Feasibility study on solving the Helmholtz equation in 3D with PINNs

Room acoustic simulations at low frequencies often face significant uncertainties of material parameters and boundary conditions due to absorbing material. We discuss the application of Physics-Informed Neural Networks (PINNs) to solve the (forward) Helmholtz equation in three dimensions (3D), employing mini-batch stochastic gradient descent with periodic resampling every 100 iterations for memory-efficient training. Addressing the computational challenges posed by the extension of PINNs from 2D to 3D for acoustics, DeepXDE is used for implementing the forward PINN. The proposed numerical method is benchmarked against an analytical solution of a standing wave field in 3D. The PINN results are also compared to the Finite Element Method (FEM) solutions for a 3D wave field computed with openCFS. The alignment between PINN-generated solutions and analytical/FEM solutions shows the feasibility of PINNs modeling 3D acoustic applications for future inverse problems, and validating the accuracy and reliability of the proposed approach. Compared to FEM, establishing the PINN model took few hours (similar to the setup of a FEM simulation), the training took 38h to 42.8h (which is longer than the solution of the FEM simulation, which took 17min-19min), and the inference took 0.05 seconds being more than 20,000 times faster than the FEM benchmark openCFS using the same number of degrees of freedomwhen producing the results. Thereby, the insight is gained that 3D acoustic wave simulations in the frequency domain are feasible for forward PINNs and can predict complex wave behaviors in real-world applications.