Adjoints

There are multiple ways to backpropagate through a differential equation (to compute the gradient of the solution with respect to its initial condition and any parameters).

Info

Why are there multiple ways of backpropagating through a differential equation? Suppose we are given an ODE

\(\frac{\mathrm{d}y}{\mathrm{d}t} = f(t, y(t))\)

on \([t_0, t_1]\), with initial condition \(y(0) = y_0\). So \(y(t)\) is the (unknown) exact solution, to which we will compute some numerical approxiation \(y_N \approx y(t_1)\).

We may directly apply autodifferentiation to calculate \(\frac{\mathrm{d}y_N}{\mathrm{d}y_0}\), by backpropagating through the internals of the solver. This is known a "discretise then optimise", is the default in Diffrax, and corresponds to diffrax.RecursiveCheckpointAdjoint below.

Alternatively we may compute \(\frac{\mathrm{d}y(t_1)}{\mathrm{d}y_0}\) analytically. In doing so we obtain a backwards-in-time ODE that we must numerically solve to obtain the desired gradients. This is known as "optimise then discretise", and corresponds to diffrax.BacksolveAdjoint below.

diffrax.AbstractSolver

diffrax.AbstractAdjoint

Abstract base class for all adjoint methods.

loop(self, *, args, terms, solver, stepsize_controller, discrete_terminating_event, saveat, t0, t1, dt0, max_steps, throw, init_state, passed_solver_state, passed_controller_state) abstractmethod

Runs the main solve loop. Subclasses can override this to provide custom backpropagation behaviour; see for example the implementation of diffrax.BacksolveAdjoint.

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diffrax.RecursiveCheckpointAdjoint (AbstractAdjoint)

Backpropagate through diffrax.diffeqsolve by differentiating the numerical solution directly. This is sometimes known as "discretise-then-optimise", or described as "backpropagation through the solver".

For most problems this is the preferred technique for backpropagating through a differential equation.

In addition a binomial checkpointing scheme is used so that memory usage is low. (This checkpointing can increase compile time a bit, though.)

diffrax.NoAdjoint (AbstractAdjoint)

Disable backpropagation through diffrax.diffeqsolve. Forward-mode autodifferentiation (jax.jvp) will continue to work as normal. If you do not need to differentiate the results of diffrax.diffeqsolve then this may sometimes improve the speed at which the differential equation is solved.

diffrax.ImplicitAdjoint (AbstractAdjoint)

Backpropagate via the implicit function theorem.

This is used when solving towards a steady state, typically using diffrax.SteadyStateEvent. In this case, the output of the solver is \(y(θ)\) for which \(f(t, y(θ), θ) = 0\). (Where \(θ\) corresponds to all parameters found through terms and args, but not y0.) Then we can skip backpropagating through the solver and instead directly compute \(\frac{\mathrm{d}y}{\mathrm{d}θ} = - (\frac{\mathrm{d}f}{\mathrm{d}y})^{-1}\frac{\mathrm{d}f}{\mathrm{d}θ}\) via the implicit function theorem.

diffrax.BacksolveAdjoint (AbstractAdjoint)

Backpropagate through diffrax.diffeqsolve by solving the continuous adjoint equations backwards-in-time. This is also sometimes known as "optimise-then-discretise", the "continuous adjoint method" or simply the "adjoint method".

This method implies very low memory usage, but the computed gradients will only be approximate. As such other methods are generally preferred unless exceeding memory is a concern.

This will compute gradients with respect to the terms, y0 and args arguments passed to diffrax.diffeqsolve. If you attempt to compute gradients with respect to anything else (for example t0, or arguments passed via closure), then a CustomVJPException will be raised. See also this FAQ entry.

Note

This was popularised by this paper. For this reason it is sometimes erroneously believed to be a better method for backpropagation than the other choices available.

Warning

Using this method prevents computing forward-mode autoderivatives of diffrax.diffeqsolve. (That is to say, jax.jvp will not work.)

__init__(self, **kwargs)

Arguments:

• **kwargs: The arguments for the diffrax.diffeqsolve operations that are called on the backward pass. For example use

  BacksolveAdjoint(solver=Dopri5())
  

  to specify a particular solver to use on the backward pass.

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diffrax.adjoint_rms_seminorm(x: Tuple[PyTree, PyTree, PyTree, PyTree]) -> Scalar

Defines an adjoint seminorm. This can frequently be used to increase the efficiency of backpropagation via diffrax.BacksolveAdjoint, as follows:

adjoint_controller = diffrax.PIDController(norm=diffrax.adjoint_rms_seminorm)
adjoint = diffrax.BacksolveAdjoint(stepsize_controller=adjoint_controller)
diffrax.diffeqsolve(..., adjoint=adjoint)

Note that this means that any stepsize_controller specified for the forward pass will not be automatically used for the backward pass (as adjoint_controller overrides it), so you should specify any custom rtol, atol etc. for the backward pass as well.

Reference

@article{kidger2021hey,
    author={Kidger, Patrick and Chen, Ricky T. Q. and Lyons, Terry},
    title={``{H}ey, that's not an {ODE}'': {F}aster {ODE} {A}djoints via
           {S}eminorms},
    year={2021},
    journal={International Conference on Machine Learning}
}