Universal Approximation

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Overview

Universal approximation asks when neural networks and related models can approximate target functions, operators, or probability transformations, a prerequisite for learning what those models can express. Common approaches prove density under conditions on activation functions and depth, derive error rates for smoothness classes, and study architectures such as residual networks, invariant models, Kolmogorov–Arnold networks, and signature-based models. Results now extend beyond compact finite-dimensional domains to derivatives, path-dependent processes, variable-size sets, partial differential equations, and measure-valued maps. A central tension is that universality may fail or require substantial resources under restrictions such as narrow architectures, small embeddings, atomic inputs, finite precision, or physical noise, making quantitative and practical guarantees important open concerns.

Latest papers

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29 papers4 papers in the last four weeks, against 1 the four weeks before. 0.0% of all new papers.

  1. Proofs of strong forms of the Bovdi--Geche universality conjecture and a complex analogue

    Oct 26, 2026Victor Bovdi, Ambrus PalUniversal ApproximationNeural Network Approximation Theory

  2. Two hidden layer suffice: the caterpillar net

    Oct 19, 2026Jakob Paul ZimmermannReLU Neural NetworksShallow Neural Networks

    FU Berlin · TU Berlin

  3. Wavelength Diversity Expands the Expressive Power of Optical Computing

    Oct 16, 2026Shiqi Chen, Che-Yung Shen, Parnian Ghapandar Kashani +2Photonic ComputingUniversal Approximation

    Electrical and Computer Engineering Department, University of California, Los Angeles, CA, 90095, USA · Bioengineering Department, University of California, Los Angeles, CA, 90095, USA · California NanoSystems Institute (CNSI), University of California, Los Angeles, CA, 90095, USA

  4. Kolmogorov-Arnold Classifier Systems as Universal Approximators

    Sep 29, 2026Hiroki Shiraishi, Hisao Ishibuchi, Masaya NakataUniversal ApproximationFunction Approximation

    Graduate School of Engineering Science, Yokohama National University, Yokohama 240-8501, Japan · Digital Healthcare Research Department, Hitachi, Ltd., Tokyo 185-8601, Japan · Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China +1

  5. Universal Approximation of Measure-to-Measure Operators by Pushforwards

    Sep 28, 2026Takashi Furuya, Nicholas H. Nelsen, Frank ColeTransformerUniversal Approximation

    Doshisha University, RIKEN AIP · UT Austin · UCLA

  6. Kolmogorov--Arnold stability for discontinuous functions

    Sep 7, 2026Sviatoslav V. DzhenzherAdversarial RobustnessKolmogorov-Arnold Networks

    Moscow Institute of Physics and Technology 141701, Institutskii lane, 9, Dolgoprudny, Russia.

  7. Expressivity In Multimodal Contrastive Learning

    Aug 17, 2026Andrew Stuart, Florian WolfMultimodal Contrastive LearningNeural Network Expressivity

    California Institute of Technology

  8. A Single Atom in Front of a Mirror is a Universal Reservoir Computer

    Aug 11, 2026Peter J. Ehlers, Phi Hung Nguyen, Kanu Sinha +4Reservoir ComputingUniversal Approximation

    Wyant College of Optical Sciences, University of Arizona, Tucson, AZ, USA · Department of Physics, University of Arizona, Tucson, AZ, USA · School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney, Australia

  9. Universal Function Approximation via Diffractive Optical Processors: Physical Limits, Error Bounds, and Learnability

    Aug 5, 2026Md Sadman Sakib Rahman, Che-Yung Shen, Aydogan OzcanDiffractive Neural NetworksUniversal Approximation

    Electrical and Computer Engineering Department, University of California, Los Angeles, CA, 90095, USA · Bioengineering Department, University of California, Los Angeles, CA, 90095, USA · California NanoSystems Institute (CNSI), University of California, Los Angeles, CA, 90095, USA

  10. Minimum Block Width for Universal Approximation by Residual Neural Networks with Inner Width One

    Jul 6, 2026Qi Zhou, Xuan Zhou, Xiao-Song YangReLU Neural NetworksResidual Learning

    School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, P.R. China. · Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, Hubei, P.R. China.

  11. Algorithmic Foundations of Deep Learning: Complexity-Theoretic Rates and a Characterization of Universal Approximation

    Jun 25, 2026Anastasis Kratsios, Simone Brugiapaglia, Bum Jun Kim +2Universal ApproximationNeural Network Approximation Theory

    McMaster University & Vector Institute, Toronto, ON, Canada · Concordia University, Montréal, QC, Canada · University of Tokyo, Tokyo, Japan +2

  12. EML Trees Are Universal Approximators

    Jun 22, 2026Joe Germany, Elie Abdo, Joseph BakarjiUniversal ApproximationNeural Network Approximation Theory

    Department of Mathematics American University of Beirut Beirut, Lebanon · Department of Mechanical Engineering American University of Beirut Beirut, Lebanon

  13. Learning universal approximations for partial differential equations with Physics-Informed Broad Learning System

    Jun 18, 2026Zhiwen Yu, Derong Yang, Liujian Zhang +5PDE SolvingUniversal Approximation

    School of Computer Science and Engineering, South China University of Technology, Guangzhou 510006, China · Peng Cheng Laboratory, Shenzhen 518000, China · School of Future Technology, South China University of Technology, Guangzhou 510006, China +2

  14. Sobolev Approximation by Fixed-Size Neural Networks with Arbitrary Accuracy

    Jun 15, 2026Baicheng Li, Haizhao Yang, Shijun ZhangNeural Network Activation FunctionsUniversal Approximation

    Department of Mathematics, University of Maryland, College Park, MD, USA · Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

  15. Weighted universal approximation of differentiable maps on infinite-dimensional manifolds

    Jun 8, 2026Philipp Schmocker, Josef TeichmannUniversal ApproximationNeural Network Approximation Theory

  16. Expressive Power of Floating-Point Neural Networks with Arbitrary Reduction Orders and Inexact Activation Implementations

    May 27, 2026Yeachan Park, Geonho Hwang, Wonyeol Lee +1Neural Network Activation FunctionsUniversal Approximation

  17. Any-Dimensional Invariant Universality

    May 22, 2026Shengtai Yao, Eitan Levin, Mateo DíazUniversal ApproximationNeural Network Approximation Theory

    Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD 21218, USA. · Department of Computing and Mathematical Sciences, Caltech, Pasadena, CA 91125, USA.

  18. Approximation Theory for Neural Networks: Old and New

    May 20, 2026Soumendu Sundar Mukherjee, Himasish TalukdarUniversal ApproximationNeural Network Approximation Theory

    Statistics and Mathematics Unit, Indian Statistical Institute, Kolkata

  19. Embedding Dimension Lower Bounds for Universality of Deep Sets and Janossy Pooling

    May 8, 2026Ali Syed, Aditya Nambiar, Jonathan W. SiegelRepresentation LearningDeep Sets

    Department of Mathematics Texas A&M University College Station, TX 77843

  20. Progressive Approximation in Deep Residual Networks: Theory and Validation

    Apr 27, 2026Wei Wang, Xiao-Yong Wei, Qing LiEfficient Neural Network InferenceResidual Learning

    Department of Computing, the Hong Kong Polytechnic University.

  21. Necessary and sufficient conditions for universality of Kolmogorov-Arnold networks

    Apr 26, 2026Vugar IsmailovKolmogorov-Arnold NetworksUniversal Approximation

  22. Universal Approximation Constraints of Narrow ResNets: The Tunnel Effect

    Mar 30, 2026Christian Kuehn, Sara-Viola Kuntz, Tobias WöhrerUniversal ApproximationNeural Network Approximation Theory

    Technical University of Munich, School of Computation, Information and Technology, Department of Mathematics, Boltzmannstraße 3, 85748 Garching, Germany · Munich Data Science Institute (MDSI), Garching, Germany · Munich Center for Machine Learning (MCML), München, Germany +1

  23. Global universality via discrete-time signatures

    Mar 10, 2026Mihriban Ceylan, David J. PrömelPath SignaturesUniversal Approximation

  24. Global universal approximation with Brownian signatures

    Dec 18, 2025Mihriban Ceylan, David J. PrömelUniversal ApproximationPath Signatures

  25. SUPN: Shallow Universal Polynomial Networks for Efficient Function Approximation

    Nov 26, 2025Zachary Morrow, Michael Penwarden, Brian Chen +3Polynomial Neural NetworksNeural Network Approximation Theory

  26. Autonomous-Flow-Based Generation

    Nov 13, 2025Hossein Rouhvarzi, Anastasis KratsiosUniversal ApproximationContinuous Normalizing Flows

    McMaster University and the Vector Institute Department of Mathematics 1280 Main Street West, Hamilton, Ontario, L8S 4K1, Canada

  27. Distributionally robust approximation property of neural networks

    Oct 10, 2025Mihriban Ceylan, David J. PrömelNeural Network Approximation TheoryUniversal Approximation

  28. Re-uploading quantum data: a universal function approximator for quantum inputs

    Sep 23, 2025Hyunho Cha, Daniel K. Park, Jungwoo LeeQuantum Machine LearningUniversal Approximation

    NextQuantum and Department of Electrical and Computer Engineering, Seoul National University, Seoul 08826, Republic of Korea · Department of Statistics and Data Science, Yonsei University, Seoul 03722, Republic of Korea · Department of Applied Statistics, Yonsei University, Seoul 03722, Republic of Korea +1

  29. On shallow feedforward neural networks with inputs from a topological space

    Apr 3, 2025Vugar IsmailovFeedforward Neural NetworksNeural Network Approximation Theory

    Institute of Mathematics and Mechanics, Baku, Azerbaijan · Center for Mathematics and its Applications, Khazar University, Baku, Azerbaijan