functorch.hessian(func, argnums=0)[source]

Computes the Hessian of func with respect to the arg(s) at index argnum via a forward-over-reverse strategy.

The forward-over-reverse strategy (composing jacfwd(jacrev(func))) is a good default for good performance. It is possible to compute Hessians through other compositions of jacfwd() and jacrev() like jacfwd(jacfwd(func)) or jacrev(jacrev(func)).

  • func (function) – A Python function that takes one or more arguments, one of which must be a Tensor, and returns one or more Tensors

  • argnums (int or Tuple[int]) – Optional, integer or tuple of integers, saying which arguments to get the Hessian with respect to. Default: 0.


Returns a function that takes in the same inputs as func and returns the Hessian of func with respect to the arg(s) at argnums.


PyTorch’s forward-mode AD coverage on operators is not very good at the moment. You may see this API error out with “forward-mode AD not implemented for operator X”. If so, please file us a bug report and we will prioritize it.

A basic usage with a R^N -> R^1 function gives a N x N Hessian:

>>> from functorch import hessian
>>> def f(x):
>>>   return x.sin().sum()
>>> x = torch.randn(5)
>>> hess = jacfwd(jacrev(f))(x)
>>> assert torch.allclose(hess, torch.diag(-x.sin()))