Open smartalecH opened 4 years ago
By the chain rule, shouldn't we also multiply the objective function derivative by the partial of the basis expansion functions?
For example, if our cost function is a function of the fields and the permittivity:
J(E_x,E_y, ...,ϵ)
usually, the permittivity is also a function of our basis functions (and maybe some other normalizing functions):
ϵ = f(g(h(p)))
where f, g, and h are all functions that relate our design vector p to the permittivity.
f
g
h
p
To go from ∂J / ∂ϵ to ∂J / ∂p, we need to apply the chain rule:
∂J / ∂p = (∂J / ∂ϵ) (∂ϵ / ∂p) = (∂J / ∂ϵ) (∂J / ∂ϵ) (∂f / ∂g) (∂g / ∂h) * (∂h / ∂p)
Note we can right multiply or left multiply (backpropagate) and get the same result. We would still need to project, but after we've backpropagated through the full chain.
By the chain rule, shouldn't we also multiply the objective function derivative by the partial of the basis expansion functions?
For example, if our cost function is a function of the fields and the permittivity:
J(E_x,E_y, ...,ϵ)
usually, the permittivity is also a function of our basis functions (and maybe some other normalizing functions):
ϵ = f(g(h(p)))
where
f,g, andhare all functions that relate our design vectorpto the permittivity.To go from ∂J / ∂ϵ to ∂J / ∂p, we need to apply the chain rule:
∂J / ∂p = (∂J / ∂ϵ) (∂ϵ / ∂p) = (∂J / ∂ϵ) (∂J / ∂ϵ) (∂f / ∂g) (∂g / ∂h) * (∂h / ∂p)
Note we can right multiply or left multiply (backpropagate) and get the same result. We would still need to project, but after we've backpropagated through the full chain.