使用自动微分库计算任意张量的偏导数 [英] Using automatic differentiation libraries to compute partial derivatives of an arbitrary tensor

问题描述

(注意:这不是有关反向传播的问题.) 我正在尝试在GPU上使用PyTorch张量代替Numpy数组来解决非线性PDE问题.我想计算任意张量的偏导数，类似于中心有限差分的作用<​​a href ="https://docs.scipy.org/doc/numpy/reference/generation/numpy.gradient.html "rel =" noreferrer> numpy.gradient 函数.我有其他方法可以解决此问题，但是由于我已经在使用PyTorch，所以我想知道是否可以使用autograd模块(或者通常是任何其他自动分化模块)来执行此操作.

(Note: this is not a question about back-propagation.) I am trying so solve on a GPU a non-linear PDE using PyTorch tensors in place of Numpy arrays. I want to calculate the partial derivatives of an arbitrary tensor, akin to the action of the center finite-difference numpy.gradient function. I have other ways around this problem, but since I am already using PyTorch, I'm wondering if it is possible use the autograd module (or, in general, any other autodifferentiation module) to perform this action.

我创建了numpy.gradient函数的张量兼容版本-运行速度快得多.但是，也许有更优雅的方法可以做到这一点.我找不到其他任何资料可以解决这个问题，无论是表明这是可能的还是不可能的.也许这反映了我对自动分化算法的无知.

I have created a tensor-compatible version of the numpy.gradient function - which runs a lot faster. But perhaps there is a more elegant way of doing this. I can't find any other sources that address this question, either to show that it's possible or impossible; perhaps this reflects my ignorance with the autodifferentiation algorithms.

推荐答案

我本人也遇到过同样的问题:在对PDE进行数值求解时，我们需要访问所有的空间梯度(numpy.gradients函数可以为我们提供空间梯度).时间-可以使用自动微分来计算梯度，而不是使用有限差分或某种形式的梯度吗?

I've had this same question myself: when numerically solving PDEs, we need access to spatial gradients (which the numpy.gradients function can give us) all the time - could it be possible to use automatic differentiation to compute the gradients, instead of using finite-difference or some flavor of it?

我想知道是否可以使用autograd模块(或者通常是任何其他自动分化模块)来执行此操作."

答案是否定的:一旦您在空间或时间上离散问题，时间和空间就会变成具有网格状结构的离散变量，而不是您要输入的显式变量一些函数来计算PDE的解.

The answer is no: as soon as you discretize your problem in space or time, then time and space become discrete variables with a grid-like structure, and are not explicit variables which you feed into some function to compute the solution to the PDE.

例如，如果我想计算某些流体流u(x，t)的速度场，我将在空间和时间上离散化，并且我将具有u[:,:]，其中的索引表示空间和时间上的位置.

For example, if I wanted to compute the velocity field of some fluid flow u(x,t), I would discretize in space and time, and I would have u[:,:] where the indices represent positions in space and time.

自动微分可以计算函数u(x，t)的导数.那么为什么不能在这里计算空间或时间导数呢?因为您离散化了您的问题.这意味着您没有任意x的u函数，而是在某些网格点具有u函数.您无法根据网格点的间距自动进行区分.

Automatic differentiation can compute the derivative of a function u(x,t). So why can't it compute the spatial or time derivative here? Because you've discretized your problem. This means you don't have a function for u for arbitrary x, but rather a function of u at some grid points. You can't differentiate automatically with respect to the spacing of the grid points.

据我所知，您编写的与张量兼容的函数可能是最好的选择.您可以在PyTorch论坛上看到类似的问题此处.或者，您可以做类似

As far as I can tell, the tensor-compatible function you've written is probably your best bet. You can see that a similar question has been asked in the PyTorch forums here and here. Or you could do something like

dx = x[:,:,1:]-x[:,:,:-1]

如果您不担心端点.

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