LiNO: New neural operator solves multiscale PDEs with adaptive wavelet lifting

Neural operators have emerged as powerful tools for learning solution operators of differential equations directly from data, but existing models often struggle to balance global dynamics with fine-scale structures. To address this, Pandey et al. developed LiNO (Lifting Neural Operator), a multiresolution operator using the second-generation wavelet lifting scheme. Unlike traditional wavelet transforms, LiNO learns the lifting transform from data, making it adaptive to the underlying solution function and exactly invertible. This information-preserving decomposition allows the operator to evolve coarse and directional detail coefficients separately, achieving scale-aware modeling that captures both large-scale patterns and local variations. LiNO thus offers a principled way to handle multiscale physics without sacrificing resolution or invertibility.

LiNO was evaluated on five challenging benchmarks spanning diverse physical behaviors: Darcy flow (elliptic PDE), Poisson equation (steady-state), Allen-Cahn (phase separation), compressible Navier-Stokes (fluid dynamics with shocks), and Gray-Scott reaction-diffusion (pattern formation). Across these scenarios, LiNO demonstrated strong performance compared to state-of-the-art neural operators such as FNO, U-Net, and DeepONet. Notably, it excelled on multiscale and chaotic systems where fixed-resolution operators typically degrade. The adaptive, data-driven lifting transform allows LiNO to automatically discover the relevant scales in the data, making it a promising direction for scientific machine learning – from climate modeling to turbulence simulation – where capturing both large-scale trends and fine-grained details is critical.