## 1 Introduction
While this assumption is reasonable for normal vocal fold vibrations, it becomes erroneous in many irregular glottal shapes in which the intraglottal flow can be severely curved and the glottis often presents a multiple channel configuration. The Navier-Stokes (N-S) equation can compute the correct flow pressures in these irregular shapes; however, this approach is computationally extremely expensive and as such, is not suitable for clinical use. A major clinical application of a computational voice simulator would be simulation based surgery management, such as predicting the outcome of surgical interventions, optimizing the surgical adjustments, providing patient-specific solutions based on patient-specific anatomy and needs. It has been shown that self-sustained oscillation of vocal folds is dominated by a few modes of vibration, even when the motion is abnormal [23; 24; 25; 26]. This high predictability of the vibratory pattern of the vocal folds stimulated the use of a machine-learning approach to model glottal flow dynamics based on glottal shapes. In this paper, the model is based on the Bernoulli equation with a viscous loss term predicted by a deep neural network (DNN) model. The DNN model is used for two reasons. First, the viscous loss of glottal flow is mainly affected by its channel shape; yet, the relationship is highly complex and nonlinear. The DNN model is well known for its ability in learning complex relationships using a large amount of data [27], which makes it well suitable for learning the relationship in the current problem. Second, vocal fold vibration patterns are dominated by a few vibration modes; therefore, the shapes of the glottal flow channel are highly predictable. By generating flow solutions using a large amount of combinations of vibration modes, vast data can be provided for training the DNN model. The training data of the DNN model is the N-S equation based three-dimensional simulation of glottal flows in various glottal shapes generated by a synthetic shape function. The input parameters of the DNN model are the geometric and flow parameters extracted from discretized cross sections of the glottal shapes and the output target is the corresponding flow resistance coefficient which determines the viscous loss term of the modified Bernoulli equation. The target value of the flow resistance coefficient is calculated from the modified Bernoulli equation where the values of the flow rate and pressure distribution are obtained from the N-S solution. \(K\)-fold cross validation [27] is performed to fine tune the architecture and hyperparameters of the DNN. With this trained DNN-Bernoulli model, the flow resistance coefficient as well as the flow rate and pressure distribution in any given glottal shape generated by the synthetic shape function can be predicted. Furthermore, in order to assess the dynamical prediction performance of the DNN-Bernoulli model, a specific fluid-structure interaction (FSI) case of the glottal flow is studied. First, a continuum-mechanics based vocal fold model is coupled with the Bernoulli model to obtain various shapes extracted from one vibration cycle; then based on these shapes, the DNN model is trained in the same way as in the synthetic shape case. Finally, the Bernoulli model in the coupled FSI solver is replaced by the trained DNN-Bernoulli model to predict the glottal vibration dynamics. The prediction performance of the DNN-Bernoulli model in accuracy and efficiency is demonstrated by comparing with the results obtained by the Bernoulli and N-S model. The outline of the paper is organized as follows: Formulation of the reduced-order model is elaborated in Section 2.1. Implementation of the DNN model for synthetic glottal shapes is discussed in Section 2.2. Implementation of the DNN-Bernoulli model for FSI simulation of a continuum-mechanics based vocal fold model is presented in Section 2.3. Performance of the DNN-Bernoulli model for synthetic shapes and FSI simulation is evaluated in Sections 3.1 and 3.2, respectively.