Elvet is a machine learning-based differential equation and variational problem solver.
- It can solve any system of **coupled ODEs or PDEs** with any boundary conditions.
- It can go beyond differential equations, and solve variational problems that consist of the **minimization of a given functional**, without computing the corresponding Euler-Lagrange equations.
- It can also be used for **fitting** any family of functions (which are viewed as a machine learning model) to a set of multi-dimensional data points.
By default, **Elvet** uses **neural networks** to solve these problems. A version 2 providing other methods, including **tensor networks** and **quantum computing**, is currently under development.
# Quick start
Try Elvet online in Google Colaboratory through the [example notebooks](https://elvet.gitlab.io/elvet/examples.html).
Install Elvet by running `pip install elvet`. Tensorflow is required, with version between 2.4 and 2.10 (both included).
The following code solves the [logistic differential equation](https://en.wikipedia.org/wiki/Logistic_function#Logistic_differential_equation):
```python
import elvet
def equation(x, f, df_dx):
return df_dx - f * (1 - f)
bc = elvet.BC(0, lambda x, f, df_dx: f - 1/2)
domain = elvet.box((-5, 5, 101))
result = elvet.solver(equation, bc, domain, epochs=5e3)
```
Where have defined the equation, in the form `equation(x, f, df_dx) == 0`, the "boundary" condition `f(0) - 1/2 == 0`, and the domain: the interval (or "box" in elvet's terms) `[-5, 5]`, with 101 equally spaced points. Then we use the `solver` function to generate a solver, which contains a machine learning model. This model is, by default, a fully connected neural net with one hidden layer with 10 units and 1 unit in the input and output layers. The `epochs` argument specifies over how many epochs this model is to be trained.
The predictions of the trained model, which give the solution to the differential equations, can be obtained through `result.prediction()`. They can also be plotted, and compared with the analytic solution, which is just the sigmoid function
```python
import elvet.plotting
def analytic_solution(x):
return 1 / (1 + elvet.math.exp(-x))
elvet.plotting.plot_prediction(result, true_function=analytic_solution)
```
This code should produce the plot
# Documentation
The [documentation](https://elvet.gitlab.io/elvet) contains a detailed specification of Elvet's API.
# Citation
If you use elvet, please cite [arXiv:2103.14575](https://arxiv.org/abs/2103.14575)
Bibtex:
```latex
@misc{araz2021elvet,
title={Elvet -- a neural network-based differential equation and variational problem solver},
author={Jack Y. Araz and Juan Carlos Criado and Michael Spannwosky},
year={2021},
eprint={2103.14575},
archivePrefix={arXiv},
primaryClass={cs.LG}
}
```
