Hermite neural operator for solving partial differential equations on unbounded domains ^*

Abstract Accurately simulating physical systems such as wave propagation and quantum mechanics often requires solving partial differential equations on unbounded domains. This poses a fundamental challenge: classical numerical methods rely on artificial boundary conditions, while state-of-the-art neural operators such as the Fourier neural operator (FNO) may introduce nonphysical artifacts due to inherent periodicity assumptions. To address these challenges, we propose the Hermite neural operator (HNO), a novel framework designed to learn solution operators specifically tailored for problems on unbounded domains. HNO produces continuous outputs that inherently satisfy decay conditions at infinity, effectively overcoming the limitations of grid-based and periodic approaches. We demonstrate the efficacy of HNO on several challenging benchmarks, including the heat and nonlinear Schrödinger equations. Across these tasks, HNO consistently achieves significantly higher accuracy than leading baselines such as FNO, proper orthogonal decomposition-deep operator network, and latent spectral model. We further verify HNO’s ability to learn a true operator through dedicated off-grid generalization tests. By making accurate inferences at arbitrary locations far beyond the training data support, HNO shows remarkable stability, an advantage absent in grid-dependent models. Overall, HNO offers a robust and physically principled framework for tackling complex simulation tasks in unbounded domains.