# Polynomial-Spline Neural Networks with Exact Integrals

arXiv: https://arxiv.org/abs/2110.14055 · HTML: https://mlsanity.com/paper/2110.14055/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 27 references that are in the map.
- First posted on arXiv 26 Oct 2021.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Understanding Deep Neural Networks with Rectified Linear Units](https://mlsanity.com/paper/1611.01491/) (2016)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Mad Max: Affine Spline Insights into Deep Learning](https://mlsanity.com/paper/1805.06576/) (2018)
- [Robust Training and Initialization of Deep Neural Networks: An Adaptive Basis Viewpoint](https://mlsanity.com/paper/1912.04862/) (2019)
- [Full error analysis for the training of deep neural networks](https://mlsanity.com/paper/1910.00121/) (2019)
- [Partition of unity networks: deep hp-approximation](https://mlsanity.com/paper/2101.11256/) (2021)

## Abstract (arXiv)

Using neural networks to solve variational problems, and other scientific machine learning tasks, has been limited by a lack of consistency and an inability to exactly integrate expressions involving neural network architectures. We address these limitations by formulating a novel neural network architecture that combines a polynomial mixture-of-experts model with free knot B1-spline basis functions. Effectively, our architecture performs piecewise polynomial approximation on each cell of a trainable partition of unity. Our architecture exhibits both h- and p- refinement for regression problems at the convergence rates expected from approximation theory, allowing for consistency in solving variational problems. Moreover, this architecture, its moments, and its partial derivatives can all be integrated exactly, obviating a reliance on sampling or quadrature and enabling error-free computation of variational forms. We demonstrate the success of our network on a range of regression and variational problems that illustrate the consistency and exact integrability of our network architecture.