论文标题
具有Helmholtz方程的平面波激活的神经网络
A Neural Network with Plane Wave Activation for Helmholtz Equation
论文作者
论文摘要
本文提出了一个基于平面波激活的神经网络（PWNN），用于求解Helmholtz方程，即表示波传播的基本偏微分方程，例如声波，电磁波和地震波。与使用基于传统激活的神经网络（TANN）或基于$ sin $激活的神经网络（Siren）来求解一般偏微分方程不同，我们推出了复杂的激活函数$ e^{\ MATHBF {i} {i} {x}}} $，是helmholtz equarey的基本组件的平面波。通过简单的推导，我们进一步发现，PWNN实际上是对统一方法（PWPUM）平面波分区的概括，并通过添加振幅和方向的学习基础来更好地表征潜在的解决方案。首先，我们研究解决方案问题的性能是平面波与所有已知方向的组成部分。该实验表明：PWNN在不同的体系结构或训练样本的数量上的效果要比Tann和Siren好得多，这意味着飞机波的激活确实有助于增强神经网络对Helmholtz方程求解的表示能力； PWNN比PWPUM具有竞争性能，例如相同的收敛顺序，但相对误差较少。此外，我们集中了一个更实用的问题，其解决方案仅将平面波与某些未知方向集成在一起。我们发现，在这种情况下，PWNN的效果要比PWPUM好得多。与使用PWPUM中固定方向的平面波基相比，PWNN可以学习一组优化的平面波基础，该平面波可以更好地预测溶液的未知方向。提出的方法可以在将深度学习应用于Helmholtz方程式中提供一些新的见解。
This paper proposes a plane wave activation based neural network (PWNN) for solving Helmholtz equation, the basic partial differential equation to represent wave propagation, e.g. acoustic wave, electromagnetic wave, and seismic wave. Unlike using traditional activation based neural network (TANN) or $sin$ activation based neural network (SIREN) for solving general partial differential equations, we instead introduce a complex activation function $e^{\mathbf{i}{x}}$, the plane wave which is the basic component of the solution of Helmholtz equation. By a simple derivation, we further find that PWNN is actually a generalization of the plane wave partition of unity method (PWPUM) by additionally imposing a learned basis with both amplitude and direction to better characterize the potential solution. We firstly investigate our performance on a problem with the solution is an integral of the plane waves with all known directions. The experiments demonstrate that: PWNN works much better than TANN and SIREN on varying architectures or the number of training samples, that means the plane wave activation indeed helps to enhance the representation ability of neural network toward the solution of Helmholtz equation; PWNN has competitive performance than PWPUM, e.g. the same convergence order but less relative error. Furthermore, we focus a more practical problem, the solution of which only integrate the plane waves with some unknown directions. We find that PWNN works much better than PWPUM at this case. Unlike using the plane wave basis with fixed directions in PWPUM, PWNN can learn a group of optimized plane wave basis which can better predict the unknown directions of the solution. The proposed approach may provide some new insights in the aspect of applying deep learning in Helmholtz equation.
