Machine learning non-reciprocity of a linear waveguide with a local nonlinear, asymmetric gate: Case of weak coupling

This work considers the acoustics of an infinite, one-dimensional, linear waveguide composed of weakly coupled grounded oscillators, and incorporating a local nonlinear gate with cubic stiffness nonlinearity. A break of configurational symmetry is realized by the weak detuning of the two linear natural frequencies of the oscillators within the nonlinear gate. When harmonic excitation is applied to an oscillator of the waveguide, the synergy of nonlinearity and asymmetry at the local gate yields global effects in the acoustics, namely, in the form of strong non-reciprocity that is tunable with the frequency and intensity of the excitation. It is desirable to select the parameters for this system (such as mass, springs, nonlinear springs, etc.) to maximize its performance. Towards this objective, the nonlinear acoustics is studied by training two neural network (NN) simulators to evaluate (i) the harmonic contents of the transmitted waves through the nonlinear gate and (ii) the global non-reciprocity and transmissibility features of the waveguide. The training data is obtained by direct numerical simulations of the governing differential equations of motion. The predictions of the NN, which agree well with the direct simulations, drastically reduce simulation time, making practical the parametric study of the nonlinear acoustics and allowing for the detailed study the waveguide response at high-dimensional parametric domains. Using the NN, we demonstrate robust regions of non-reciprocal acoustic responses and achieve predictive design of the waveguide for maximum acoustic non-reciprocity. Indeed, it is shown that, when appropriately designed, the waveguide acts as an acoustic diode, allowing wave transmission only in one (preferred) direction and preventing wave propagation in the other direction. Moreover, depending on the system and excitation parameters, the non-reciprocally transmitted waves through the nonlinear gate can be either monochromatic or strongly modulated, i.e., possessing multiple frequency components. As a result, the non-reciprocal waveguide studied in this work may act either as a frequency converter or, conversely, as an acoustic diode with minimal frequency distortion of the transmitted waves. The developed machine learning approach is also applicable to a broad class of nonlinear waveguides with different configurations, asymmetries, and system properties, as well as in more than one dimensions.