Multiwavelet-based Operator Learning for Differential Equations
Multiwavelet-based Operator Learning for Differential Equations
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发表时间：
2021-09
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通讯作者：
Gaurav Gupta;Xiongye Xiao;P. Bogdan
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作者：
Gaurav Gupta;Xiongye Xiao;P. Bogdan
偏微分方程的解可以通过计算输入和解空间之间的逆算子映射得到。为此，我们引入了一种\textit{基于多小波的神经算子学习方案，该方案}使用细粒度小波压缩相关算子的核。通过显式嵌入逆多小波滤波器，我们学习核在固定多小波多项式基上的投影。通过对多小波变换的重复计算，在多个尺度上对投影核进行训练。这允许在不同的尺度上学习复杂的依赖关系，并产生一个独立于分辨率的方案。与先前的工作相比，我们利用了算子核的基本性质，使其能够在数字上有效地表示。我们对Korteweg-de Vries （KdV）方程、Burgers方程、Darcy Flow和Navier-Stokes方程进行了实验。与现有的神经算子方法相比，我们的模型显示出更高的精度，并在一系列数据集上达到了最先进的水平。对于时变方程，所提出的方法表现出（$2X-10X$）改进（$0.0018$ （$0.0033$）相对于Burgers’（KdV）方程的$L2$误差）。通过学习函数空间之间的映射，该方法能够在学习低分辨率数据后找到高分辨率输入的解。
The solution of a partial differential equation can be obtained by computing the inverse operator map between the input and the solution space. Towards this end, we introduce a \textit{multiwavelet-based neural operator learning scheme} that compresses the associated operator's kernel using fine-grained wavelets. By explicitly embedding the inverse multiwavelet filters, we learn the projection of the kernel onto fixed multiwavelet polynomial bases. The projected kernel is trained at multiple scales derived from using repeated computation of multiwavelet transform. This allows learning the complex dependencies at various scales and results in a resolution-independent scheme. Compare to the prior works, we exploit the fundamental properties of the operator's kernel which enable numerically efficient representation. We perform experiments on the Korteweg-de Vries (KdV) equation, Burgers' equation, Darcy Flow, and Navier-Stokes equation. Compared with the existing neural operator approaches, our model shows significantly higher accuracy and achieves state-of-the-art in a range of datasets. For the time-varying equations, the proposed method exhibits a ($2X-10X$) improvement ($0.0018$ ($0.0033$) relative $L2$ error for Burgers' (KdV) equation). By learning the mappings between function spaces, the proposed method has the ability to find the solution of a high-resolution input after learning from lower-resolution data.