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.

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:

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

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 contains a detailed specification of Elvet's API.

Citation

If you use elvet, please cite arXiv:2103.14575

Bibtex:

@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}
}