Multifrequency Encoding in PINNs for Precision Wave Equation Modeling in Inhomogeneous Media

Physics-informed Neural Networks (PINNs) have emerged as powerful tools for solving partial differential equations (PDEs) by incorporating physical laws into the neural network training process. Traditional PINNs often use simple feedforward architectures for approximator networks; however, they can struggle with complex wave equation modeling in inhomogeneous media. In this paper, we introduce MSFme, a PINN variant that utilizes a Multiscale Fourier (MSF) approximator neural network with multiple frequency encodings to enhance precision. Each frequency encoding focuses on a single differential term of the wave equation depending on its complexity, allowing the network to better capture the intricate behavior of waveforms in such environments. Our results show that MSFme significantly reduces the L2 Relative Error (L2RE) compared to traditional Feed Forward Neural Networks (FNN) and Sinusoidal Representation Networks (SIREN) across diverse configurations. MSFme achieved error reductions of 5.35% in the two-region domain and 22.37% in the oval-shaped domain, demonstrating its potential for high-fidelity wave equation modeling in complex, inhomogeneous media. This advance highlights the importance of frequency-specific network design in improving the performance of PINNs in challenging physical simulations.