  This work introduces a new Control Neural Network (Ctrl.NN) method to uncover
evidence of exotic quantum state, \textit{i.e.}, the breathing modes in
3-$\alpha$ resonant states of $^{12}$C nucleus. We provide the most precise
microscopic description to date for the $^{12}$C energy spectrum, identify two
new exotic breathing states, and uncover strong evidence that directly connects
the recent experimental observations to the breathing modes. The Ctrl.NN method
significantly simplifies numerical calculations of quantum systems under
multiple constraints and offers a new perspective for solving the nuclear
many-body problem.
