Physics-informed neural networks (PINNs) have shown remarkable achievements in solving partial differential equations (PDEs). However, their performance is limited when encountering oscillatory part in the solutions of PDEs. Therefore, this paper proposes a multi-scale deep neural network with periodic activation function to achieve high-frequency to low-frequency conversion, which can capture the oscillation part of the solution of PDEs. Moreover, the use of adaptive sampling method can adaptively change the location and distribution of residual points, improving the performance of the network. Additionally, the gradient-enhanced strategy is also utilized to embed the gradient information of the PDEs into the loss function of the neural network, which further improves the accuracy of PINNs. Through the numerical experiments verification, it is found that our method is better than PINNs in terms of accuracy and efficiency.

中文翻译：

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基于自适应采样点的多尺度残差网络求解偏微分方程


物理信息神经网络（PINN）在求解偏微分方程（PDE）方面取得了显着的成就。然而，当遇到偏微分方程解中的振荡部分时，它们的性能受到限制。因此，本文提出了一种具有周期性激活函数的多尺度深度神经网络来实现高频到低频的转换，可以捕获偏微分方程解的振荡部分。而且，采用自适应采样方法可以自适应地改变残差点的位置和分布，提高网络的性能。此外，还利用梯度增强策略将偏微分方程的梯度信息嵌入到神经网络的损失函数中，进一步提高了PINN的准确性。通过数值实验验证，我们的方法在精度和效率方面都优于PINNs。