Operator learning for PDEs on unstructured meshes with varying geometries via second-generation wavelets

Learning solution operators for partial differential equations on unstructured meshes with varying geometry remains a fundamental challenge for neural operators. Existing approaches either transform scattered data onto a regular grid as a preprocessing step, discarding what the grid cannot represent, or encode data on the mesh graph, where cost and receptive field are tied to a constructed neighborhood. We propose LiGNO, a lifting-based geometric neural operator built on a second-generation wavelet in which we treat the point cloud itself as the finest level of a multiresolution hierarchy. A geometric lifting step decomposes nodal features into a coarse field on a fixed lattice and a nodal residual carrying the fine-scale detail. Together, they reconstruct the nodal data exactly, at any lattice resolution, so no information is discarded before the operator acts. A multi-level parameterized second-generation wavelet transform then recursively decomposes coarse and directional detail coefficients, enabling the operator to model physical interactions at multiple scales. We evaluate LiGNO against state-of-the-art methods on Darcy flow, the Poisson equation, and aerodynamic flows on DrivAer and airfoil geometries, with deformed domains and varying numbers of nodal points.