Higher order derivatives in importance sampling


I am considering several functions to be used in the importance sampling scheme proposed by Modulus. In the lid driven cavity example, the 2-norm of the velocity derivative is used. In case higher-order derivatives are to be needed, how should one proceed? This might be of interest if one wanted to use the residual as a sampling measure (as it might be for the heat equation or Navier-Stokes where a second-order term is found).
In the list of required outputs of the graph, it is possible to specify which key (e.g., T in the heat equation) holds the derivative with respect to another key ( e.g., x ). How do you specify that you want a larger order derivative with respect to that key?


Hi @nripamont

For importance sampling in the LDC example, the first order derivatives are used. Changing to higher-order should be straight forward (assuming the gradients can be calculated in the graph). High-order diffs can be specified in the output keys which will then show up in the output dictionary.

Keys that are derivatives are converted into Keys using the diff_str, so u__x is du/dx = Key('u', diff=[Key('x')]), u__x__x is d2u/dx2 = Key('u', diff=[Key('x'), Key('x')]), u__x__y is d2u/dxdy = Key('u', diff=[Key('x'), Key('y')]), etc.

So if you want to importance sample in the LDC example with second order diffs:

importance_model_graph = Graph(
        invar=[Key("x"), Key("y")],
            Key("u", derivatives=[Key("x"), Key("x")]),

    def importance_measure(invar):
        outvar = importance_model_graph(
            Constraint._set_device(invar, device=device, requires_grad=True)
        importance = (
            outvar["u__x__x"] ** 2
        ) ** 0.5 + 10
        return importance.cpu().detach().numpy()

(You could try going higher to third order if you want, will be much slower but Modulus should do the autodiffs for you)

can we go up to fourth order, but for solving our equation we do not need first and third order, we only need second and fourth to solve equation.