A-Alaa / diffrax

Numerical differential equation solvers in JAX. Autodifferentiable and GPU-capable. https://docs.kidger.site/diffrax/

Diffrax

Numerical differential equation solvers in JAX. Autodifferentiable and GPU-capable.

Diffrax is a JAX-based library providing numerical differential equation solvers.

Features include:

• ODE/SDE/CDE (ordinary/stochastic/controlled) solvers;
• lots of different solvers (including Tsit5, Dopri8, symplectic solvers, implicit solvers);
• vmappable everything (including the region of integration);
• using a PyTree as the state;
• dense solutions;
• multiple adjoint methods for backpropagation;
• support for neural differential equations.

From a technical point of view, the internal structure of the library is pretty cool -- all kinds of equations (ODEs, SDEs, CDEs) are solved in a unified way (rather than being treated separately), producing a small tightly-written library.

Installation

pip install diffrax

Requires Python >=3.7 and JAX >=0.3.4.

Documentation

Available at https://docs.kidger.site/diffrax.

Quick example

from diffrax import diffeqsolve, ODETerm, Dopri5
import jax.numpy as jnp

def f(t, y, args):
    return -y

term = ODETerm(f)
solver = Dopri5()
y0 = jnp.array([2., 3.])
solution = diffeqsolve(term, solver, t0=0, t1=1, dt0=0.1, y0=y0)

Here, Dopri5 refers to the Dormand--Prince 5(4) numerical differential equation solver, which is a standard choice for many problems.

Citation

If you found this library useful in academic research, please cite: (arXiv link)

@phdthesis{kidger2021on,
    title={{O}n {N}eural {D}ifferential {E}quations},
    author={Patrick Kidger},
    year={2021},
    school={University of Oxford},
}

(Also consider starring the project on GitHub.)

See also

Neural networks: Equinox.

Type annotations and runtime checking for PyTrees and shape/dtype of JAX arrays: jaxtyping.

SymPy<->JAX conversion; train symbolic expressions via gradient descent: sympy2jax.