## Abstract

We study nonreciprocity in a passive linear waveguide augmented with a local asymmetric, dissipative, and strongly nonlinear gate. Strong coupling between the constituent oscillators of the waveguide is assumed, resulting in broadband capacity for wave transmission. The local nonlinearity and asymmetry at the gate can yield strong global nonreciprocal acoustics, in the sense of drastically different acoustical responses depending on which side of the waveguide a harmonic excitation is applied. Two types of highly nonreciprocal responses are observed: (i) Monochromatic responses without frequency distortion compared to the applied harmonic excitation, and (ii) strongly modulated responses (SMRs) with strong frequency distortion. The complexification averaging (CX-A) method is applied to analytically predict the monochromatic solutions of this strongly nonlinear problem, and a stability analysis is performed to study the governing bifurcations. In addition, we build a machine learning framework where neural net (NN) simulators are trained to predict the performance measures of the gated waveguide in terms of certain transmissibility and nonreciprocity measures. The NN drastically reduces the required simulation time, enabling the determination of parameter ranges for desired performance in a high-dimensional parameter space. In the predicted desirable parameter space for nonreciprocity, the maximum transmissibility reaches 40%, and the transmitted energy varies by up to three orders of magnitude depending on the direction of wave transmission. The machine learning tools along with the analytical methods of this work can inform predictive designs of practical nonreciprocal waveguides and acoustic metamaterials that incorporate local nonlinear gates.

## 1 Introduction

Reciprocity is a basic feature in linear time-invariant elastodynamics and acoustics. Following the Maxwell-Betti reciprocity theorem, the response of a linear time-invariant (LTI) elastodynamic system is invariant when switching the points of excitation and measurement [1,2]. A break of acoustic reciprocity enables, e.g., wave propagation only in a preferred direction paving the way for various types of novel acoustic elements such as acoustic diodes [3–6], rectifiers [7], frequency convertors [8], topological insulators [9], and resonators with reconfigurable bandwidth properties [10]. Specifically, nonreciprocal acoustics have been widely studied in kinetic [11,12], active [13], and nonlinear media [14–18], where break of nonreciprocity is achieved by applying external biases, introducing time-variant properties and imposing nonlinearity coupled with asymmetry [19], respectively. Passive nonreciprocal acoustics through application of intentional nonlinearity has been achieved by imposing local asymmetric acoustic elements such as vibro-impactors [3], bilinear oscillators [8], or a hierarchical series of nonlinear oscillators [4]. These local nonlinear elements integrated into an otherwise LTI elastodynamic waveguide can result in global nonreciprocal acoustics without requiring any external source of energy, i.e., in a passive way.

However, a typical feature of nonlinear acoustics is its dependence on energy (or the specific type of the applied excitation), whereas additional features are instabilities and bifurcations that often occur in the acoustics. Hence, it is much more challenging to solve nonlinear problems compared to linear ones. Moreover, nonreciprocal acoustic waveguides can only be predictively designed by analytically or numerically solving the governing nonlinear governing equations, which is often a formidable task, especially when in spatially extended systems that involve many subcomponents. Some examples of investigating nonlinear nonreciprocal acoustics involve asymptotic analysis [20], and the implementation of nonlinear normal modes [21,22], reduced order models [23] or Poincaré maps [24]. However, these methods are only applicable to systems with simplifications such as homogeneous structure (i.e., with no disorder), weak nonlinearity, or possessing local nonlinear elements with a few degrees-of-freedom. In addition, since the acoustics of nonlinear waveguides is dependent on energy, the computational load associated with direct numerical simulations is often prohibitive for predictive design. These limitations of traditional analytical and numerical approaches motivate for an alternative and more efficient way for predictive design of nonlinear nonreciprocal waveguides.

Despite the traditional approaches of acoustical analysis and simulations for ordinary differential equations, machine learning based simulators have attracted the attention of researchers as an alternative way for cost-efficient engineering simulation. Machine learning algorithms build models that can be “trained” (or that they “learn”) by feeding them appropriate data sets. Compared with direct numerical simulations, a trained machine learning simulator is much more efficient at the cost of acceptable errors; hence, machine learning methods have been widely applied to various fields of engineering computation, e.g., in finite elements [25], computational fluid dynamics [26] and particle-based simulations [27]. However, the applications of machine learning techniques in field of nonreciprocal acoustics are still limited. Recently, Wang et al. [28] developed a machine learning model that is applicable to a class of *weakly coupled* LTI waveguides that were excited by harmonic excitations and were locally augmented by nonlinear “gates.” With local nonlinearity and asymmetry, these weakly coupled waveguides exhibited strong acoustic nonreciprocity with mainly two types of responses, that is, monochromatic and strongly modulated responses (SMRs). The machine learning model correctly predicted the acoustics in a parameter space, enabling the parametric study and optimized design for nonreciprocity and transmissibility. However, the study in Ref. [28] was restricted only to waveguides with weak coupling and weak nonlinearity, so that the nonreciprocity was realized only in a narrow frequency band. Moreover, the group velocity of propagating waves was slow due to the weak coupling, and therefore, the acoustics was sensitive to structural damping. This motivates us to extend the study in Ref. [28] to the case of *strong coupled* LTI waveguides augmented by local nonlinear and asymmetric gates, supporting broad-band wave transmission and expanding the design space of the system parameters. This is the aim of the present work.

We start with a numerical study to illustrate that the strongly coupled waveguide, when augmented by a local nonlinear asymmetric gate features monochromatic responses and SMRs—as in the weakly coupled case [28]. Then, the bifurcations that generate the nonreciprocal acoustics in this system are discussed. These are based on analytical solutions by employing complexification averaging (CX-A) analysis, which yields closed form solutions for the monochromatic responses and enables stability analysis that reveals the bifurcations governing the transition from nonreciprocal monochromatic responses to SMRs. Extending the study to SMRs, machine learning is employed. To this end, the system and excitation parameters are considered as the inputs of the machine learning model, whereas the objective performance measures (in terms of nonreciprocity and transmissibility) are considered as outputs. An artificial neural net (NN) is trained based on numerical simulation results on a randomly-sampled dataset. The trained NN simulator drastically decreases the computational effort with acceptable errors of prediction. The NN and analytical predictions are then verified by direct numerical simulations. Finally, we combine the NN predicted results and the analytical results to investigate the robust range of system and excitation parameters for strongly nonreciprocal acoustics of the nonlinear gated waveguide.

## 2 Numerical Investigation of the Non-Reciprocal Acoustics

*upstream*one, while the nondirectly excited subwaveguide (which is initially at rest) is named as the

*downstream*one. The oscillators are ordered sequentially from the gate to the far field, with the oscillators named as $En\u2009(n=1,2,3,\u2026)$ in the upstream subwaveguide and $An\u2009(n=1,2,3,\u2026)$ in the downstream subwaveguide. Moreover, the oscillators in the nonlinear gate are named as $E0$ and $A0$, connected to the upstream and downstream linear subwaveguides, respectively. The equations of motion are expressed as follows, with all initial conditions assumed to be zero

where $xn$ and $yn$ denote the displacements of oscillators $En$ and $An$ with $n=1,2,3,\u2026$. Also, overdot denotes differentiation with respect to the time $t$, and $m$, $k$, $c$ and $kl$ denote the mass, linear grounding stiffness, viscous damping coefficient, and linear coupling stiffness of the oscillators in the linear semi-infinite waveguides, respectively. The grounding stiffnesses of the gate oscillators $E0$ and $A0$ are detuned from $k$ and denoted as $k1$ and $k2$, respectively, whereas the coupled stiffness of these oscillators is purely cubic (without a linear term) with coefficient $kc$. The harmonic excitation is applied at oscillator $Ep$ of the upstream subwaveguide, with frequency $\omega $ and amplitude $2Fp$, and $\delta $ denotes the Dirac function indicating that all other oscillators are not excited by an external force.

As shown in Ref. [28] where weak coupling in the subwaveguides of Fig. 1 was studied, nonreciprocal acoustic responses were realized in the corresponding narrow pass band, with energy transfer allowed in one direction and prevented from the other. Moreover, the nonreciprocal responses were in either monochromatic or strongly modulated. We show here that similar responses can be realized in the strongly coupled nonlinear waveguide. As examples of the nonreciprocal responses, the equations of motion were simulated for the parameters shown in Table 1, a total normalized simulation time equal to 1500, and for 1000 oscillators in total. Also note that exchanging the excitation position from the oscillator $Ep$ to $Ap$ is equivalent to simply exchanging the detuning parameters $\sigma 1$ and $\sigma 2$; therefore, we will study the nonreciprocal acoustics by fixing the excitation position and comparing the responses when the detuning parameters $(\sigma 1,\sigma 2)$ are interchanged.

Parameter | $d$ | ${\sigma 1,\sigma 2}$ | $\zeta $ | $\theta $ | $\alpha $ | $Ap$ | $p$ |
---|---|---|---|---|---|---|---|

System 1 | 0.5 | ${\u22121,\u22121.5}$ | 0.03 | $(1/6)\pi $ | 2 | 0.25 | 4 |

System 2 | 0.5 | ${1,0}$ | 0.02 | $(2/3)\pi $ | 2 | 0.25 | 4 |

Parameter | $d$ | ${\sigma 1,\sigma 2}$ | $\zeta $ | $\theta $ | $\alpha $ | $Ap$ | $p$ |
---|---|---|---|---|---|---|---|

System 1 | 0.5 | ${\u22121,\u22121.5}$ | 0.03 | $(1/6)\pi $ | 2 | 0.25 | 4 |

System 2 | 0.5 | ${1,0}$ | 0.02 | $(2/3)\pi $ | 2 | 0.25 | 4 |

In Fig. 2, we consider the responses for the first set of parameters (System 1) listed in Table 1. The results for $(\sigma 1,\sigma 2)=(\u22121,\u22121.5)$ are shown in Figs. 2(a), 2(c), and 2(e), while the results of the excitation from the other side, i.e., $(\sigma 1,\sigma 2)=(\u22121.5,\u22121)$, are depicted in Figs. 2(b), 2(d), and 2(f). In Figs. 2(a) and 2(b), we compare the displacement time series of the two oscillators of the nonlinear gate, $x0$ and $y0$. For $(\sigma 1,\sigma 2)=(\u22121,\u22121.5)$, the steady-state oscillation amplitudes of the gate oscillators are similar, so it is inferred that energy transmission occurs through the nonlinear gate. This is not the case when $(\sigma 1,\sigma 2)$ are flipped though (i.e., interchanging the site of the excitation between the left and right subwaveguides), since drastically diminished energy propagates through the nonlinear gate as the amplitude of $y0$ is much smaller than the amplitude of $x0$ at steady-state. Therefore, we observe the realization of strong nonreciprocity in the steady-state acoustics, with energy being transmitted only in one direction.

These energy measures are plotted as functions of time in Figs. 2(e) and 2(f) for $(\sigma 1,\sigma 2)=(\u22121,\u22121.5)$ and $(\u22121.5,\u22121)$, respectively. From Fig. 2(e), we note that the transmitted energy is about 25% of the input energy for $(\sigma 1,\sigma 2)=(\u22121,\u22121.5)$, and almost zero in the opposite direction. Besides the time series and the frequency spectra, these energy measures provide another prospective to understand the physics of acoustic nonreciprocity in the gated waveguide.

Another case is shown in Fig. 3, for parameters indicated as System 1 listed in Table 1. The simulations for $(\sigma 1,\sigma 2)=(1,0)$ are shown in Figs. 3(a), 3(c), 3(e), while for $(\sigma 1,\sigma 2)=(0,1)$ in Figs. 3(b), 3(d), and 3(f). From Figs. 3(e) and 3(f), we note that again energy transmission is only allowed only in one direction, revealing strong nonreciprocity in terms of energy. Moreover, in this case, the responses of the waveguide are strongly modulated for $(\sigma 1,\sigma 2)=(1,0)$; as in Ref. [28], this type of responses will be referred to as strongly modulated responses (SMRs). The amplitude of the discrete Fourier transformation of $y0$ is plotted in Fig. 3(c), showing equally-spaced peaks (side bands) centered at the excitation frequency which is the highest (dominant) harmonic; this indicates that the SMR is quasi-periodic. However, for flipped $(\sigma 1,\sigma 2)$ the transmitted wave is monochromatic. In this case, the acoustics are nonreciprocal in terms of both transmitted energy and harmonic composition of the steady-state responses.

In Eq. (5), $T$ denotes the time interval where the measurements are recorded, $\eta $ is a measure of transmissibility and $\delta $ a measure of nonreciprocity. Note that in a reciprocal system, $Edown(\sigma 1,\sigma 2,T)=Edown(\sigma 2,\sigma 1,T),$ which corresponds to $\delta =0$ [1]. Therefore, $\delta $ measures the difference (in logarithmic scale) in the transmitted energy when the excitation is switched between the two subwaveguides. We also focus on optimizing the nonreciprocal transmissibility $\eta $, which is defined as the ratio of the transmitted energy to input energy. We aim at maximizing the nonreciprocal transmissibility from one direction with negligible nonreciprocal transmissibility from the other direction, thus requiring large transmissibility and nonreciprocity measures $\eta $ and $\delta $. As the acoustics reaches the steady-state, the energy measures $\eta $ and $\delta $ converge for sufficiently large simulation times. Hence, the time interval $T$ is selected such as the acoustics reach steady-state.

Intuitively, for sufficiently large damping to coupling ratio, $\zeta $, the dissipation effect overweighs wave propagation and the energy is confined in the leading oscillators close to the excited oscillator $Ep$; therefore, this ratio is carefully selected and assigned moderate values. In Fig. 4, we show the nonreciprocal transmissibility (linear scale) and downstream energy (logarithmic scale) measures when the normalized damping ratio $\zeta $ varies from 0 to 0.07 for the parameters listed in Table 1. In the schematic shown in Fig. 1, the upstream subwaveguide is located at the left side, while the downstream subwaveguide at the right side. Hence for the frequency detuning values $(\sigma 1,\sigma 2)$ listed in Table 1, we refer to the simulated responses as “left-to-right” propagating waves, whereas for the flipped values $(\sigma 2,\sigma 1)$ as “right-to-left” propagating waves.

Considering the results for system parameters corresponding to System 1 (cf. Figs. 4(a) and 4(b)), the transmissibility and downstream energy measures decrease when the normalized damping ratio $\zeta $ increases. Moreover, steep descents in downstream energy are observed for both “left-to-right” propagation and “right-to-left” propagation at certain critical thresholds of $\zeta $, with the corresponding decreases being nearly two to three orders of magnitude. We note that the critical damping ratio $\zeta $ is around 0.05 for “left-to-right” propagation, and 0.015 for “right-to-left” propagation. This disparity in the critical damping ratios provides a region of strong nonreciprocity, where energy can propagate from left to right but is prohibited from propagating right to left.

Another interesting acoustic feature of the gated waveguide is observed for system parameters corresponding to Simulation 2. In Fig. 4(c), when $\zeta <0.02$, the “left-to-right” transmissibility increases as the damping ratio increases, which is counterintuitive since damping typically restricts energy transfer. Moreover, In Fig. 4(d), we note that the “left-to-right” transmitted energy decreases linearly in the logarithmic plot for $\zeta >0.02$, indicating that the transmitted energy exponentially decreases for increasing damping. This is not the case for $\zeta <0.02$, where the transmitted energy is nearly constant. Therefore, we hypothesize that in this case the “left-to-right” acoustics exhibit drastically different features before and after the critical threshold $\zeta =0.02$. To further explore this phenomenon, we investigate the responses of the gated waveguide by means of numerical simulations corresponding to values of $\zeta $ close to the critical threshold 0.02.

In Fig. 5, we plot the time series of $x0$, $y0$ and the corresponding Morlet wavelet transforms for system parameters corresponding to Simulation 2 of Table 1 with $\zeta =0.02,\u20090.022,$ and $0.024$. In these results the mother wavelet (normalized) frequency is equal to 16. We observe that a small variation in damping leads to different types of steady-state responses. As shown in Figs. 5(a) and 5(b) ($\zeta =0.02$), a strongly modulated quasi-periodic response is generated, involving multiple harmonics at steady-state. However, in Figs. 5(e) and 5(f) ($\zeta =0.024$) the steady-state response is monochromatic involving only one frequency component. A transition state is observed for $\zeta =0.022$ in Figs. 5(c) and 5(d), where multiple harmonics are noted in the response; however, in this case it takes a relatively long duration for the response to converge to steady-state where a single harmonic survives. For this particular case, at the critical damping ratio there occurs a *bifurcation* (transition) from an SMR to a monochromatic steady-state response as damping increases. We then hypothesize that this critical damping ratio applies to other cases which support SMRs. This interesting feature paves the way for predictive design of the nonlinear nonreciprocal acoustics of the gated waveguide. To this end, we proceed to the analytical study of the strongly nonlinear and strongly coupled system (2) in order to recover the previous numerical results and study the governing mechanism of the different nonlinear acoustic regimes at steady-state; this will represent the first step toward predictive design and optimization of the nonreciprocal acoustics of the gated waveguide.

## 3 Analytical Study of the Non-Reciprocal Acoustics

The analytically study of the normalized Eq. (2) is based on the complexification-averaging (CX-A) method which is suitable for the analysis of this strongly nonlinear problem [31,32]. Generally, the CX-A method requires that the responses only possess one or a finite number of “fast” dominant frequencies [33,34], so in this section, only monochromatic steady-state solutions are considered. To this end, we assume that the responses of all oscillators of the gated waveguide possess a single dominant harmonic component at the excitation frequency. Note, however, that this *ansatz* is not valid for SMRs with multiple harmonics (side bands).

*ansatz*) so the complex amplitudes $\phi x0$, $\phi y0$, $\phi xn$, and $\phi yn$ are constants; this yields the following set of nonlinear algebraic equations governing the monochromatic responses at the steady-state:

We note that due to the monochromatic feature of the applied excitation, the governing equation at the steady-state is ultimately reduced to a 2DOF reduced-order system; note that this is an *exact* reduction since no approximation was involved. Moreover, the reduced system has the form of two linear damped oscillators coupled with a cubic nonlinear spring, but the system parameters are dependent on $\lambda 1$, which, in turn, is dependent on the excitation frequency. Therefore, the reduction of the acoustics of the infinite-dimensional gated waveguide to the 2DOF reduced system is only valid for monochromatic responses. This is not the case for nondispersive semi-infinite waveguides, which are equivalent to linear dampers with constant damping coefficients regardless of the excitation frequency [3].

Note that Eq. (22) is an *infinite set of linear differential equations* and represents the variational system for stability analysis. Therefore, we truncate the number of equations and consider the finite version of Eq. (22) as an approximation for the stability analysis. In this work, 1000 oscillators are considered in total for the stability analysis, and the eigenvalues of Eq. (22) are studied with the details provided in Appendix B. Clearly, the monochromatic solution is deemed stable as long as the real parts of *all* eigenvalues are negative.

From Figs. 2(e), 3(e), 2(f), and 3(f), we notice that the durations of the transient effects are much smaller compared with the total simulation time. Therefore, we approximate the analytical transmitted energy downstream by the product of the averaged power predicted in Eq. (24) and the total simulation time $T$, and compare it with the corresponding numerically computed transmitted energy, $Edown$, given by Eq. (4).

In Fig. 6, we reconsider the numerical results of the transmitted energy shown in Figs. 4(a) and 4(b) and superimpose onto the analytical predictions based on the previous variational analysis. We notice that the analytical results agree well with the numerical simulations in the regime of stable monochromatic steady-state solutions. S-shape curves are observed in Figs. 6(a) and 6(b) with three branches of solutions as predicted by the cubic algebraic equation (20). We notice that the middle branch is always unstable. As the damping ratio increases, the simulation results switch from the upper to the lower branch of stable analytical solutions, with the transmitted energies after the transitions being lower by nearly two orders of magnitude. The predicted bifurcations lead to the steep reduction of the transmitted energy at a critical damping ratio. The critical damping ratios vary when switching the harmonic excitation (i.e., flipping $\sigma 1,\sigma 2$), which gives rise to nonreciprocal acoustics for damping ratios between the critical values. It is interesting to point out that the disparities in the critical damping ratios observed in Figs. 6(a) and 6(b) are successfully captured by the analytical approximations.

We also notice that the analytical solutions lose stability for sufficiently low damping ratios in all cases considered. This is because in these regimes there is a unique monochromatic solution and it is unstable. The most robust unstable region is observed in Fig. 6(c), with damping ratio $\zeta $ ranging from approximately 0 to 0.02, and the numerical simulations differ from the analytical results. Since the single monochromatic solution is unstable, the responses at the steady-state are not monochromatic but rather SMRs are generated. From Figs. 3(a) and 5(a) we notice that at $\zeta =0.02$, SMRs are realized at the steady-state. In this case, the monochromatic *ansatz* is not valid and therefore the analytical prediction is not accurate (as expected). Moreover, when steady-state monochromatic solutions governed by Eq. (20) are realized, the transmissibility decreases as the damping ratio increases. This is not necessarily valid for the SMR responses, which are not governed by Eq. (20). Lastly, we conclude that the SMRs are generated due to the loss of stability of the monochromatic solutions. Hence, by increasing damping ratios, the monochromatic solutions tend to be stable and the SMRs vanish. Moreover, a critical damping ratio exists where a bifurcation occurs, and beyond which the monochromatic solutions become stable. These analytical results validate our hypothesis regarding the existence of a critical damping ratio that governs the transition between different steady-state acoustic regimes and enable the prediction of the corresponding downstream transmitted energies.

We note that the analytical method succeeds in predicting the stability of the monochromatic responses and the bifurcations. However, the SMRs cannot be modeled by this analysis, and are difficult to predict through analytical methods due to the dispersive nature of the linear subwaveguides. Therefore, in the next section we proceed to the machine learning approach first developed in Ref. [1] in order to computationally predict the performance in parameter space of the gated waveguide in terms of transmissibility and nonreciprocity.

## 4 Machine Learning Approach for Predicting the Non-Reciprocal Acoustics

where $X0=x0/Ap$, $Y0=y0/Ap$, $Xn=xn/Ap$, and $Yn=yn/Ap$. We note from Eq. (25) that the nonlinear acoustics of the gated waveguide are governed by six normalized independent parameters, that is, the normalized detuning parameters $\sigma 1$, $\sigma 2$, the excitation frequency $\omega \u0302$, the normalized (strong) coupling $d$, the nonlinear coupling parameter $\alpha Ap2$, and finally the damping ratio $\xi $. Without loss of generality, in what follows we assume that $\alpha =0.5$ and vary the amplitude $Ap$ to tune the nonlinear coefficient. The aim for constructing the learnable simulator is to predictively design the parameters of the waveguide so that is suitable for experimental design or engineering applications. Since the precise control of damping in the experiments is difficult [29,30], in practice nonuniform damping coefficients are not avoidable. As a theoretical preliminary study, we assume that the damping ratios of the grounding stiffnesses are uniform along the waveguides with $\xi =$ 0.01. Here, machine learning methods are employed to study the effects of the parameters $\sigma 1$, $\sigma 2$, $d$, $Ap$, and $\omega \u0302$ on the previously defined transmissibility and nonreciprocity measures.

A five-dimensional dataset is created for varying parameters $\sigma 1$, $\sigma 2$, $d$, $Ap,$ and $\omega \u0302$. Since the excitation frequency $\omega \u0302$ is located inside the passband, that is, in the range of Eq. (3) with the wavenumber $\theta $ ranging from 0 to $\pi $. If the excitation frequency is close to the boundary of the pass band, the group velocity approaches zero and it takes a long time for the system to reach the steady-state. Therefore, the values of $\omega \u0302$ are selected for wavenumbers $\theta $ between $16\pi $ and $56\pi $. The detuning parameters $\sigma 1$, $\sigma 2$ range from $\u22122$ to 2 with an increment of 0.2, while $d$ and $Ap$ range from 0.2 to 0.8 with an increment of 0.2. Since the grounding stiffnesses of the nonlinear gate are positive, the detuning parameters follow the constraints that $(1+d\sigma 1,2)>0$. The dataset is generated by randomly sampling the coefficients with a uniform distribution in the five-dimensional domain. At the end, the random dataset incorporating the constraints involves 31,768 sets of parameters in total.

where $ai$ denotes the vector of the values at the *i*th layer, $wi$ and $bi$ are the learnable weight matrix and the offset vector, respectively, and $f$ is the activation function. In this work, the activation functions of the hidden layers are rectified linear unit (ReLU) functions, where $f(x)=max(0,x)$, and the activation function of the output layer is linear (identity transformation).

The overall dataset is randomly divided into three subsets with 50% used for training, 15% for validation and 35% for testing. The Levenberg–Marquardt backpropagation algorithm is adopted to train the weights $wi$ and the offsets $bi$ with the mean square error adopted as the loss function. The loss function for the validation subset is evaluated after each epoch (the number of passes of the entire training set), and the training stops if the loss function on the validation subset does not get improved in six consecutive epochs. The loss function at each epoch is shown in Appendix D. The neural network is developed and trained based on the platform of matlab deep learning toolbox. In order to evaluate the efficacy of the trained simulator, the nonreciprocity and transmissibility measures $\delta $ and $\eta $ are plotted in Figs. 8(a) and 8(b) with the comparison of the NN prediction and the direct numerical simulations on the testing subset. We notice that the neural network accurately predicts the nonreciprocity and the transmissibility, with the correlation $R$ between the targets generated by the simulations and the outputs predicted by the NN simulator are close to 0.99.

Data points in the test set: $N=11,119$ . | Predicted condition . | Performance . | ||
---|---|---|---|---|

Actual condition | Desirable | Undesirable | ||

Desirable | TP = 81 | FN = 17 | Sensitivity: 0.827 | |

Undesirable | FP = 15 | TN = 11,006 | Specificity: 0.999 | |

Precision: 0.844 |

Data points in the test set: $N=11,119$ . | Predicted condition . | Performance . | ||
---|---|---|---|---|

Actual condition | Desirable | Undesirable | ||

Desirable | TP = 81 | FN = 17 | Sensitivity: 0.827 | |

Undesirable | FP = 15 | TN = 11,006 | Specificity: 0.999 | |

Precision: 0.844 |

From Table 2, we note that the sensitivity, precision, and specificity measures are all close to 1, and therefore, the NN simulator well classifies the waveguides with large transmissibility and nonreciprocity.

The classification results are visualized in Fig. 9. Note that the performance is dependent on the five independent parameters $\sigma 1,\sigma 2,\u2009Ap,\u2009\theta ,\u2009d$. The desired region predicted by the NN simulator is shown in Fig. 9(a) by fixing $Ap=0.4$, $\theta =7/12\pi $ and $d=0.6$. Only the blue region in Fig. 9(a) is well defined due to the constraints $1+d\sigma 1,2>0$. A region with desired performance classified by the NN simulator where $\sigma 1$ is close to zero and $\sigma 2$ is close to −1. To visualize the desired region in higher dimensional space, we fit a square in the desired region. An auxiliary measure, the kernel size, is defined as the side length of the square, and is used to quantify the robustness of the desired performance given $\theta ,\u2009Ap$ and $d$. In Fig. 9(b), we fix $d=0.6$ and plot the kernel size versus $Ap$ and $\theta $. We note that the most robust region occurs at around $Ap=0.4$ and $\theta =712\pi $, where the kernel size is close to 1. In order to visualize classification results in the three-dimensional space involving varying $Ap$, $\theta $, and $d$, we proceed to animate the contour plots of the kernel sizes with variable $d$ where each frame depicts a contour of kernel sizes for fixed $d$ similar to Fig. 9(b). The animation is available through the link given in Appendix E.

The transmissibility and downstream energy are computed by substituting Eqs. (24) and (28) into Eq. (5). Note that the analytical approximations are only applicable for monochromatic steady-state responses. To this end, the analytical monochromatic desired responses are identified through the following two steps:

Step 1: Obtain the analytical solution from Eq. (20) and check its stability via Eq. (22). If there is only one stable solution, then we proceed to the next step.

Step 2: Compute the analytical approximation of the nonreciprocity measure $\delta $ and nonreciprocal transmissibility measure $\eta $, and check whether $\eta >0.2$ and $\delta >1.$

Similar to the results of the NN prediction shown in Fig. 9, the kernel size of the (monochromatic) analytical predictions is defined as the side length of the square fit to the desired region in the $(\sigma 1,\sigma 2)$ plane given the parameters $Ap,\theta $ and $d$. The animation of the analytical (monochromatic) kernel sizes for the desired regions is available in Appendix E. In Fig. 10, we show the comparison of the analytical (monochromatic) results and the NN predicted results for the three representative values $d=0.2,\u20090.5,$ and $0.8$. Since the analytical results are only applicable for the monochromatic response regime, we expect that the analytical desired region is a subset of the full desired region. Indeed, in Fig. 10, the monochromatic desired region is mainly located in the NN desired region with smaller kernel sizes, validating the previous inclusion predictions for the NN and the analytical predicted results.

In a final validation step, we verify the predicted results by the NN output and the CX-A analysis by considering three cases denoted with crosses in Fig. 10 and labeled as A, B, and C, respectively. The corresponding system parameters for these cases are summarized in Table 3. Case A corresponds to $(Ap,\theta ,d)=(0.35,13\pi ,0.5)$ with NN and analytically predicted kernel sizes being close to 0.5. Case B corresponds to $(Ap,\theta ,d)=(0.35,712\pi ,0.5)$ and is predicted to be desirable by the analytical solution but undesirable by the NN output. Lastly, case C corresponds to $(Ap,\theta ,d)=(0.35,712\pi ,0.8)$ and the NN predicted kernel size is much larger than the analytically predicted kernel size.

Cases | Case A | Case B | Case C |
---|---|---|---|

Case parameters $(Ap,\theta ,d)$ | $(0.35,13\pi ,0.5)$ | $(0.35,712\pi ,0.5)$ | $(0.35,712\pi ,0.8)$ |

Systems | System A1 | System B1 | System C1 |

Gate parameters $(\sigma 1,\sigma 2)$ | $(\u22121,\u22121.5)$ | $(0,\u22121.5)$ | $(0.3,\u22120.5)$ |

Cases | Case A | Case B | Case C |
---|---|---|---|

Case parameters $(Ap,\theta ,d)$ | $(0.35,13\pi ,0.5)$ | $(0.35,712\pi ,0.5)$ | $(0.35,712\pi ,0.8)$ |

Systems | System A1 | System B1 | System C1 |

Gate parameters $(\sigma 1,\sigma 2)$ | $(\u22121,\u22121.5)$ | $(0,\u22121.5)$ | $(0.3,\u22120.5)$ |

In Figs. 11(a) and 11(b), we compare the desired regions predicted by the CX-A method and the NN output for case A. The desired regions are in the third quadrant of the $(\sigma 1,\sigma 2)$ domain in both cases with similar areas and positions. To verify the results of NN and analytical predictions, we present the numerical results of one representative case with $(\sigma 1,\sigma 2)=(\u22121,\u22121.5)$ denoted as A1 in the desired regions. To analyze the nonreciprocal acoustics, the case of flipped $(\sigma 1,\sigma 2)=(\u22121.5,\u22121)$ is simulated also, corresponding to the excitation moved to the opposite side. The time series of $x0$ and $y0$ are shown in Figs. 11(c) and 11(d) for the direct and flipped $(\sigma 1,\sigma 2)$ cases, respectively. In both cases the system reaches the steady-state at approximately $\tau =400$ and the amplitudes of the responses are not modulated. Hence, the monochromatic response *ansatz* is valid in this case. The amplitude of $y0$ is comparable to $x0$ for $(\sigma 1,\sigma 2)=(\u22121,\u22121.5)$, while the amplitude of $y0$ is much smaller than that of $x0$ for $(\sigma 1,\sigma 2)=(\u22121.5,\u22121)$. The corresponding input and transmitted energies are shown in Figs. 11(e) and 11(f). A strong disparity is observed between the two simulation cases in terms of the transmitted energy as well as the amplitudes of the oscillator after the gate, indicating the strong nonreciprocal acoustics, namely, with strong energy transfer from left to right, but weak from right to left. The resulting transmissibility and nonreciprocity measures $\eta $ and $\delta $ are listed in Table 4 for the numerical simulations in comparison with the analytical and NN predictions. Based on the numerical simulations of system A1, $\eta =0.395$ and $\delta =1.923$, indicating the strong nonreciprocity and transmissibility which agrees with the NN and analytical predictions. Since the steady-state responses are monochromatic, the analytical solutions accurately predict the transmissibility and nonreciprocity measures with small relative errors. As a comparison, the NN also captures the nonreciprocal results.

System A1 | System B1 | System C1 | ||||
---|---|---|---|---|---|---|

Performance of the waveguide | $\eta $ | $\delta $ | $\eta $ | $\delta $ | $\eta $ | $\delta $ |

Numerical simulations | 0.395 | 1.923 | 0.248 | 2.501 | 0.220 | 1.159 |

NN outputs | 0.381 | 1.862 | 0.164 | 2.256 | 0.212 | 1.122 |

Analytical predictions | 0.410 | 1.930 | 0.253 | 2.460 | No stable solution |

System A1 | System B1 | System C1 | ||||
---|---|---|---|---|---|---|

Performance of the waveguide | $\eta $ | $\delta $ | $\eta $ | $\delta $ | $\eta $ | $\delta $ |

Numerical simulations | 0.395 | 1.923 | 0.248 | 2.501 | 0.220 | 1.159 |

NN outputs | 0.381 | 1.862 | 0.164 | 2.256 | 0.212 | 1.122 |

Analytical predictions | 0.410 | 1.930 | 0.253 | 2.460 | No stable solution |

In Figs. 12(a) and 12(b), we plot the desired regions predicted by the analysis and the NN for case B. We observe a desired region predicted by the analytical results for $\sigma 1$ close to zero and $\sigma 2$ in the approximate range −1.3 to −1.8. This region, however, is not predicted by the NN. Therefore, we present numerical simulation results at one representative value in the analytical desired region denoted by B1 with $(\sigma 1,\sigma 2)=(0,\u22121.5)$. The time series of the displacements of the gate oscillators and the energy flow are shown in Figs. 12(c) and 12(e), respectively. To highlight the strong nonreciprocal acoustics in this case, the simulation results of B1 for the case of flipped $(\sigma 1,\sigma 2)$ are shown in Figs. 12(d) and 12(f). We notice the nonreciprocal energy transfer from left to right but not for right-to-left. The numerical transmissibility and nonreciprocity measures are listed in Table 4 with $\eta =0.248$ and $\delta =2.501$, indicating the desired nonreciprocal performance. We note that the desired performance is captured by the analytical solutions but not by the NN (with predicted $\eta =0.164$ and $\delta =2.256$). Despite the false negative prediction at B1, we note that the predicted transmissibility measure is close to the threshold $\eta =0.2$. Since the sensitivity of the NN classification is 0.827, the NN captures most of the desired responses in the parameter domain, and the prediction of the major robust desired region should be acceptable. From Fig. 10(d), we note that the desired region at point B is sensitive to the excitation parameters, and therefore the NN prediction is not necessarily valid. This reveals the limitation of the NN in predicting the nonreciprocal responses that are sensitive to input coefficients, which does not conflict with its good overall performance.

In Figs. 13(a) and 13(b), we consider the desired regions for case C. We note that the desired region predicted by the NN is mainly located in the fourth quadrant of the $(\sigma 1,\sigma 2)$ domain. However, the major part of the desired region is not captured by the analytical prediction. Therefore, we present the numerical simulations for $(\sigma 1,\sigma 2)=(0.3,\u22120.5)$ denoted as C1, which is a representative case where the NN and analytical solutions provide different predictions. The direct numerical simulations are performed for C1 and the case of flipped $(\sigma 1,\sigma 2)$. The time series of $x0$ and $y0$ are shown in Fig. 13(c) for left to right wave propagation. In this case, the steady-state amplitudes of both responses are strongly modulated. However, in Fig. 13(d) the amplitudes are not modulated for right to left wave propagation and flipped $(\sigma 1,\sigma 2)$. Therefore, the nonreciprocity is observed in terms of the frequency contents of the responses. The energy flows for the excitation applied from both sides are shown in Figs. 13(e) and 13(f) with the strong transmissibility and nonreciprocity evidenced by the measures $\eta =0.220$ and $\delta =1.159$ in Table 4. This desired performance is captured by the NN but not by the analytical solutions. Based on the eigenvalue analysis of Eq. (22), the analytical solution at point C1 is unstable. Therefore, the monochromatic ansatz is invalid, and the responses exhibit multiple frequency components since the response amplitudes are modulated as shown in Fig. 13(c).

In synopsis, the NN simulator is trained and applied to predict the responses of the waveguide and classify the desired results based on the transmissibility and nonreciprocity measures. A high overall performance of the classification is achieved, which is verified through the correlations of the NN output and target data, as well as the confusion matrix. The analytical solutions are also adopted to classify the desired region for monochromatic responses. Through the comparison with direct numerical simulations, we observe that the results agree well. However, high accuracy is achieved in the analytical predictions only in the regime of monochromatic responses. As a comparison, the NN simulator predicts all types of desired responses, with lower accuracy, however, compared to the analytical results. The errors of NN prediction in sensitive regions represent a possible limitation of the machine learning approach.

## 5 Concluding Remarks

We studied the nonreciprocal acoustics of an infinite linear waveguide subject to a single harmonic excitation, and incorporating a local, asymmetric, nonlinear gate. The grounded oscillators comprising the linear waveguide were strongly coupled, and, as a result, it supported broadband wave transmission in its single passband. This posed distinct challenges in our study, compared to an earlier work [28] where the weakly coupled waveguide was considered, supporting narrowband wave transmission. A strongly nonlinear problem was formulated to explore the nonreciprocal acoustics. First, we numerically showed that this system enabled strong nonreciprocity resulting in energy propagation only in one direction. The transmitted waves exhibited either a single frequency (case of monochromatic response), or multiple frequency components (case of strongly modulated responses, SMRs). Transitions between these two response types were governed by bifurcations, which were analytically studied by CX-A analysis.

The CX-A method, valid for this strongly nonlinear problem, was employed based on a monochromatic response *ansatz*, yielding analytical approximations based on slow flow (modulation) equations. Stability analysis was also performed, and the stable solutions agreed well with the numerical results. The dramatic change in nonreciprocal transmitted energy originated from the transition between two branches of analytical solutions, while SMRs were generated at the loss of stability of the monochromatic results. The governing mechanism of the transition between different types of responses was clearly explained through the analysis, enabling predictive design of the nonreciprocal gated waveguide.

Since the analytical model was only applicable to monochromatic responses, we proceeded to predict and classify the general class of nonreciprocal waveguides through a machine learning approach which was applicable for both monochromatic responses and SMRs. The NN model first introduced for the case of weak coupling and weak nonlinearity [28] was adopted and extended to the case of strong coupling and strong nonlinearity. Reasonable damping ratios were taken into considerations as well. Following the preliminary training process, the machine learning model well captured the waveguide responses corresponding to strong nonreciprocity and transmissibility. This was verified through comparisons with analytical solutions and direct numerical simulations. Hence, the machine learning model, along with the analytical model, act as powerful tools for the predictive design of the nonreciprocal waveguide for desired nonreciprocity and transmissibility features. In the desired region, the transmissibility measure reaches 0.4 and the nonreciprocity measure reaches as high as 3.

This work demonstrates the design of an otherwise linear dissipative broadband waveguide featuring strong break of acoustic reciprocity, achieved merely by a local nonlinear gate. The machine learning approach can be extended to the design of nonreciprocal waveguides with multiple passbands, and in higher dimensions. The approaches developed in this work can be applied to a broad class of linear waveguides incorporating a local nonlinear element. Therefore, a promising prospect is to design the local dynamics of the nonlinear element to be capable of complicated functions, such as frequency conversion or acoustic logic operation. The main limitations in these studies would be the high dimensionality of the parameter spaces where the NN would be applied which requires a larger training set. Another promising prospect is to effectively presample the data through analytical predictions, thus optimizing the neural net efficiency and reducing the training costs.

## Acknowledgment

We would like to acknowledge the useful discussions on the contents of the paper with Ali Kanj at the University of Illinois at Urbana-Champaign. Also, we appreciate the reviewers' comments that help us improve the performance of the neural networks.

## Funding Data

Division of Emerging Frontiers (Award ID: 1741565; Funder ID: 10.13039/100000156).

### A Closed Form Solution of the Slow Flow Equations at the Nonlinear Gate

We note that Eq. (A5) is a real cubic equation, where the closed form solution for $|v|2=vv\xaf$ is given by Cardano's formula. Substituting the closed form solution for $vv\xaf$ into Eq. (A1) we get linear equations with respect to $x$ and $y$, and the closed form solution of equation (A1) is completed.

### Stability Analysis of the Truncated Version of Eq. (22)

Given the nonzero dissipation effects, the energy attenuation is not avoidable and the amplitudes at infinity tend to zero. Therefore, it is reasonable to assume the “truncated” boundary conditions $\varphi xN=\varphi yN=0$ for sufficiently large $N$. Hence, the infinite-dimensional dynamical system (22) is approximated with a dynamical system of finite dimensions.

where Eq. (B3) is in the general form of a linear autonomous dynamical system with constant coefficients. The response of the dynamical system is stable as long as all the eigenvalues of the matrix $(ABB\xafA\xaf)$ have negative real parts.

### The Efficacy of Neural Nets With Different Hyperparameters

The performance of different NN configurations (in terms of layers and number of neurons in each layer) with different hyperparameters is considered here. Multiple NN simulators were trained, ranging from 2 to 4 layers, and 20 to 50 neurons per layer. The sensitivity, specificity, precision, and correlations for the different neural nets are listed in Tables 5–9.

Sensitivity | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.714 | 0.729 | 0.746 | 0.773 |

3 Layers | 0.743 | 0.809 | 0.743 | 0.752 |

4 Layers | 0.795 | 0.800 | 0.750 | 0.827 |

Sensitivity | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.714 | 0.729 | 0.746 | 0.773 |

3 Layers | 0.743 | 0.809 | 0.743 | 0.752 |

4 Layers | 0.795 | 0.800 | 0.750 | 0.827 |

Specificity | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.998 | 0.998 | 0.998 | 0.999 |

3 Layers | 0.998 | 0.998 | 0.999 | 0.998 |

4 Layers | 0.999 | 0.998 | 0.998 | 0.999 |

Specificity | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.998 | 0.998 | 0.998 | 0.999 |

3 Layers | 0.998 | 0.998 | 0.999 | 0.998 |

4 Layers | 0.999 | 0.998 | 0.998 | 0.999 |

Precision | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.789 | 0.796 | 0.773 | 0.860 |

3 Layers | 0.781 | 0.776 | 0.840 | 0.759 |

4 Layers | 0.824 | 0.777 | 0.796 | 0.844 |

Precision | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.789 | 0.796 | 0.773 | 0.860 |

3 Layers | 0.781 | 0.776 | 0.840 | 0.759 |

4 Layers | 0.824 | 0.777 | 0.796 | 0.844 |

Correlation of transmissibility | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.990 | 0.990 | 0.990 | 0.990 |

3 Layers | 0.989 | 0.990 | 0.990 | 0.988 |

4 Layers | 0.992 | 0.992 | 0.992 | 0.992 |

Correlation of transmissibility | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.990 | 0.990 | 0.990 | 0.990 |

3 Layers | 0.989 | 0.990 | 0.990 | 0.988 |

4 Layers | 0.992 | 0.992 | 0.992 | 0.992 |

Correlation of nonreciprocity | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.989 | 0.989 | 0.990 | 0.990 |

3 Layers | 0.990 | 0.990 | 0.990 | 0.991 |

4 Layers | 0.992 | 0.992 | 0.992 | 0.993 |

Correlation of nonreciprocity | 20 Neurons | 30 Neurons | 40 Neurons | 50 Neurons |
---|---|---|---|---|

2 Layers | 0.989 | 0.989 | 0.990 | 0.990 |

3 Layers | 0.990 | 0.990 | 0.990 | 0.991 |

4 Layers | 0.992 | 0.992 | 0.992 | 0.993 |

From these tables, we note that the NN with 4 layers and 50 neurons per layer has the best overall performance in terms of the regression and classification measures. Therefore, the neural network with 4 layers and 50 neurons was adopted in this work to predict the acoustic transmissibility and nonreciprocity of the gated waveguide.

Note that by adding more layers and more neurons, the performance measures are not significantly enhanced. Therefore, we did not train deeper neural networks.

### Animations of the Desired Regions

In Fig. 14, the loss function at each epoch is plotted. The loss function of the validation set does not decrease for consecutive 6 epochs after epoch 32. At epoch 32, the normalized mean square error decreases from $104$ to 0.028, revealing that the neural network is well trained.

### Animations of the Desired Regions

Here, we provide the links for animations for the kernel sizes of the desired regions in the $(\sigma 1,\sigma 2)$ domain for varying. $(Ap,\theta ,d)$.

The link for the animations of the kernel sizes of the desired regions predicted by the NN simulator can be found below.^{1}

The link for the animations of the kernel sizes of the desired regions predicted by the analytical solutions can be found below.^{2}