Modeling of the Forward Wave Propagation Using Physics-Informed Neural Networks

Partial Differential Equations (PDEs) are used in modeling problems in nature and are commonly solved using classical methods like Finite Element Method (FEM), Finite Volume Method (FVM), or Finite Difference Method (FDM). However, solving high-dimensional PDEs has been notoriously difficult due to the Curse of Dimensionality (CoD). Among the pool of hyperbolic PDEs, the wave equation in particular is the base for modeling various clinical applications and designing many solutions in the medical fields of therapeutic and diagnostic ultrasound. This draws attention to the importance of accurate and efficient simulation. In recent years, deep neural networks have been proposed to predict numerical solutions of PDEs. Within that context, Physics-Informed Neural Networks (PINNs) have surfaced as a powerful tool for modeling PDEs. We simulate a linear wave equation with a single time-dependent sinusoidal source function e.g.: sin(pi t) using PINNs to model one of the most fundamental modeling equations in medical ultrasound applications. Results achieved are validated by an FDM solution with the same problem setup. After training, the PINN prediction takes an average time 47% of the FDM time performed by MATLAB for the same simulation metrics (IC, BC, and domain range) on the same machine. Being a mesh-free approach, PINNs overcome the CoD which is one of the main challenges in traditional modeling methods.