Neural Operators with Adaptive Spectral and Low-Rank Representations

Neural operators provide a data-driven framework for learning mappings between function spaces and have shown strong performance in scientific computing and surrogate modeling. Existing architectures, however, typically rely on a single representation of the input function—either purely pointwise, as in DeepONet, or purely spectral, as in Fourier Neural Operators—which limits their ability to simultaneously capture local variability and global structure. In this work, we propose NOASLRR, a neural operator that integrates three complementary branches within a unified DeepONet-style formulation: a pointwise MLP embedding, a spectral branch based on Chebyshev polynomial coefficients, and a low-rank linear embedding. The three representations are fused through two learnable sigmoid gating mechanisms that adaptively balance structured inductive biases against unstructured expressivity in an input-dependent manner. We provide a theoretical guarantee showing that the proposed architecture can uniformly approximate any continuous operator admitting a decomposable structure, and we validate the method on three canonical PDE benchmarks: the heat equation, the viscous Burgers equation, and the Laplace equation with Dirichlet boundary conditions. On all three benchmarks NOASLRR converges faster and reaches substantially lower mean squared error than DeepONet and FNO baselines—for the heat equation the final MSE is reduced from 1.1 $\times$ ${10}^{-3}$ (DeepONet) and 2.9 $\times$ ${10}^{-2}$ (FNO) to 3.4 $\times$ ${10}^{-4}$, with comparable improvements on the Burgers and Laplace equations. Ablation experiments further confirm that each of the three branches contributes to the accuracy gains, while the parameter overhead relative to DeepONet remains moderate thanks to the low-rank factorization.