Prasanta K. Ghosh

D. Veerababu, Prasanta K. Ghosh

Physics-informed neural networks offered an alternate way to solve several differential equations that govern complicated physics. However, their success in predicting the acoustic field is limited by the vanishing-gradient problem that occurs when solving the Helmholtz equation. In this paper, a formulation is presented that addresses this difficulty. The problem of solving the two-dimensional Helmholtz equation with the prescribed boundary conditions is posed as an unconstrained optimization problem using trial solution method. According to this method, a trial neural network that satisfies the given boundary conditions prior to the training process is constructed using the technique of transfinite interpolation and the theory of R-functions. This ansatz is initially applied to the rectangular domain and later extended to the circular and elliptical domains. The acoustic field predicted from the proposed formulation is compared with that obtained from the two-dimensional finite element methods. Good agreement is observed in all three domains considered. Minor limitations associated with the proposed formulation and their remedies are also discussed.

D. Veerababu, Ashwin A. Raikar, Prasanta K. Ghosh

Training neural networks can be challenging, especially as the complexity of the problem increases. Despite using wider or deeper networks, training them can be a tedious process, especially if a wrong choice of the hyperparameter is made. The learning rate is one of such crucial hyperparameters, which is usually kept static during the training process. Learning dynamics in complex systems often requires a more adaptive approach to the learning rate. This adaptability becomes crucial to effectively navigate varying gradients and optimize the learning process during the training process. In this paper, a dynamic learning rate scheduler (DLRS) algorithm is presented that adapts the learning rate based on the loss values calculated during the training process. Experiments are conducted on problems related to physics-informed neural networks (PINNs) and image classification using multilayer perceptrons and convolutional neural networks, respectively. The results demonstrate that the proposed DLRS accelerates training and improves stability.

D. Veerababu, Prasanta K. Ghosh

Neural networks constrained by the physical laws emerged as an alternate numerical tool. In this paper, the governing equation that represents the propagation of sound inside a one-dimensional duct carrying a heterogeneous medium is derived. The problem is converted into an unconstrained optimization problem and solved using neural networks. Both the acoustic state variables: acoustic pressure and particle velocity are predicted and validated with the traditional Runge-Kutta solver. The effect of the temperature gradient on the acoustic field is studied. Utilization of machine learning techniques such as transfer learning and automatic differentiation for acoustic applications is demonstrated.