A collocation point sampling method for solving the Helmholtz equation using physics-informed neural networks

Background In the field of seismic wavefield simulation, physics-informed neural networks (PINNs) have emerged as a new method for efficiently solving the Helmholtz equation due to their characteristic of grid-free computation. However, for the forward modeling of complex medium models using the Helmholtz equation, traditional uniform grid-based methods for collocation point sampling are insufficient to efficiently provide gradient information, leading to prolonged training time and high resource consumption.

Objective and Method  To improve the forward modeling efficiency and accuracy based on PINNs and overcome the anomalous concentration of collocation points in the seismic source during dynamic sampling, this study proposed a hybrid residual-based adaptive distribution (RAD-H) method and then compared this method with six fixed and four dynamic sampling methods. In the RAD-H method, fixed sampling is employed for the seismic source area to ensure the capture of the source features, and residual-based adaptive distribution (RAD) is used for other areas to ensure the consistency between the resampled collocation points and the spatial sampling of the high-residual target area.

Results Numerical experiments indicate that for a homogeneous medium model, the RAD-H method reduced the L2 error of the wavefield by 3.050% using only 18.57% of collocation points compared to the classical baseline method. Moreover, the RAD-H method exhibited a model training time of approximately 4192.5 s, suggesting an improvement of 231.74% in computational efficiency compared to the classical baseline method. For the Marmousi2 model, the RAD-H strategy remained the error within the same order of magnitude as the classical baseline method while reducing the number of collocation points by 57.75%.

Conclusions The RAD-H method overcomes oversampling for seismic source areas and undersampling for other areas of other dynamic sampling methods, significantly enhancing the sampling efficiency of collocation points of PINNs during the forward modeling involving seismic sources. This study holds significant implications for efficient, high-precision wavefield simulation of complex geological models.