U-HNO: A U-shaped Hybrid Neural Operator with Sparse-Point Adaptive Routing for Non-stationary PDE Dynamics

2026-05-13

AI Summary

This work addresses the challenge that existing neural operators struggle to simultaneously capture global smooth propagation and local sharp features—such as shocks—in solutions of partial differential equations (PDEs). To this end, we propose the U-shaped Hybrid Neural Operator (U-HNO), which introduces a novel Sparsity-aware Point Adaptive Routing (SPAR) mechanism based on local contrast. SPAR dynamically selects, at each spatial location, between a Fourier-based global branch and a multi-scale Gaussian local branch. The architecture is further enhanced with a U-shaped encoder-decoder structure featuring skip connections. Combined with an H¹ gradient loss and spectral band consistency regularization, U-HNO achieves state-of-the-art rolling prediction accuracy across multiple 1D, 2D, and 3D benchmarks in PDEBench, demonstrating particularly significant gains on problems involving sharp solution features. Ablation studies confirm the effectiveness of each proposed component.

Abstract

Solutions to many partial differential equations (PDEs) display coexisting smooth global transport and localized sharp features within a single trajectory: shock fronts, thin interfaces, and concentrated high-frequency content sit on top of slowly varying backgrounds. This poses a challenge for neural operators: Fourier-based architectures mix nonlocal interactions efficiently but tend to under-resolve localized non-smooth features, whereas spatially local architectures recover fine detail at the cost of long-range propagation and rollout stability. Existing hybrid operators paper over this tension with a fixed, spatially uniform fusion that forces the same trade-off everywhere. We propose U-HNO, a U-shaped hybrid neural operator whose central design is Sparse-Point Adaptive Routing (SPAR): at every spatial location, a per-pixel hard mask selects whether the global Fourier branch or the local multi-scale Gaussian branch should dominate, and the sparsity ratio is a function of the local contrast of the routing signal, so smooth and shock-aligned regions receive different mixtures of global and local computation. SPAR is embedded in a hierarchical encoder-bottleneck-decoder backbone with skip connections so that the dual branches and the gate operate at every resolution. Training combines pointwise supervision with a finite-difference H^1 gradient term and a band-wise spectral consistency regularizer. Across benchmarks spanning 1D Burgers, Kuramoto-Sivashinsky, KdV, 2D advection, Allen-Cahn, Navier-Stokes, Darcy flow, and 3D transonic compressible Navier-Stokes from PDEBench, U-HNO achieves state-of-the-art rollout accuracy on the majority of tasks in both relative L^2 and H^1 metrics, with the largest gains on problems dominated by sharp localized features. Ablations show that removing any single component substantially degrades rollout error.

Problem

• non-stationary PDEs
• localized sharp features
• neural operators
• global-local trade-off
• adaptive routing

Innovation

• Sparse-Point Adaptive Routing
• Hybrid Neural Operator
• Non-stationary PDEs
• U-shaped Architecture
• Spectral-Local Fusion