# PyTorch：定义新的 Autograd 函数

> 原文：<https://pytorch.org/tutorials/beginner/examples_autograd/polynomial_custom_function.html#sphx-glr-beginner-examples-autograd-polynomial-custom-function-py>
>
> 校对：DrDavidS

这里我们准备一个三阶多项式，通过最小化平方欧几里得距离来训练，并预测函数 `y = sin(x)` 在`-pi`到`pi`上的值。

这里我们不将多项式写为`y = a + bx + cx^2 + dx^3`，而是将多项式写为`y = a + bP_3(c + dx)`，其中`P_3(x) = 1/2 (5x ^ 3 - 3x)`是三次[勒让德多项式](https://en.wikipedia.org/wiki/Legendre_polynomials)。

此实现使用了 PyTorch 张量(tensor)运算来实现前向传播，并使用 PyTorch Autograd 来计算梯度。

在此实现中，我们实现了自己的自定义 Autograd 函数来执行`P'_3(x)`。 从数学定义上讲，`P'_3(x) = 3/2 (5x ^ 2 - 1)`：

```py
import torch
import math

class LegendrePolynomial3(torch.autograd.Function):
    """
    We can implement our own custom autograd Functions by subclassing
    torch.autograd.Function and implementing the forward and backward passes
    which operate on Tensors.
    """

    @staticmethod
    def forward(ctx, input):
        """
        In the forward pass we receive a Tensor containing the input and return
        a Tensor containing the output. ctx is a context object that can be used
        to stash information for backward computation. You can cache arbitrary
        objects for use in the backward pass using the ctx.save_for_backward method.
        """
        ctx.save_for_backward(input)
        return 0.5 * (5 * input ** 3 - 3 * input)

    @staticmethod
    def backward(ctx, grad_output):
        """
        In the backward pass we receive a Tensor containing the gradient of the loss
        with respect to the output, and we need to compute the gradient of the loss
        with respect to the input.
        """
        input, = ctx.saved_tensors
        return grad_output * 1.5 * (5 * input ** 2 - 1)

dtype = torch.float
device = torch.device("cpu")
# device = torch.device("cuda:0")  # Uncomment this to run on GPU

# Create Tensors to hold input and outputs.
# By default, requires_grad=False, which indicates that we do not need to
# compute gradients with respect to these Tensors during the backward pass.
x = torch.linspace(-math.pi, math.pi, 2000, device=device, dtype=dtype)
y = torch.sin(x)

# Create random Tensors for weights. For this example, we need
# 4 weights: y = a + b * P3(c + d * x), these weights need to be initialized
# not too far from the correct result to ensure convergence.
# Setting requires_grad=True indicates that we want to compute gradients with
# respect to these Tensors during the backward pass.
a = torch.full((), 0.0, device=device, dtype=dtype, requires_grad=True)
b = torch.full((), -1.0, device=device, dtype=dtype, requires_grad=True)
c = torch.full((), 0.0, device=device, dtype=dtype, requires_grad=True)
d = torch.full((), 0.3, device=device, dtype=dtype, requires_grad=True)

learning_rate = 5e-6
for t in range(2000):
    # To apply our Function, we use Function.apply method. We alias this as 'P3'.
    P3 = LegendrePolynomial3.apply

    # Forward pass: compute predicted y using operations; we compute
    # P3 using our custom autograd operation.
    y_pred = a + b * P3(c + d * x)

    # Compute and print loss
    loss = (y_pred - y).pow(2).sum()
    if t % 100 == 99:
        print(t, loss.item())

    # Use autograd to compute the backward pass.
    loss.backward()

    # Update weights using gradient descent
    with torch.no_grad():
        a -= learning_rate * a.grad
        b -= learning_rate * b.grad
        c -= learning_rate * c.grad
        d -= learning_rate * d.grad

        # Manually zero the gradients after updating weights
        a.grad = None
        b.grad = None
        c.grad = None
        d.grad = None

print(f'Result: y = {a.item()} + {b.item()} * P3({c.item()} + {d.item()} x)')

```

**脚本的总运行时间**：（0 分钟 0.000 秒）

[下载 Python 源码：`polynomial_custom_function.py`](https://pytorch.org/tutorials/_downloads/b7ec15fd7bec1ca3f921104cfb6a54ed/polynomial_custom_function.py)

[下载 Jupyter 笔记本：`polynomial_custom_function.ipynb`](https://pytorch.org/tutorials/_downloads/0a64809624bf2f3eb497d30d5303a9a0/polynomial_custom_function.ipynb)