# Neural Approximation and Its Applications Source: [Neural Approximation and Its Applications]() Collection index: [llms.txt]() Data updated: 2026-03-18T05:00:04.466Z Authors: Wei-Hao Wu, Ting-Zhu Huang, Xi-Le Zhao, Yisi Luo, Deyu Meng Published: 2026-03-04T06:05:43.000Z ## Abstract Multivariate function approximation is a fundamental problem in machine learning. Classic multivariate function approximations rely on hand-crafted basis functions (e.g., polynomial basis and Fourier basis), which limits their approximation ability and data adaptation ability, resulting in unsatisfactory performance. To address these challenges, we introduce the neural basis function by leveraging an untrained neural network as the basis function. Equipped with the proposed neural basis function, we suggest the neural approximation (NeuApprox) paradigm for multivariate function approximation. Specifically, the underlying multivariate function behind the multi-dimensional data is decomposed into a sum of block terms. The clear physically-interpreted block term is the product of expressive neural basis functions and their corresponding learnable coefficients, which allows us to faithfully capture distinct components of the underlying data and also flexibly adapt to new data by readily fine-tuning the neural basis functions. Attributed to the elaborately designed block terms, the suggested NeuApprox enjoys strong approximation ability and flexible data adaptation ability over the hand-crafted basis function-based methods. We also theoretically prove that NeuApprox can approximate any multivariate continuous function to arbitrary accuracy. Extensive experiments on diverse multi-dimensional datasets (including multispectral images, light field data, videos, traffic data, and point cloud data) demonstrate the promising performance of NeuApprox in terms of both approximation capability and adaptability. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. - [Multi-dimensional image inpainting on MSIs Toys, Painting 256×256×31](): Multi-dimensional image inpainting; MSIs Toys, Painting 256×256×31; PSNR: 38.77. - [Multi-dimensional image inpainting on Videos (Foreman, Carphone) 144×176×150](): Multi-dimensional image inpainting; Videos (Foreman, Carphone) 144×176×150; PSNR: 31.75. - [Multi-dimensional image inpainting on Light field data Greek, Origami 128×128×240](): Multi-dimensional image inpainting; Light field data Greek, Origami 128×128×240; PSNR: 42.7. - [Traffic Data Completion on Guangzhou](): Traffic Data Completion; Guangzhou; RMSE: 9.24. - [Traffic Data Completion on PEMS07](): Traffic Data Completion; PeMS07; RMSE: 10.9685. - [Traffic Data Completion on Seattle](): Traffic Data Completion; Seattle; RMSE: 6.48. - [Point cloud data completion on Doll](): Point cloud data completion; Doll; NRMSE: 0.1373. - [Point Cloud Completion on Duck](): Point cloud data completion; Duck; NRMSE: 0.1185. - [Point Cloud Completion on Frog point cloud](): Point Cloud Completion; Frog point cloud; NRMSE: 0.1186. - [Point Cloud Completion on Mario point cloud](): Point Cloud Completion; Mario point cloud; NRMSE: 0.1985. - [Point Cloud Completion on Rabbit point cloud](): Point Cloud Completion; Rabbit point cloud; NRMSE: 0.1577. ## Original sources and implementation [Original paper]() [PDF]()