Using reservoir computing to construct scarred wave functions

Scar theory is one of the fundamental pillars in the field of quantum chaos, and scarred functions are a superb tool to carry out studies in it. Several methods, usually semiclassical, have been described to cope with these two phenomena. In this paper, we present an alternative method, based on the novel machine learning algorithm known as reservoir computing, to calculate such scarred wave functions together with the associated eigenstates of the system. The resulting methodology achieves outstanding accuracy while reducing execution times by a factor of ten. As an illustration of the effectiveness of this method, we apply it to the widespread chaotic two-dimensional coupled quartic oscillator.

Figure 1

Trajectories corresponding to the four periodic orbits in blue (light gray): (from bottom left to top right) horizontal and vertical, quadruple-loop, horizontal and vertical bowtie, and square, at $E=1$. The corresponding equipotential is shown with the red (dark gray) line.

Figure 2

Architecture of a RC model. Only the readout layer $W^{\text{out}}$ is learned during training.

Figure 3

(Left) Example of the modulus of the energy spectrum ${\left|I\left(E\right)\right|}^2$ for the system at $E=10$. (Right) Long time energy spectrum ${\left|I\left(E\right)\right|}^2$, in red (light gray), and its low-resolution version, in blue (dark gray), for the quadruple-loop scar function at energy $E=22.824$.

Figure 4

Probability density of the eigenfunctions of the coupled quartic oscillator obtained from propagating an initial wave packet with energy $E=1$. The red (light gray) solid lines depict the equipotential curves.

Figure 5

Same as Fig. 4 for $E=10$, scaled to the domain corresponding to $E=1$.

Figure 6

Same as Fig. 5 for $E=100$.

Figure 7

Probability density of the scar function for the horizontal and vertical periodic orbit, at different energies (see Table 2). The solid red (light gray) lines depict the equipotential curves, while the dashed black lines illustrate the periodic orbits.

Figure 8

Same as Fig. 7 for the quadruple-loop periodic orbit.

Figure 9

Same as Fig. 7 for the bowtie periodic orbit.

Figure 10

Same as Fig. 7 for the square periodic orbit.

Figure 11

Comparison of the semiclassical energies (red crosses) with the scar energies obtained with the reservoir computing approach (blue stars).