Machine-learning by rogue waves, dispersive shocks, and solitons
We study artificial neural networks with nonlinear waves as a computing reservoir. We discuss universality and the conditions to learn a dataset in terms of output channels and nonlinearity. A feed-forward three-layer model, with an encoding input layer, a wave layer, and a decoding readout, behaves as a conventional neural network in approximating mathematical functions, real-world datasets, and universal Boolean gates. The rank of the transmission matrix has a fundamental role in assessing the learning abilities of the wave. For a given set of training points, a threshold nonlinearity for universal interpolation exists. When considering the nonlinear Schroedinger equation, the use of highly nonlinear regimes implies that solitons, rogue, and shock waves do have a leading role in training and computing. Our results may enable the realization of novel machine learning devices by using diverse physical systems, as nonlinear optics, hydrodynamics, polaritonics, and Bose-Einstein condensates. The application of these concepts to photonics opens the way to a large class of accelerators and new computational paradigms. In complex wave systems, as multimodal fibers, integrated optical circuits, random, topological devices, and metasurfaces, nonlinear waves can be employed to perform computation and solve complex combinatorial optimization.
We show that quantum fluids enable experimental analogs of relativistic orbital precession in the presence of non-paraxial effects. The analysis is performed by the hydrodynamic limit of the Schroedinger equation. The non-commutating variables in the phase-space produce a precession and an acceleration of the orbital motion. The precession of the orbit is formally identical to the famous orbital precession of the perihelion of Mercury used by Einstein to validate the corrections of general relativity to Newton’s theory. In our case, the corrections are due to the modified uncertainty principle. The results may enable novel relativistic analogs in the laboratory, also including sub Planckian phenomenology.
In any form of wave propagation, strong spatiotemporal coupling appears when non-elementary, three-dimensional wave-packets are composed by superimposing pure plane waves, or spontaneously generated by light-matter interaction and nonlinear processes. Ultrashort pulses with orbital angular momentum (OAM), or ultrashort vortices, furnish a critical paradigm in which the analysis of the spatiotemporal coupling in the form of temporal-OAM coupling can be carried out accurately by analytical tools. By generalizing and unifying previously reported results, we show that universal and spatially heterogeneous space-time correlations occur in propagation-invariant temporal pulses carrying OAM. In regions with high intensity, the pulse duration has a lower bound fixed by the topological charge of the vortex and such that the duration must increase with the topological charge. In regions with low intensity in the vicinity of the vortex, a large blue-shift of the carrier oscillations and an increase of the number of them is predicted for strongly twisted beams. We think that these very general findings highlight the existence of a structural coupling between space and time, which is relevant at low photon numbers in quantum optics, and also in the highly nonlinear process as the high-harmonics generated with twisted beams. These results have also applications as multi-level classical and quantum free-space or satellite, communications, spectroscopy, and high-harmonic generation.
Miguel A. Porras and C. Conti in arXiv:1911.1222