partition of unity

This work addresses the inefficiency and weak theoretical guarantees of neural networks in approximating analytic functions and general $L^p$ functions. To overcome these limitations, the authors propose an efficient ReLU network architecture based on a three-dimensional design, which explicitly constructs sawtooth functions to achieve enhanced approximation capabilities. The proposed method significantly improves the exponential approximation rates for a broad class of analytic functions and, for the first time, establishes a high-order, non-asymptotic quantitative approximation theory for general $L^p$ functions. Notably, this approach achieves superior approximation performance while maintaining parameter efficiency and providing rigorous theoretical guarantees.

This study investigates the structure and applications of functions sensitive to a set of probability measures—referred to as P-sensitive functions—within the framework of non-dominated robust stochastic models. By introducing a localized representation of such functions, the work establishes, for the first time, an equivalence between P-sensitive functions and their localizations, thereby constructing a theoretical foundation tailored to non-dominated settings. Leveraging robust function spaces and modeling approaches based on sets of probability measures, the proposed theory is successfully applied to robust optimization, the construction of convex risk measures, and the analysis of no-arbitrage conditions in single-period financial markets. These contributions provide novel analytical tools and theoretical support for advancing research in these interconnected domains.

Neural network function approximation suffers from low training efficiency, poor generalization—especially in extrapolation—and non-transferable parameters. To address these limitations, we propose a reusable initialization framework based on basis-function pretraining. Our method (1) performs unsupervised pretraining of network weights using polynomial basis functions to construct a domain-agnostic parameter prior; (2) introduces an input-domain mapping mechanism that enables adaptive alignment of pretrained parameters to arbitrary function domains; and (3) supports modular training and cross-task parameter transfer. Extensive experiments on one- and two-dimensional function approximation tasks demonstrate that our approach achieves an average 2.3× speedup in training convergence, improves extrapolation accuracy by 37%–61% (measured by error reduction), and enhances model stability. This work establishes a scalable, composable paradigm for function modeling in scientific computing and machine learning.

This paper addresses the fundamental problem of unclear generalization guarantees in machine learning function approximation—specifically, the lack of theoretical foundations for model performance on unknown manifolds and unseen data. Methodologically, it departs from explicit geometric modeling of manifolds (e.g., via Laplace–Beltrami operators or atlases) and instead integrates classical approximation theory, spectral graph theory, differential geometry, and physics-informed embedding to systematically characterize the approximation mechanisms of diverse paradigms—including deep/shallow neural networks, neural operators, Transformers, and physics-informed neural surrogates. Key contributions include: (i) a precise delineation of expressive capacity boundaries across mainstream architectures; (ii) a robust generalization analysis framework that requires no prior knowledge of manifold geometry; and (iii) the first unifying theoretical perspective for manifold learning and scientific machine learning grounded explicitly in approximation theory.

Approximating high-dimensional, low-smoothness functions—common in cross-domain learning (e.g., invariant learning, transfer learning, SAR imaging)—remains challenging due to limitations of conventional symmetric or positive-definite kernels. Method: This paper proposes a neural network framework based on asymmetric, irregular kernels, breaking away from traditional kernel symmetry/positivity constraints. It systematically constructs generalized translation networks and rotationally banded function kernels—novel asymmetric kernel architectures—and establishes their approximation theory. It further introduces the ReLU^r activation (with non-integer r > 0) and derives its uniform approximation error bound for Sobolev functions. Contribution/Results: Leveraging asymmetric kernel decomposition and Sobolev space analysis, the framework yields tight approximation error estimates for low-smooth, high-dimensional functions. Empirically, it achieves significantly improved cross-domain generalization accuracy under small-sample and low-regularity conditions.

This work systematically investigates the mathematical expressivity of neural networks, with a focus on their approximation efficiency across various function spaces. By integrating tools from functional analysis, approximation theory, and Sobolev space theory, it traces the theoretical development from the universal approximation property of single-hidden-layer networks to modern insights into depth–width trade-offs, parameter efficiency, and the influence of target function smoothness on approximation rates. The study particularly highlights the advantage of deep architectures in achieving superior parameter efficiency for structured function classes. It further incorporates recent models such as Kolmogorov–Arnold Networks (KANs) into this analytical framework, establishing a unified qualitative and quantitative understanding of neural network approximation capabilities and elucidating the pivotal role of depth in enhancing approximation efficiency.

This work addresses the challenges of parameterizing convex sets in shape optimization and inverse design by proposing an implicit representation based on sublinear neural networks. The method flexibly characterizes arbitrary convex bodies by learning positively homogeneous and convex support and gauge functions. It enjoys theoretical universal approximation capabilities for convex sets and demonstrates strong empirical performance, accurately reconstructing target shapes in experiments, thereby validating its expressiveness and effectiveness. The key innovation lies in integrating convex analysis with neural networks to establish a convex set parameterization framework that simultaneously offers rigorous theoretical guarantees and practical performance.

This work addresses the absence of readily available Gaussian quadrature rules for nonclassical weight functions by proposing a general framework that constructs such rules for arbitrary weights via the method of moments and the Stieltjes procedure. Innovatively integrating type-generic programming with adaptive high-precision arithmetic, the approach effectively controls round-off errors and, for the first time, systematically introduces tailored Gaussian quadrature methods to the statistics community. Implemented in Julia as the CustomGaussQuadrature package—accessible from R through JuliaConnectoR—the resulting quadrature rules achieve exact integration of polynomials up to degree \(2n-1\) while substantially reducing the number of function evaluations, thereby offering both high accuracy and computational efficiency.

This work investigates the approximation capability of ReLU neural networks with jointly tunable width \(N\) and depth \(L\) for infinitely smooth analytic functions. By carefully constructing networks to approximate power functions, multivariate multiplication, and polynomials, the study establishes, for the first time within a joint \((N, L)\) parameterization framework, an approximation error bound of \(O(N^{-C L^\tau})\) with constants \(C > 0\) and \(\tau > 0\). Notably, when \(N \asymp L^d\), the exponent satisfies \(\tau = 1\), substantially improving upon classical results for finitely smooth functions. This finding underscores the dominant role of depth in approximating analytic functions and reveals a novel scaling relationship between width \(N\) and \(L^d\).