Higher-order error estimates for physics-informed neural networks approximating the primitive equations
Higher-order error estimates for physics-informed neural networks approximating the primitive equations
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DOI：
10.1007/s42985-023-00254-y
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发表时间：
2022-09
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影响因子：
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通讯作者：
Ruimeng Hu;Quyuan Lin;Alan Raydan;Sui Tang
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作者：
Ruimeng Hu;Quyuan Lin;Alan Raydan;Sui Tang
海洋和大气的大尺度动力学受原始方程(PES)的支配。由于非线性和非局部性，偏微分方程组的数值研究通常是具有挑战性的。神经网络已被证明是解决这一挑战的一种很有前途的机器学习工具。在这项工作中，我们使用物理信息神经网络(PINN)来逼近偏微分方程组的解，并研究误差估计。我们首先建立了具有全粘性和扩散系数的偏微分方程解的高阶正则性，或者只具有水平粘性的偏微分方程组的整体解的高阶正则性。对于仅有水平结果的情况，这种结果是新的，在PINNS框架下的分析中是必需的。然后，通过取PINN的宽度足够宽，证明了存在相应的训练误差可以任意小的两层TANH PINN，并且只要训练误差足够小，样本集足够大，则真解与其近似解之间的误差可以任意小。特别地，所有的估计都是面积先验的，我们的分析包括高阶(按空间Soblev范数)的误差估计。给出了原型系统上的数值结果，进一步说明了在训练过程中使用该方法的优越性。
Large-scale dynamics of the oceans and the atmosphere are governed by primitive equations (PEs). Due to the nonlinearity and nonlocality, the numerical study of the PEs is generally challenging. Neural networks have been shown to be a promising machine learning tool to tackle this challenge. In this work, we employ physics-informed neural networks (PINNs) to approximate the solutions to the PEs and study the error estimates. We first establish the higher-order regularity for the global solutions to the PEs with either full viscosity and diffusivity, or with only the horizontal ones. Such a result for the case with only the horizontal ones is new and required in the analysis under the PINNs framework. Then we prove the existence of two-layer tanh PINNs of which the corresponding training error can be arbitrarily small by taking the width of PINNs to be sufficiently wide, and the error between the true solution and its approximation can be arbitrarily small provided that the training error is small enough and the sample set is large enough. In particular, all the estimates area priori, and our analysis includes higher-order (in spatial Sobolev norm) error estimates. Numerical results on prototype systems are presented to further illustrate the advantage of using thenorm during the training.