This library provides ordinary differential equation (ODE) solvers implemented in PyTorch. Backpropagation through all solvers is supported using the adjoint method. For usage of ODE solvers in deep learning applications, see [1].
As the solvers are implemented in PyTorch, algorithms in this repository are fully supported to run on the GPU.
git clone https://github.com/rtqichen/torchdiffeq.git
cd torchdiffeq
pip install -e .
Examples are placed in the examples
directory.
We encourage those who are interested in using this library to take a look at examples/ode_demo.py
for understanding how to use torchdiffeq
to fit a simple spiral ODE.
This library provides one main interface odeint
which contains general-purpose algorithms for solving initial value problems (IVP), with gradients implemented for all main arguments. An initial value problem consists of an ODE and an initial value,
dy/dt = f(t, y) y(t_0) = y_0.
The goal of an ODE solver is to find a continuous trajectory satisfying the ODE that passes through the initial condition.
To solve an IVP using the default solver:
from torchdiffeq import odeint
odeint(func, y0, t)
where func
is any callable implementing the ordinary differential equation f(t, x)
, y0
is an any-D Tensor or a tuple of any-D Tensors representing the initial values, and t
is a 1-D Tensor containing the evaluation points. The initial time is taken to be t[0]
.
Backpropagation through odeint
goes through the internals of the solver, but this is not supported for all solvers. Instead, we encourage the use of the adjoint method explained in [1], which will allow solving with as many steps as necessary due to O(1) memory usage.
To use the adjoint method:
from torchdiffeq import odeint_adjoint as odeint
odeint(func, y0, t)
odeint_adjoint
simply wraps around odeint
, but will use only O(1) memory in exchange for solving an adjoint ODE in the backward call.
The biggest gotcha is that func
must be a nn.Module
when using the adjoint method. This is used to collect parameters of the differential equation.
rtol
Relative tolerance.atol
Absolute tolerance.method
One of the solvers listed below.
Adaptive-step:
dopri5
Runge-Kutta 4(5) [default].adams
Adaptive-order implicit Adams.
Fixed-step:
euler
Euler method.midpoint
Midpoint method.rk4
Fourth-order Runge-Kutta with 3/8 rule.explicit_adams
Explicit Adams.fixed_adams
Implicit Adams.
[1] Ricky T. Q. Chen, Yulia Rubanova, Jesse Bettencourt, David Duvenaud. "Neural Ordinary Differential Equations." Advances in Neural Processing Information Systems. 2018. [arxiv]
If you found this library useful in your research, please consider citing
@article{chen2018neural,
title={Neural Ordinary Differential Equations},
author={Chen, Ricky T. Q. and Rubanova, Yulia and Bettencourt, Jesse and Duvenaud, David},
journal={Advances in Neural Information Processing Systems},
year={2018}
}