I’m reading the paper HiPPO: Recurrent Memory with Optimal Polynomial Projections by Albert Gu text. In Appendix B.3, he talked about ODE Discretization, but I have several confusing questions regarding this part:

1. I can understand the forward Euler method, however, for the backward Euler, I am confused. It seems that the difference between forward and backward Euler is what point we used to estimate the integral integral, and for forward, the time point we use it t, and for backward, we use t+Δt.
   So for me, the backward method can be written as backward my equation. However, as in the paper shown backward paper, there is no Bf(t+Δt) term but only Bf(t). Why is the case?

2. When it comes to bilinear situation, I’m even more confused. To me, this time we are using timepoint [t+(t+Δt)]/2 to estimate the integral. However, the equation in the paper bilinear paper is totally something I don’t understand.

As a statistics student, I have no prior knowledge on control system. I’m highly interested in how Albert Gu applied state space model to machine learning tasks. But now I’m stucked at the discretization part. Please help me understand what’s the case here.