论文标题
动量减少了光谱偏差对物理信息的神经网络的影响
Momentum Diminishes the Effect of Spectral Bias in Physics-Informed Neural Networks
论文作者
论文摘要
物理信息神经网络（PINN）算法在解决涉及部分微分方程（PDE）的广泛问题方面显示出令人鼓舞的结果。但是，由于一种称为光谱偏差的现象，当目标函数包含高频特征时，它们通常无法收敛到理想的解决方案。在目前的工作中，我们利用神经切线内核（NTK）来研究用动量（SGDM）在随机梯度下降下演变的PINNS的训练动力学。这表明SGDM显着降低了光谱偏置的影响。我们还研究了为什么通过ADAM优化器训练模型可以在减少频谱偏置的同时加速收敛。此外，我们的数值实验已经证实，即使在存在高频特征的情况下，使用SGDM的广泛网络仍会收敛到理想的解决方案。实际上，我们表明网络的宽度在收敛中起着至关重要的作用。
Physics-informed neural network (PINN) algorithms have shown promising results in solving a wide range of problems involving partial differential equations (PDEs). However, they often fail to converge to desirable solutions when the target function contains high-frequency features, due to a phenomenon known as spectral bias. In the present work, we exploit neural tangent kernels (NTKs) to investigate the training dynamics of PINNs evolving under stochastic gradient descent with momentum (SGDM). This demonstrates SGDM significantly reduces the effect of spectral bias. We have also examined why training a model via the Adam optimizer can accelerate the convergence while reducing the spectral bias. Moreover, our numerical experiments have confirmed that wide-enough networks using SGDM still converge to desirable solutions, even in the presence of high-frequency features. In fact, we show that the width of a network plays a critical role in convergence.
