Physics-informed neural wavefields with Gabor basis functions

TL;DR

The paper addresses the bottleneck of training efficiency in physics-informed neural networks (PINNs) for high-frequency wavefields governed by the Helmholtz equation. It introduces a Gabor-based PINN where the final hidden layer computes a weighted sum of learnable Gabor basis functions that satisfy $k=\frac{\omega}{v(\boldsymbol{x})}$, with an auxiliary network predicting a center $\boldsymbol{\mu}$ to enable global coverage. This architecture yields faster convergence and improved accuracy on both simple and realistic models, particularly at higher frequencies, and benefits further from positional encoding. The contribution provides a scalable, interpretable alternative to vanilla PINNs for wavefield modeling, with practical impact for seismic and ultrasound imaging tasks where high-frequency content is essential.

Abstract

Recently, Physics-Informed Neural Networks (PINNs) have gained significant attention for their versatile interpolation capabilities in solving partial differential equations (PDEs). Despite their potential, the training can be computationally demanding, especially for intricate functions like wavefields. This is primarily due to the neural-based (learned) basis functions, biased toward low frequencies, as they are dominated by polynomial calculations, which are not inherently wavefield-friendly. In response, we propose an approach to enhance the efficiency and accuracy of neural network wavefield solutions by modeling them as linear combinations of Gabor basis functions that satisfy the wave equation. Specifically, for the Helmholtz equation, we augment the fully connected neural network model with an adaptable Gabor layer constituting the final hidden layer, employing a weighted summation of these Gabor neurons to compute the predictions (output). These weights/coefficients of the Gabor functions are learned from the previous hidden layers that include nonlinear activation functions. To ensure the Gabor layer's utilization across the model space, we incorporate a smaller auxiliary network to forecast the center of each Gabor function based on input coordinates. Realistic assessments showcase the efficacy of this novel implementation compared to the vanilla PINN, particularly in scenarios involving high-frequencies and realistic models that are often challenging for PINNs.

Figures (13)

• Figure 1: An example Gabor function of the plane wave with angle, $\theta$, wavelength, $\lambda$, and weighted by a Gaussian function with a mean located at ${\bf \mu}$, and variance, $\alpha$.
• Figure 2: A schematic diagram of the new PINN architecture with Gabor basis functions. The framework includes three parts, the network NN, and the PDE loss, with its parameters, and the boundary condition. The network takes (for training) random input source locations at the surface, $x_s$, as well as domain coordinate points, $\{x,z\}$, and outputs the real ($u_r$) and imaginary ($u_i$) parts of the wavefield that should satisfy, after training, the Helmholtz equation $F$, and the boundary condition $B$. The neurons of every layer are like the conventional PINN ones with activation functions, denoted by $\sigma$, other than the last layer that includes Gabor functions multiplied by the output of previous layer, with neural network weights $W$ and biases ${\bf b}$. Also learned are the Gabor parameters other than the frequency. The Gabor function $G$ is shared in lower left. AD stands for automatic differentiation. All parameters in the neural network function, $NN$ prior to the semi-column are inputs, and all parameters after are learnable.
• Figure 3: Gabor functions plotted schematically to show that many of them are required to properly reconstruct the wavefield corresponding to a source in the middle. ${\bf \mu_i}$ are the learned Gabor centers for these five Gabor functions $\{i=1-5\}$. The large dot in the middle represents a source.
• Figure 4: A diagram of the modified PINN network to that in Figure \ref{['fig2']}, which includes auxiliary connections with one hidden layer to predict a single Gabor center, ${\bf \boldsymbol{\mu}}=\{\mu^x,\mu^z\}$, to be used by the neurons in the Gabor layer. The activation function $\sigma_2$ stands for the Sigmoid to insure that the prediction of Gabor center location is within the domain of interest. The vectors ${\bf w^x}$ and ${\bf w^z}$ are the columns of the weight matrix connecting the hidden layer to the two outputs, and the input to the auxiliary connections are the same ${\bf x}=\{x,z\}$ as the main network. All parameters in the neural network function, NN, and the Gabor function, $G$, prior to the semi-column are inputs, and all parameters after are learnable.
• Figure 5: The velocity model and the real and imaginary parts of the scattered wavefield evaluated numerically by solving the scattered Helmholtz equation \ref{['eqn:eq2']} for a frequency of 4 Hz. The background velocity is constant and equal to 1.5 km/s, and the source is up top located in the middle at 1.25 km.