mlsanityData to 14 Oct 2026

Partition of unity networks: deep hp-approximation

arXiv 2101.11256 ↗First posted 27 Jan 2021cs.LG cs.NA math.NAcited by 16

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Abstract (arXiv)

Approximation theorists have established best-in-class optimal approximation rates of deep neural networks by utilizing their ability to simultaneously emulate partitions of unity and monomials. Motivated by this, we propose partition of unity networks (POUnets) which incorporate these elements directly into the architecture. Classification architectures of the type used to learn probability measures are used to build a meshfree partition of space, while polynomial spaces with learnable coefficients are associated to each partition. The resulting hp-element-like approximation allows use of a fast least-squares optimizer, and the resulting architecture size need not scale exponentially with spatial dimension, breaking the curse of dimensionality. An abstract approximation result establishes desirable properties to guide network design. Numerical results for two choices of architecture demonstrate that POUnets yield hp-convergence for smooth functions and consistently outperform MLPs for piecewise polynomial functions with large numbers of discontinuities.

Key facts. Cited by 16 papers in the map (data through 14 Oct 2026). Builds on 15 of its 27 references that are in the map. First posted on arXiv 27 Jan 2021.

Builds on

15 of 27 references.

References · 15

  1. 2015Deep Residual Learning for Image Recognitioncited by 52,096
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  3. 2016Error bounds for approximations with deep ReLU networkscited by 574
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  6. 2016Benefits of depth in neural networkscited by 250
  7. 2018Optimal approximation of continuous functions by very deep ReLU networkscited by 166
  8. 2020Understanding and mitigating gradient pathologies in physics-informed neural networkscited by 135
  9. 2020The gap between theory and practice in function approximation with deep neural networkscited by 54
  10. 2019Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpointcited by 42
  11. 2019Chart Auto-Encoders for Manifold Structured Datacited by 33
  12. 2020Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networkscited by 27
3 more
  1. 2019Full error analysis for the training of deep neural networkscited by 17
  2. 2019Growing axons: greedy learning of neural networks with application to function approximationcited by 6
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Cited by year (2026 to 14 Oct)

Citing papers per year: 2021: 5; 2022: 2; 2023: 2; 2024: 1; 2025: 2; 2026: 4.

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  1. 2021SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEscited by 32
  2. 2023Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategiescited by 20
  3. 2021Error Estimates for the Deep Ritz Method with Boundary Penaltycited by 14
  4. 2023Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanicscited by 4
  5. 2021Machine-learning custom-made basis functions for partial differential equationscited by 3
  6. 2021Probabilistic partition of unity networks: clustering based deep approximationcited by 2
  7. 2022Probabilistic partition of unity networks for high-dimensional regression problemscited by 2
  8. 2024Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticitycited by 2
  9. 2025RSR-NF: Neural Field Regularization by Static Restoration Priors for Dynamic Imagingcited by 2
  10. 2025AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kineticscited by 2
  11. 2021Polynomial-Spline Neural Networks with Exact Integralscited by 0
  12. 2022Parameter-varying neural ordinary differential equations with partition-of-unity networkscited by 0
4 more
  1. 2026Partition of Unity Neural Networks for Interpretable Classification with Explicit Class Regionscited by 0
  2. 2026Escaping Spectral Bias without Backpropagation: Fast Implicit Neural Representations with Extreme Learning Machinescited by 0
  3. 2026Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decompositioncited by 0
  4. 2026Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networkscited by 0