Custom Backward for `nn.Module` Function other than `forward`

I am re-implementing the Invertible Residual Networks architecture. This model has, beyond the forward pass, also an inverse function:

class iResNetBlock(nn.Module):
    def __init__(self, input_size, hidden_size):
        self.bottleneck = nn.Sequential(
            LinearContraction(input_size, hidden_size),
            LinearContraction(hidden_size, input_size),
    def forward(self, x):
        return x + self.bottleneck(x)
    def inverse(self, y):
        x = y.clone()
        while not converged:
            # fixed point iteration
            x = y - self.bottleneck(x)
        return x

I want to add a custom backward pass to the inverse function. Since it is a fixed point iteration, one can make use of the implicit function theorem to avoid unrolling of the loop, and instead compute the gradient by solving a linear system. This is for example done in the Deep Equilibrium Models architecture.

How do I register my custom backwards pass for this function? I would have hoped that there is some decorator (this is for example possible in Jax) or other simple way of telling PyTorch to use a certain function as the backward of another function.

class iResNetBlock(nn.Module):

    def inverse_backwards(self, grad_output)
         # How to tell pytorch to use this as the backward for the .inverse?

In particular, I want that, when I later define some loss such as r = loss(y, model.inverse(other_model(model(x)))), that r.backwards() correctly uses my custom gradient for the inverse call.

Ideally the solution should be torchscript-compatible. A colleague suggested putting a inverse=False switch in the forward pass, but (a) I am not sure if this key-val would get passed to the backward pass and (b) this would force me to also implement a backward for the forward function, which I do not want to do. Another Idea came up to create an “Inverse” module, but it seems the parameter sharing between modules is not possible.