A note on the adjoint method for neural ordinary differential equation network
This clarifies theoretical foundations for neural ODE training, addressing inconsistencies in gradient computation for researchers in differential equation-based machine learning.
The paper rigorously analyzes the adjoint method for neural ODEs, showing that the loss gradient is an integral rather than an ODE and that the traditional adjoint form is not equivalent to backpropagation results, with equivalence only occurring when discrete schemes match.
Perturbation and operator adjoint method are used to give the right adjoint form rigourously. From the derivation, we can have following results: 1) The loss gradient is not an ODE, it is an integral and we shows the reason; 2) The traditional adjoint form is not equivalent with the back propagation results. 3) The adjoint operator analysis shows that if and only if the discrete adjoint has the same scheme with the discrete neural ODE, the adjoint form would give the same results as BP does.