How to create a neural network from a set of equations?

Question

Say I have these equations:

$$x_1 = x_2 + 2y_1 + b$$ $$x_2 = y_2 + c$$ $$y_1 = z + a$$ $$y_2 = y_3 + d$$ $$z = z_1 + e$$

$x_1$ depends on $x_2$ (depends on $y_2$ (depends on $y_3$)) and $y_1$ (depends on $z$ (depends on $z_1$)).

$x_1$ is my final equation and $y_3$ and $z_1$ are my initial variables.

How do I represent them in a neural network? My final aim is to backtrack from $x_1$, and see what change of an amount $n$ in $x_1$ resulted from which of $y_3$ or $z_1$.

All these variables are item prices in the real world.

My inputs are $z_1$ and $y_3$ and my output is $x_1. z_1$ and $y_3$ are prices and the final output $x_1$ is also a price.

Answer 1

All modern frameworks for deep learning (PyTorch, Jax, Tensorflow) support automatic differentation. These operations can be easily implemented. Here I write, how it would look like in PyTorch:

class Net(nn.Module):

    def __init__(self):
         super().__init__()

         self.a = nn.Parameter(torch.randn(1))
         self.b = nn.Parameter(torch.randn(1))
         self.c = nn.Parameter(torch.randn(1))
         self.d = nn.Parameter(torch.randn(1))
         self.e = nn.Parameter(torch.randn(1))

   def forward(self, z1, y3):
       z = z1 + self.e
       y2 = y3 + self.d
       y1 = z + self.a
       x2 = y2 + self.c
       x1 = x2 + 2 * y1 + self.b
       return x1

And the use case is the following, say:

net = Net()
net(torch.ones(1), 2 * torch.ones(1))