# Machine-learning custom-made basis functions for partial differential equations

arXiv: https://arxiv.org/abs/2111.05307 · HTML: https://mlsanity.com/paper/2111.05307/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 39 references that are in the map.
- First posted on arXiv 9 Nov 2021.

## Builds on (most cited first, 9 of 9)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [VarNet: Variational Neural Networks for the Solution of Partial Differential Equations](https://mlsanity.com/paper/1912.07443/) (2019)
- [Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control](https://mlsanity.com/paper/2105.14094/) (2021)
- [Partition of unity networks: deep hp-approximation](https://mlsanity.com/paper/2101.11256/) (2021)
- [Probabilistic partition of unity networks: clustering based deep approximation](https://mlsanity.com/paper/2107.03066/) (2021)

## Built on by (most cited first, 3 of 3)

- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [Deep Learning for Subspace Regression](https://mlsanity.com/paper/2509.23249/) (2025)
- [Fiber neural networks for the intelligent optical fiber communications](https://mlsanity.com/paper/2408.12602/) (2024)

## Abstract (arXiv)

Spectral methods are an important part of scientific computing's arsenal for solving partial differential equations (PDEs). However, their applicability and effectiveness depend crucially on the choice of basis functions used to expand the solution of a PDE. The last decade has seen the emergence of deep learning as a strong contender in providing efficient representations of complex functions. In the current work, we present an approach for combining deep neural networks with spectral methods to solve PDEs. In particular, we use a deep learning technique known as the Deep Operator Network (DeepONet), to identify candidate functions on which to expand the solution of PDEs. We have devised an approach which uses the candidate functions provided by the DeepONet as a starting point to construct a set of functions which have the following properties: i) they constitute a basis, 2) they are orthonormal, and 3) they are hierarchical i.e., akin to Fourier series or orthogonal polynomials. We have exploited the favorable properties of our custom-made basis functions to both study their approximation capability and use them to expand the solution of linear and nonlinear time-dependent PDEs.