$$\frac{\partial p(z(t), t)}{\partial t} = \frac{\partial p(z(t), t)}{\partial z(t)} \frac{\partial z(t)}{\partial t} + \frac{\partial p(z(t), t)}{\partial t}$$
Shouldn't the LHS be total derivative of p wrt to t? (source page 14 of Neural ODE Paper)