# Elvet -- a neural network-based differential equation and variational problem solver

arXiv: https://arxiv.org/abs/2103.14575 · HTML: https://mlsanity.com/paper/2103.14575/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 39 references that are in the map.
- First posted on arXiv 26 Mar 2021.

## Builds on (most cited first, 10 of 10)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Neural Networks Trained to Solve Differential Equations Learn General Representations](https://mlsanity.com/paper/1807.00042/) (2018)
- [A Discussion on Solving Partial Differential Equations using Neural Networks](https://mlsanity.com/paper/1904.07200/) (2019)
- [dNNsolve: an efficient NN-based PDE solver](https://mlsanity.com/paper/2103.08662/) (2021)

## Built on by (most cited first, 7 of 7)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [IDRLnet: A Physics-Informed Neural Network Library](https://mlsanity.com/paper/2107.04320/) (2021)
- [Quantum Optimisation of Complex Systems with a Quantum Annealer](https://mlsanity.com/paper/2105.13945/) (2021)
- [Qade: Solving Differential Equations on Quantum Annealers](https://mlsanity.com/paper/2204.03657/) (2022)
- [jinns: a JAX Library for Physics-Informed Neural Networks](https://mlsanity.com/paper/2412.14132/) (2024)
- [Learning Biomolecular Motion: The Physics-Informed Machine Learning Paradigm](https://mlsanity.com/paper/2511.06585/) (2025)

## Abstract (arXiv)

We present Elvet, a Python package for solving differential equations and variational problems using machine learning methods. Elvet can deal with any system of coupled ordinary or partial differential equations with arbitrary initial and boundary conditions. It can also minimize any functional that depends on a collection of functions of several variables while imposing constraints on them. The solution to any of these problems is represented as a neural network trained to produce the desired function.