Learning mapping between two function spaces has attracted considerable research attention. However, learning the solution operator of partial differential equations (PDEs) remains a challenge in scientific computing. Therefore, in this study, we propose a novel pseudo-differential integral operator (PDIO) inspired by a pseudo-differential operator, which is a generalization of a differential operator and characterized by a certain symbol. We parameterize the symbol by using a neural network and show that the neural-network-based symbol is contained in a smooth symbol class. Subsequently, we prove that the PDIO is a bounded linear operator, and thus is continuous in the Sobolev space. We combine the PDIO with the neural operator to develop a pseudo-differential neural operator (PDNO) to learn the nonlinear solution operator of PDEs. We experimentally validate the effectiveness of the proposed model by using Burgers' equation, Darcy flow, and the Navier-Stokes equation. The results reveal that the proposed PDNO outperforms the existing neural operator approaches in most experiments.
翻译:两个功能空间之间的学习映射吸引了相当多的研究关注。然而,学习部分差异方程式(PDEs)的解决方案操作员仍然是科学计算中的一个挑战。因此,在本研究中,我们提议由假差异操作员(PDIO)启发的新型伪差异整体操作员(PDIO),这是对差异操作员的概括,具有某种符号的特征。我们使用神经网络将符号参数化,并显示神经网络符号包含在一个光滑的符号类中。随后,我们证明PDIO是一个受约束的线性操作员,因此在Sobolev空间是连续的。我们把PDIO与神经操作员(PDNO)结合起来,以学习PDEs的非线性解决方案操作员。我们通过使用Burgers的方程式、达西流和Navier-Stokes方程式实验验证了拟议模型的有效性。结果显示,拟议的PDNO在大多数实验中都比现有的神经操作员的方法。