LiNO: New Neural Operator for Multiscale Differential Equations.

Researchers have developed LiNO, the Lifting Neural Operator, a new multiresolution operator designed to learn solutions to differential equations directly from data. Unlike existing neural operators that often struggle to simultaneously capture both broad global dynamics and intricate fine-scale structures, LiNO addresses this challenge through a hierarchical multiscale decomposition framework. It is built upon the second-generation wavelet lifting scheme, allowing it to learn an adaptive multiresolution decomposition directly from the input data. A key feature of LiNO is its ability to parameterize the lifting transform, which is exactly invertible by construction, ensuring information-preserving multiscale operator learning. In this lifted multiresolution space, the operator processes coarse and directional detail coefficients separately, enabling scale-aware modeling of underlying physical phenomena. Evaluated across diverse benchmarks, including Darcy flow, Navier-Stokes, and reaction-diffusion systems, LiNO consistently demonstrates strong performance, suggesting that adaptive multiresolution operators are a promising direction for scientific machine learning.