conditions on the regularity of the diffusion coefficient. We also present numerical experiments that illustrate the accuracy and stability of the method for the approximation of sparse and compressible solutions.Comment: 33 pages, 9 figure

Learning Theory and Approximation

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Applications of classical approximation theory to periodic basis function networks and computational harmonic analysis

In this paper, we describe a novel approach to classical approximation theory of periodic univariate and multivariate functions by trigonometric polynomials. While classical wisdom holds that such approximation is too sensitive to the lack of smoothness of the target functions at isolated points, our constructions show how to overcome this problem. We describe applications to approximation by periodic basis function networks, and indicate further research in the direction of Jacobi expansion and approximation on the Euclidean sphere. While the paper is mainly intended to be a survey of our recent research in these directions, several results are proved for the first time here

Optimal approximation of CkC^k-functions using shallow complex-valued neural networks

We prove a quantitative result for the approximation of functions of regularity $C^k$ (in the sense of real variables) defined on the complex cube $\Omega_n := [-1,1]^n +i[-1,1]^n\subseteq \mathbb{C}^n$ using shallow complex-valued neural networks. Precisely, we consider neural networks with a single hidden layer and $m$ neurons, i.e., networks of the form $z \mapsto \sum_{j=1}^m \sigma_j \cdot \phi\big(\rho_j^T z + b_j\big)$ and show that one can approximate every function in $C^k \left( \Omega_n; \mathbb{C}\right)$ using a function of that form with error of the order $m^{-k/(2n)}$ as $m \to \infty$, provided that the activation function $\phi: \mathbb{C} \to \mathbb{C}$ is smooth but not polyharmonic on some non-empty open set. Furthermore, we show that the selection of the weights $\sigma_j, b_j \in \mathbb{C}$ and $\rho_j \in \mathbb{C}^n$ is continuous with respect to $f$ and prove that the derived rate of approximation is optimal under this continuity assumption. We also discuss the optimality of the result for a possibly discontinuous choice of the weights

Error estimates for DeepOnets: A deep learning framework in infinite dimensions

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 particular, we extend the universal approximation property of DeepOnets to include measurable mappings in non-compact spaces. By a decomposition of the error into encoding, approximation and reconstruction errors, we prove both lower and upper bounds on the total error, relating it to the spectral decay properties of the covariance operators, associated with the underlying measures. We derive almost optimal error bounds with very general affine reconstructors and with random sensor locations as well as bounds on the generalization error, using covering number arguments. We illustrate our general framework with four prototypical examples of nonlinear operators, namely those arising in a nonlinear forced ODE, an elliptic PDE with variable coefficients and nonlinear parabolic and hyperbolic PDEs. In all these examples, we prove that DeepOnets break the curse of dimensionality, thus demonstrating the efficient approximation of infinite-dimensional operators with this machine learning framework

Approximation Theory and Related Applications

In recent years, we have seen a growing interest in various aspects of approximation theory. This happened due to the increasing complexity of mathematical models that require computer calculations and the development of the theoretical foundations of the approximation theory. Approximation theory has broad and important applications in many areas of mathematics, including functional analysis, differential equations, dynamical systems theory, mathematical physics, control theory, probability theory and mathematical statistics, and others. Approximation theory is also of great practical importance, as approximate methods and estimation of approximation errors are used in physics, economics, chemistry, signal theory, neural networks and many other areas. This book presents the works published in the Special Issue "Approximation Theory and Related Applications". The research of the world’s leading scientists presented in this book reflect new trends in approximation theory and related topics

Representational Capabilities of Feed-forward and Sequential Neural Architectures

Despite the widespread empirical success of deep neural networks over the past decade, a comprehensive understanding of their mathematical properties remains elusive, which limits the abilities of practitioners to train neural networks in a principled manner. This dissertation provides a representational characterization of a variety of neural network architectures, including fully-connected feed-forward networks and sequential models like transformers. The representational capabilities of neural networks are most famously characterized by the universal approximation theorem, which states that sufficiently large neural networks can closely approximate any well-behaved target function. However, the universal approximation theorem applies exclusively to two-layer neural networks of unbounded size and fails to capture the comparative strengths and weaknesses of different architectures. The thesis addresses these limitations by quantifying the representational consequences of random features, weight regularization, and model depth on feed-forward architectures. It further investigates and contrasts the expressive powers of transformers and other sequential neural architectures. Taken together, these results apply a wide range of theoretical tools—including approximation theory, discrete dynamical systems, and communication complexity—to prove rigorous separations between different neural architectures and scaling regimes

Neural and spectral operator surrogates: unified construction and expression rate bounds

Approximation rates are analyzed for deep surrogates of maps between infinite-dimensional function spaces, arising e.g. as data-to-solution maps of linear and nonlinear partial differential equations. Specifically, we study approximation rates for Deep Neural Operator and Generalized Polynomial Chaos (gpc) Operator surrogates for nonlinear, holomorphic maps between infinite-dimensional, separable Hilbert spaces. Operator in- and outputs from function spaces are assumed to be parametrized by stable, affine representation systems. Admissible representation systems comprise orthonormal bases, Riesz bases or suitable tight frames of the spaces under consideration. Algebraic expression rate bounds are established for both, deep neural and spectral operator surrogates acting in scales of separable Hilbert spaces containing domain and range of the map to be expressed, with finite Sobolev or Besov regularity. We illustrate the abstract concepts by expression rate bounds for the coefficient-to-solution map for a linear elliptic PDE on the torus

Localized linear polynomial operators and quadrature formulas on the sphere

The purpose of this paper is to construct universal, auto--adaptive, localized, linear, polynomial (-valued) operators based on scattered data on the (hyper--)sphere \SS^q ($q\ge 2$). The approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings. Numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete Fourier projection polynomial approximations. An essential ingredient in our construction is the construction of quadrature formulas based on scattered data, exact for integrating spherical polynomials of (moderately) high degree. Our formulas are based on scattered sites; i.e., in contrast to such well known formulas as Driscoll--Healy formulas, we need not choose the location of the sites in any particular manner. While the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18, we are able to construct formulas exact for spherical polynomials of degree 178.Comment: 24 pages 2 figures, accepted for publication in SIAM J. Numer. Ana