Built on

Nonlinear approximation and (deep) ReLU networks
Original
Daubechies, I.; DeVore, R.; Foucart, S.; Hanin, B.; and Petrova, G. 2019 · 1905
Earlier work this paper cites.
Full error analysis for the training of deep neural networks
Original
Beck, C.; Jentzen, A.; and Kuckuck, B. 2019 · 1910
Earlier work this paper cites.
Growing axons: greedy learning of neural networks with application to function approximation
Original
Fokina, D.; and Oseledets, I. 2019 · 1910
Earlier work this paper cites.
Robust training and initialization of deep neural networks: An adaptive basis viewpoint
Original
Cyr, E. C.; Gulian, M. A.; Patel, R. G.; Perego, M.; and Trask, N. A. 2019 · 1912
Earlier work this paper cites.
Chart Auto-Encoders for Manifold Structured Data
Original
Schonsheck, S.; Chen, J.; and Lai, R. 2019 · 1912
Earlier work this paper cites.
Radial basis functions, multi-variable functional interpolation and adaptive networks
Broomhead, D. S.; and Lowe, D. 1988 · 1988
Earlier work this paper cites.
Radial basis function network configuration using genetic algorithms
Billings, S. A.; and Zheng, G. L. 1995 · 1995
Earlier work this paper cites.
Taking on the curse of dimensionality in joint distributions using neural networks
Bengio, S.; and Bengio, Y. 2000 · 2000
Earlier work this paper cites.
Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities , volume 5
Hebey, E. 2000 · 2000
Earlier work this paper cites.
The gap between theory and practice in function approximation with deep neural networks
Original
Adcock, B.; and Dexter, N. 2020 · 2001
Earlier work this paper cites.

Similar

The generalized finite element method
Strouboulis, T.; Copps, K.; and Babuška, I. 2001 · 2001
Cited alongside, same era.
Understanding and mitigating gradient pathologies in physics-informed neural networks
Original
Wang, S.; Teng, Y.; and Perdikaris, P. 2020 · 2001
Cited alongside, same era.
Fast evaluation of radial basis functions: Methods based on partition of unity
Wendland, H. 2002 · 2002
Cited alongside, same era.
Numerical solution of the parametric diffusion equation by deep neural networks
Original
Geist, M.; Petersen, P.; Raslan, M.; Schneider, R.; and Kutyniok, G. 2020 · 2004
Cited alongside, same era.
A block coordinate descent optimizer for classification problems exploiting convexity
Original
Patel, R. G.; Trask, N. A.; Gulian, M. A.; and Cyr, E. C. 2020 · 2006
Cited alongside, same era.

Then

Benefits of depth in neural networks
Original
Telgarsky, M. 2016 · 2016
Later among the works it cites.
Breaking the curse of dimensionality with convex neural networks
Bach, F. 2017 · 2017
Later among the works it cites.
Deep learning with Python
Chollet, F. 2017 · 2017
Later among the works it cites.
Error bounds for approximations with deep ReLU networks
Yarotsky, D. 2017 · 2017
Later among the works it cites.
Solving high-dimensional partial differential equations using deep learning
Han, J.; Jentzen, A.; and Weinan, E. 2018 · 2018
Later among the works it cites.
ReLU deep neural networks and linear finite elements
Original
He, J.; Li, L.; Xu, J.; and Zheng, C. 2018 · 2018

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2021

Partition of unity networks: deep hp-approximation

Lee, Kookjin, Trask, Nathaniel A., Patel, Ravi G. et al.

Understand

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.

Built on

Nonlinear approximation and (deep) ReLU networks
Original
Daubechies, I.; DeVore, R.; Foucart, S.; Hanin, B.; and Petrova, G. 2019 · 1905
Earlier work this paper cites.
Full error analysis for the training of deep neural networks
Original
Beck, C.; Jentzen, A.; and Kuckuck, B. 2019 · 1910
Earlier work this paper cites.
Growing axons: greedy learning of neural networks with application to function approximation
Original
Fokina, D.; and Oseledets, I. 2019 · 1910
Earlier work this paper cites.
Robust training and initialization of deep neural networks: An adaptive basis viewpoint
Original
Cyr, E. C.; Gulian, M. A.; Patel, R. G.; Perego, M.; and Trask, N. A. 2019 · 1912
Earlier work this paper cites.
Chart Auto-Encoders for Manifold Structured Data
Original
Schonsheck, S.; Chen, J.; and Lai, R. 2019 · 1912
Earlier work this paper cites.
Radial basis functions, multi-variable functional interpolation and adaptive networks
Broomhead, D. S.; and Lowe, D. 1988 · 1988
Earlier work this paper cites.
Radial basis function network configuration using genetic algorithms
Billings, S. A.; and Zheng, G. L. 1995 · 1995
Earlier work this paper cites.
Taking on the curse of dimensionality in joint distributions using neural networks
Bengio, S.; and Bengio, Y. 2000 · 2000
Earlier work this paper cites.
Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities , volume 5
Hebey, E. 2000 · 2000
Earlier work this paper cites.
The gap between theory and practice in function approximation with deep neural networks
Original
Adcock, B.; and Dexter, N. 2020 · 2001
Earlier work this paper cites.

Similar

The generalized finite element method
Strouboulis, T.; Copps, K.; and Babuška, I. 2001 · 2001
Cited alongside, same era.
Understanding and mitigating gradient pathologies in physics-informed neural networks
Original
Wang, S.; Teng, Y.; and Perdikaris, P. 2020 · 2001
Cited alongside, same era.
Fast evaluation of radial basis functions: Methods based on partition of unity
Wendland, H. 2002 · 2002
Cited alongside, same era.
Numerical solution of the parametric diffusion equation by deep neural networks
Original
Geist, M.; Petersen, P.; Raslan, M.; Schneider, R.; and Kutyniok, G. 2020 · 2004
Cited alongside, same era.
A block coordinate descent optimizer for classification problems exploiting convexity
Original
Patel, R. G.; Trask, N. A.; Gulian, M. A.; and Cyr, E. C. 2020 · 2006
Cited alongside, same era.

Then

Benefits of depth in neural networks
Original
Telgarsky, M. 2016 · 2016
Later among the works it cites.
Breaking the curse of dimensionality with convex neural networks
Bach, F. 2017 · 2017
Later among the works it cites.
Deep learning with Python
Chollet, F. 2017 · 2017
Later among the works it cites.
Error bounds for approximations with deep ReLU networks
Yarotsky, D. 2017 · 2017
Later among the works it cites.
Solving high-dimensional partial differential equations using deep learning
Han, J.; Jentzen, A.; and Weinan, E. 2018 · 2018
Later among the works it cites.
ReLU deep neural networks and linear finite elements
Original
He, J.; Li, L.; Xu, J.; and Zheng, C. 2018 · 2018