When I use the normalized grid corresponding to the original coordinate system, as expected, the output should be the same as the original input.
However, the actual output results are slightly different. Perhaps it is a precision issue?
h_coord, w_coord = torch.meshgrid(torch.arange(7), torch.arange(7))
h_coord = h_coord.reshape(1, 7, 7, 1)
w_coord = w_coord.reshape(1, 7, 7, 1)
grid = torch.cat(
    [
        2 * torch.clamp(w_coord, 0, 6) / (7 - 1) - 1,
        2 * torch.clamp(h_coord, 0, 6) / (7 - 1) - 1,
    ],
    dim=-1,
)
grid_sampled_shift = F.grid_sample(pad_x[:, :2], grid, mode="bilinear", align_corners=True)

print(grid)
print(pad_x[:, :2])
print(grid_sampled_shift)


'''
tensor([[[[-1.0000, -1.0000],
          [-0.6667, -1.0000],
          [-0.3333, -1.0000],
          [ 0.0000, -1.0000],
          [ 0.3333, -1.0000],
          [ 0.6667, -1.0000],
          [ 1.0000, -1.0000]],

         [[-1.0000, -0.6667],
          [-0.6667, -0.6667],
          [-0.3333, -0.6667],
          [ 0.0000, -0.6667],
          [ 0.3333, -0.6667],
          [ 0.6667, -0.6667],
          [ 1.0000, -0.6667]],

         [[-1.0000, -0.3333],
          [-0.6667, -0.3333],
          [-0.3333, -0.3333],
          [ 0.0000, -0.3333],
          [ 0.3333, -0.3333],
          [ 0.6667, -0.3333],
          [ 1.0000, -0.3333]],

         [[-1.0000,  0.0000],
          [-0.6667,  0.0000],
          [-0.3333,  0.0000],
          [ 0.0000,  0.0000],
          [ 0.3333,  0.0000],
          [ 0.6667,  0.0000],
          [ 1.0000,  0.0000]],

         [[-1.0000,  0.3333],
          [-0.6667,  0.3333],
          [-0.3333,  0.3333],
          [ 0.0000,  0.3333],
          [ 0.3333,  0.3333],
          [ 0.6667,  0.3333],
          [ 1.0000,  0.3333]],

         [[-1.0000,  0.6667],
          [-0.6667,  0.6667],
          [-0.3333,  0.6667],
          [ 0.0000,  0.6667],
          [ 0.3333,  0.6667],
          [ 0.6667,  0.6667],
          [ 1.0000,  0.6667]],

         [[-1.0000,  1.0000],
          [-0.6667,  1.0000],
          [-0.3333,  1.0000],
          [ 0.0000,  1.0000],
          [ 0.3333,  1.0000],
          [ 0.6667,  1.0000],
          [ 1.0000,  1.0000]]]])

tensor([[[[0., 0., 0., 0., 0., 0., 0.],
          [0., 0., 0., 0., 0., 0., 0.],
          [1., 1., 1., 1., 1., 1., 1.],
          [2., 2., 2., 2., 2., 2., 2.],
          [3., 3., 3., 3., 3., 3., 3.],
          [4., 4., 4., 4., 4., 4., 4.],
          [4., 4., 4., 4., 4., 4., 4.]],

         [[0., 0., 1., 2., 3., 4., 4.],
          [0., 0., 1., 2., 3., 4., 4.],
          [0., 0., 1., 2., 3., 4., 4.],
          [0., 0., 1., 2., 3., 4., 4.],
          [0., 0., 1., 2., 3., 4., 4.],
          [0., 0., 1., 2., 3., 4., 4.],
          [0., 0., 1., 2., 3., 4., 4.]]]])
          
tensor([[[[0.0000e+00, 0.0000e+00, 0.0000e+00, 0.0000e+00, 0.0000e+00,
           0.0000e+00, 0.0000e+00],
          [1.1921e-07, 1.1921e-07, 1.1921e-07, 1.1921e-07, 1.1921e-07,
           1.1921e-07, 1.1921e-07],
          [1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00, 1.0000e+00,
           1.0000e+00, 1.0000e+00],
          [2.0000e+00, 2.0000e+00, 2.0000e+00, 2.0000e+00, 2.0000e+00,
           2.0000e+00, 2.0000e+00],
          [3.0000e+00, 3.0000e+00, 3.0000e+00, 3.0000e+00, 3.0000e+00,
           3.0000e+00, 3.0000e+00],
          [4.0000e+00, 4.0000e+00, 4.0000e+00, 4.0000e+00, 4.0000e+00,
           4.0000e+00, 4.0000e+00],
          [4.0000e+00, 4.0000e+00, 4.0000e+00, 4.0000e+00, 4.0000e+00,
           4.0000e+00, 4.0000e+00]],

         [[0.0000e+00, 1.1921e-07, 1.0000e+00, 2.0000e+00, 3.0000e+00,
           4.0000e+00, 4.0000e+00],
          [0.0000e+00, 1.1921e-07, 1.0000e+00, 2.0000e+00, 3.0000e+00,
           4.0000e+00, 4.0000e+00],
          [0.0000e+00, 1.1921e-07, 1.0000e+00, 2.0000e+00, 3.0000e+00,
           4.0000e+00, 4.0000e+00],
          [0.0000e+00, 1.1921e-07, 1.0000e+00, 2.0000e+00, 3.0000e+00,
           4.0000e+00, 4.0000e+00],
          [0.0000e+00, 1.1921e-07, 1.0000e+00, 2.0000e+00, 3.0000e+00,
           4.0000e+00, 4.0000e+00],
          [0.0000e+00, 1.1921e-07, 1.0000e+00, 2.0000e+00, 3.0000e+00,
           4.0000e+00, 4.0000e+00],
          [0.0000e+00, 1.1921e-07, 1.0000e+00, 2.0000e+00, 3.0000e+00,
           4.0000e+00, 4.0000e+00]]]])
'''