Laplace neural operator for complex geometries

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.