Achieving stable convergence of neural networks for estimating acoustic field in uniform ducts

The ability of the neural networks as function approximators can be exploited to solve several governing differential equations. In this work, 1-D Helmholz equation is solved to predict the acoustic pressure in a uniform duct. Solving the Helmholtz equation across a range of frequencies, especially at higher frequencies is challenging as the loss function destabilizes the training process, thereby preventing it from converging to the true solution with the desired accuracy. To overcome this issue, a dynamic learning rate technique is proposed that helps to stabilize the training process and improve overall accuracy of the network. The efficiency of the method is demonstrated by comparing the results with a static learning rate method and the analytical solutions. A good agreement is observed between the predicted solution with dynamic learning rate and the analytical solution up to 2000 Hz. Without dynamic learning rate, the relative errors are observed tobe 2% and 58% at 500 and 2000 Hz, respectively, whereas they reduced to 0.6% and 0.1%, respectively, with the dynamic learning rate at the same frequencies. The proposed dynamic learning rate method is found to be effective for different types of boundary conditions.