🤖 AI Summary
This work addresses the inefficiency of conventional neural spectral methods for time-dependent partial differential equations, which require learning the entire vector field. The authors propose a residual decomposition framework that splits the solution into a low-fidelity background field and a high-resolution perturbation, modeling only the unresolved dynamics. Their approach integrates a fixed spectral operator, a background-dependent correction term, and an optional neural closure component. Crucially, the neural closure is formulated as a conditional refinement dependent on background fidelity, residual structure, and model compatibility, thereby explicitly separating physical mechanisms from neural components. On 2D Burgers, Klein-Gordon, and heterogeneous wave equations, their deterministic solver surpasses the NeuSA baseline without any training; for the Burgers equation, it achieves 24-fold and 44-fold reductions in training and extrapolation errors, respectively.
📝 Abstract
Neural spectral PDE solvers often learn an entire unresolved vector field even when an inexpensive approximate model can already capture most of the trajectory. Here we introduce Perturbative-NeuSA, a residual formulation that decomposes the target solution into a low-fidelity background and a high-resolution perturbation, so that only the unresolved dynamics is learned. Starting from the exact perturbation equation, the method combines a fixed spectral operator, a background-dependent correction, the background defect in the target PDE, and an optional neural closure. This construction makes the roles of physical structure and neural closure separately measurable. Across 2D Burgers, Klein-Gordon, and heterogeneous 2D wave equations, the deterministic structured solver outperforms the trained NeuSA baseline while requiring no neural-network training. The largest gains occur on Burgers, where the deterministic correction reduces training and extrapolation errors by factors of 24 and 44, respectively. In addition, a Klein-Gordon sweep over seven background resolutions shows that the effect of the closure is conditional: it improves a poor background by 3.6 times, becomes neutral at intermediate resolutions, and degrades a well-resolved background. For the wave equation, however, the closure provides an additional 18% reduction when the remaining residual is interface-localized. Multi-initial-condition diagnostics further show that the useful closure regime depends on the initial-condition spectrum and can disappear in extrapolation when structured correction already captures the dominant Burgers dynamics. Perturbative-NeuSA therefore reframes neural closure as a conditional, diagnosable correction governed by background fidelity, residual organization, and compatibility with the closure model.