Interpret neural network like the linear regression equation such as how much will Y change if we change X1 and keep the other variables fixed In linear regression, assume we get the following equation : Y = 0.8X1+1.9X2+2.4X3+4X4.
We can interpret the linear equation: Keep the other predictors fix, one unit change in X1 will increase 0.8 units in Y in average.
Now, I use the neuralnet package in R and built a neural network model. Then, make a plot like the following picture.

From my perspective, all variables start with W such as W11,W21,W31 are the weight that previous variable or neural. The blue line and letters starts with b such as b11,b21,b31 are the bias weight.
Ignore the other coefficients I did not notice, just focus on the X1, can I write the equation of this neural network plot like:
Y = W31*W21*W11*X1 + W31*W21*W12*X1 + W31*W21*W13*X1 + W32*W21*W11*X1 + W32*W21*W12*X1 + W32*W21*W13*X1 + W32*W22*W11*X1 + W32*W22*W12*X1 + W32*W22*W13*X1
It looks quite ugly, but what I am curious is can I interpret the formula like the previous linear model ? Such as how Y changes if X1 increases one unit.
Or are there any other ways to interpret neural network plot (neuralnet) like this?

You can interpret NN like that but it is pointless generally. The seeming utility in doing so for OLS is in decoupling of the partial derivative from other variables, I.e. $\partial/\partial x_1 f(x_1,\dots,x_2)=\beta_1$.
With a useful NN you won’t get this decoupling and the partial ends up being a nonlinear function of all inputs $\beta(x_1,\dots,x_2) $ ,which you may have thousands of. So, the sensitivity to the variables is not the constant and as such not a simple intuitive concept to start with. Consider also that if you add interaction and polynomial terms to the OLS the sensitivity becomes more complicated and less intuitive for a linear regression too.
Note, that your example is NN with linear activation in inner layers, not a useful contraption as it turns out.
