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.

翻译：最近，物理信息神经网络（PINNs）因其在求解偏微分方程（PDEs）时出色的插值能力而备受关注。尽管潜力巨大，但其训练过程计算负担沉重，尤其是针对波场这类复杂函数。这主要源于基于神经网络的（可学习的）基函数偏向低频成分，因为它们由多项式计算主导，本质上不利于波场建模。为此，我们提出一种方法，通过将神经网络的波场解建模为满足波动方程的Gabor基函数的线性组合，从而提高其效率与精度。具体而言，针对亥姆霍兹方程，我们在全连接神经网络模型中引入一个自适应的Gabor层作为最终隐藏层，通过对这些Gabor神经元进行加权求和来计算预测（输出）。这些Gabor函数的权重/系数从包含非线性激活函数的先前隐藏层中学习得到。为确保Gabor层在整个模型空间中的有效利用，我们引入一个小型辅助网络，根据输入坐标预测每个Gabor函数的中心。实际评估表明，与标准PINN相比，这种新型实现方法在涉及高频和现实模型（通常对PINNs具有挑战性）的情况下表现出显著优越性。