# Partition of unity networks: deep hp-approximation

arXiv: https://arxiv.org/abs/2101.11256 · HTML: https://mlsanity.com/paper/2101.11256/

## 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 (most cited first, 15 of 15)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Benefits of depth in neural networks](https://mlsanity.com/paper/1602.04485/) (2016)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [The gap between theory and practice in function approximation with deep neural networks](https://mlsanity.com/paper/2001.07523/) (2020)
- [Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpoint](https://mlsanity.com/paper/1912.04862/) (2019)
- [Chart Auto-Encoders for Manifold Structured Data](https://mlsanity.com/paper/1912.10094/) (2019)
- [Numerical Solution of the Parametric Diffusion Equation by Deep Neural Networks](https://mlsanity.com/paper/2004.12131/) (2020)
- [Full error analysis for the training of deep neural networks](https://mlsanity.com/paper/1910.00121/) (2019)
- [Growing axons: greedy learning of neural networks with application to function approximation](https://mlsanity.com/paper/1910.12686/) (2019)
- [A block coordinate descent optimizer for classification problems exploiting convexity](https://mlsanity.com/paper/2006.10123/) (2020)

## Built on by (most cited first, 16 of 16)

- [SPINN: Sparse, Physics-based, and partially Interpretable Neural Networks for PDEs](https://mlsanity.com/paper/2102.13037/) (2021)
- [Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies](https://mlsanity.com/paper/2306.17648/) (2023)
- [Error Estimates for the Deep Ritz Method with Boundary Penalty](https://mlsanity.com/paper/2103.01007/) (2021)
- [Neural-Integrated Meshfree (NIM) Method: A differentiable programming-based hybrid solver for computational mechanics](https://mlsanity.com/paper/2311.12915/) (2023)
- [Machine-learning custom-made basis functions for partial differential equations](https://mlsanity.com/paper/2111.05307/) (2021)
- [Probabilistic partition of unity networks: clustering based deep approximation](https://mlsanity.com/paper/2107.03066/) (2021)
- [Probabilistic partition of unity networks for high-dimensional regression problems](https://mlsanity.com/paper/2210.02694/) (2022)
- [Differentiable Neural-Integrated Meshfree Method for Forward and Inverse Modeling of Finite Strain Hyperelasticity](https://mlsanity.com/paper/2407.11183/) (2024)
- [RSR-NF: Neural Field Regularization by Static Restoration Priors for Dynamic Imaging](https://mlsanity.com/paper/2503.10015/) (2025)
- [AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kinetics](https://mlsanity.com/paper/2510.12999/) (2025)
- [Polynomial-Spline Neural Networks with Exact Integrals](https://mlsanity.com/paper/2110.14055/) (2021)
- [Parameter-varying neural ordinary differential equations with partition-of-unity networks](https://mlsanity.com/paper/2210.00368/) (2022)
- [Partition of Unity Neural Networks for Interpretable Classification with Explicit Class Regions](https://mlsanity.com/paper/2602.00511/) (2026)
- [Escaping Spectral Bias without Backpropagation: Fast Implicit Neural Representations with Extreme Learning Machines](https://mlsanity.com/paper/2602.07603/) (2026)
- [Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decomposition](https://mlsanity.com/paper/2605.25115/) (2026)
- [Localized Operator Learning with Adaptive Partition-of-Unity Mixture-of-Expert Networks](https://mlsanity.com/paper/2610.04708/) (2026)

## 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.