3 research outputs found
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.Comment: 21 pages, 15 figure