  We propose the Automatic-differentiated Physics-Informed Echo State Network
(API-ESN). The network is constrained by the physical equations through the
reservoir's exact time-derivative, which is computed by automatic
differentiation. As compared to the original Physics-Informed Echo State
Network, the accuracy of the time-derivative is increased by up to seven orders
of magnitude. This increased accuracy is key in chaotic dynamical systems,
where errors grows exponentially in time. The network is showcased in the
reconstruction of unmeasured (hidden) states of a chaotic system. The API-ESN
eliminates a source of error, which is present in existing physics-informed
echo state networks, in the computation of the time-derivative. This opens up
new possibilities for an accurate reconstruction of chaotic dynamical states.
