Neural operators have emerged as a new area of machine learning for learning mappings between function spaces. Recently, an expressive and efficient architecture, Fourier neural operator (FNO) has been developed by directly parameterising the integral kernel in the Fourier domain, and achieved significant success in different parametric partial differential equations. However, the Fourier transform of FNO requires the regular domain with uniform grids, which means FNO is inherently inapplicable to complex geometric domains widely existing in real applications. The eigenfunctions of the Laplace operator can also provide the frequency basis in Euclidean space, and can even be extended to Riemannian manifolds. Therefore, this research proposes a Laplace Neural Operator (LNO) in which the kernel integral can be parameterised in the space of the Laplacian spectrum of the geometric domain. LNO breaks the grid limitation of FNO and can be applied to any complex geometries while maintaining the discretisation-invariant property. The proposed method is demonstrated on the Darcy flow problem with a complex 2d domain, and a composite part deformation prediction problem with a complex 3d geometry. The experimental results demonstrate superior performance in prediction accuracy, convergence and generalisability.

翻译：神经算子已成为机器学习中学习函数空间映射的新领域。近期，一种高效且表达能力强的架构——傅里叶神经算子（FNO）通过直接在傅里叶域中参数化积分核被提出，并在多种参数化偏微分方程中取得了显著成功。然而，FNO的傅里叶变换要求规则域具有均匀网格，这导致FNO本质上无法应用于实际应用中广泛存在的复杂几何域。拉普拉斯算子的本征函数同样能提供欧几里得空间中的频率基，甚至可以推广至黎曼流形。因此，本研究提出了一种拉普拉斯神经算子（LNO），其核积分可在几何域的拉普拉斯谱空间中进行参数化。LNO突破了FNO的网格限制，可在保持离散化不变性的同时应用于任意复杂几何。该方法在含复杂二维域的达西流问题以及含复杂三维几何的复合材料变形预测问题上得到了验证。实验结果表明，该方法在预测准确性、收敛性和泛化能力方面均展现出优越性能。