How to formulate the problem with the semi-implicit Euler Formula?

[J2thearo]

Hello Ben,

how would you tackle the whole thing if you determine the next position and velocity with the Euler method? This approach is mainly used to simulate physics in video games and produces an approximate discrete solution by iterating position and velocity. Here, b is the damping, which is typically between 0 and 1, and d is the position in force equilibrium.

function step(x_0, v_0) {
  a_spring = k * (d - x_0)
  v_1 = b * v_0 + a_spring * delta_t
  x_1 = x_0 + v_1 * delta_t
  return [x_1, v_1]
}

What I would like to get in the end is a neural network that does the following:

x_n+1, v_n+1 = NN(x_n, v_n)

I think I would treat it essentially like a time series forecasting but how would I formulate the "physics loss"?