An Optimized Control Approach for HIFU Tissue Ablation Using PDE Constrained Optimization Method

A Combination of Deep Neural Networks and Physics to Solve the Inverse Problem of Burger's Equation

One of the most basic nonlinear Partial Differential Equations (PDEs) to model the effects of propagation and diffusion is Burger's equation. This puts great emphasize on seeking efficient versatile methods for finding a solution to the forward and inverse problems of this equation. The focus of this paper is to introduce a method for solving the inverse problem of Burger's equation using neural networks. With recent advances in the area of deep learning, a Physics-Informed Neural Network (PINN) is a category of neural networks that proved efficient for handling PDEs. In our work, the 1D and 2D Burger's equations are simulated by applying a PINN to a set of domain points. The training process of PINNs is governed by the PDE formula, the initial conditions (ICs), the Boundary Conditions (BCs), and the loss minimization algorithm. After training the network to predict the coefficients of the nonlinear PDE, the inverse problem of the 1D and 2D Burger's equations are solved with an error as low as 0.047 and 0.2 for 1D and 2D case studies, respectively. The wave propagation model is accomplished with an approximate training loss value of 1×e^-4. The utilization of PINNs for modeling Burger's equation is a mesh-free approach that competes with the commonly used numerical methods as it overcomes the curse of dimensionality. Training the PINN model to predict the propagation and diffusion effects can also be generalized to address further detailed applications of Burger's equation with complex domains. This contributes to clinical applications such as ultrasound therapeutics.