Adaptive multiscale operator correction via learned spectral subspace and physics-informed optimization.

Neural operators have shown strong performance in learning solution mappings for differential equations and offer fast inference across parameter spaces. However, they often suffer from spectral bias, leading to poor resolution of high-frequency and localized features, especially in out-of-distribution settings. In this work, we propose adaptive multiscale operator correction (AMOC), a hybrid framework that combines neural operator learning with physics-constrained optimization in a low-dimensional, instance-specific adaptive spectral subspace. First, a neural operator learns a coarse solution, which is then projected onto a sparse multiscale basis using either wavelet family or Fourier modes, where dominant components are selected based on energy. Finally, a physics-informed optimization is performed on adaptive spectral subspace, significantly lowering computational cost while improving accuracy. The proposed approach enables efficient correction of neural operator predictions by leveraging the sparsity and localization of multiscale representations, while bypassing automatic differentiation in the residual loss through closed-form derivatives of the chosen basis. The proposed AMOC is evaluated on the heat, Poisson, Darcy, and Helmholtz equations, achieving up to four orders-of-magnitude error reduction over FNO, with particularly strong gains observed in challenging out-of-distribution regimes.