Automatic differentiation of composition of variables

[ululi1970]

Consider this snippet of code

def pde(x,y):
    u_x = dde.grad.jacobian(y,x, i=0, j=0)
    a = y[:,1:2]
    C= a * u_x
    C_x = dde.grad.jacobian(C. x, j = 0)
 # do something with C

Will C_x evaluate to a_x * u_x + a * u_xx or do I need to apply the jacobian to a, calculate the jacobian and hessian of u and apply the chain rule explicitly?
Thank you for your answers.

[LordPontiag]

Consider this snippet of code

def pde(x,y):
    u_x = dde.grad.jacobian(y,x, i=0, j=0)
    a = y[:,1:2]
    C= a * u_x
    C_x = dde.grad.jacobian(C. x, j = 0)
 # do something with C

Will C_x evaluate to a_x * u_x + a * u_xx or do I need to apply the jacobian to a, calculate the jacobian and hessian of u and apply the chain rule explicitly? Thank you for your answers.

C_x will evaluate to a_x * u_x + a * u_xx (or the equivalent in higher dimensions) because DeepXDE uses automatic differentiation (via the backend like TensorFlow) to handle the chain rule and product rule implicitly when computing the Jacobian of the composed tensor C. You do not need to manually apply the chain rule or compute the individual Jacobians and Hessians explicitly.