Bridging Spectral Operator Learning and U-Net Hierarchies: SpectraNet for Stable Autoregressive PDE Surrogates

2026-05-09

AI Summary

This work addresses the exponential error growth in long-term predictions by neural operators for time-dependent partial differential equations, which arises from one-step Lipschitz constants. To mitigate this, the authors propose SpectraNet, an autoregressive neural operator that integrates truncated spectral convolutions with a U-Net multiscale architecture. By parameterizing residual target spectral blocks, the method reduces error accumulation from exponential to linear drift and introduces a semigroup consistency loss to decouple model parameters from spatial grid resolution. Evaluated on PDEs such as Navier–Stokes, SpectraNet achieves a relative L² error of 0.0822 with only 2.04M parameters—20% lower error and 2.33× fewer parameters than the baseline—and further improves to 0.0724 at 128² resolution. The model remains stable over long rollouts up to T=100 and requires less than 200ms for CPU inference.

Abstract

Neural operators for time-dependent PDEs face a structural tension: spectral architectures (FNO and descendants) inherit exponential rollout-error growth from their one-step Lipschitz constant, while hierarchical U-Net operators trade resolution invariance for multi-scale detail. We introduce SpectraNet, an autoregressive neural operator that composes truncated spectral convolutions inside a U-Net hierarchy with a Residual-Target Spectral Block trained under a Semigroup-Consistency Loss. The residual-target parametrization replaces L^T stability blow-up with linear T*delta drift, and the spectral path's parameter count is Theta(L w^2 M^2), independent of grid N. Under a single unified protocol against 16 published neural-operator baselines on Navier-Stokes nu=1e-5 at 64x64, SpectraNet reaches test relative L2 = 0.0822 at 2.04M parameters -- 2.33x fewer than canonical FNO at ~20% lower error -- and wins five of six rows in a cross-PDE comparison against FNO (NS at nu in {1e-4, 1e-3}, PDEBench Shallow-Water 2D and Diffusion-Reaction, with the Active-Matter row going to FNO inside its seed spread). Trained from scratch at native 128^2 under the same protocol, SpectraNet improves to 0.0724 while FNO regresses to 0.3080. Free rollout stays bounded for T=100 where FNO diverges across all 200 test trajectories. On consumer CPU at B=1, SpectraNet runs sub-200ms while the full-attention Transformer that wins raw L2 pays ~60x latency; we do not claim to beat that Transformer on raw L2, only to dominate the lightweight (<=5M parameter, sub-200ms CPU) Pareto frontier. Source code: https://github.com/Enrikkk/spectranet

Problem

• neural operators
• time-dependent PDEs
• spectral methods
• U-Net hierarchies
• rollout-error growth

Innovation

• SpectraNet
• neural operator
• spectral convolution
• U-Net hierarchy
• semigroup-consistency loss