SpectraNet融合光谱与U-Net结构，实现稳定高效的偏微分方程预测。
Bridging Spectral Operator Learning and U-Net Hierarchies: SpectraNet for Stable Autoregressive PDE Surrogates
- 在U-Net框架中嵌入截断光谱卷积，用残差目标块抑制误差累积。
- 测试相对L2误差0.0822，参数量仅FNO的2.33倍，且在100步滚动预测中不发散。
- 适合对轻量级、低延迟、高稳定性有要求的物理模拟场景。
针对时间依赖偏微分方程的神经算子面临结构矛盾：光谱架构（如FNO）继承了一阶利普希茨常数带来的指数级滚动误差增长，而层级U-Net算子则牺牲分辨率不变性以换取多尺度细节。本文提出SpectraNet，一种自回归神经算子，将截断光谱卷积嵌入U-Net层级结构，并采用半群一致性损失训练残差-目标光谱块。残差-目标参数化将L^T稳定性爆炸替换为线性T*delta漂移，光谱路径参数量为Theta(L w^2 M^2)，与网格大小N无关。在16个公开基线模型上统一评估，于64×64的Navier-Stokes（nu=1e-5）测试中，参数量204万时相对L2误差达0.0822，比标准FNO少2.33倍参数且误差低20%；在六组跨PDE对比中胜出五项（包括不同黏度NS、浅水波2D、扩散-反应），仅主动物质模型由FNO胜出。在128²原生训练下，误差进一步降至0.0724，而FNO退化至0.3080。在所有200条测试轨迹中，自由滚动可稳定至T=100，FNO全部发散。在消费级CPU（B=1）上，SpectraNet推理<200ms，而全注意力Transformer虽原始误差更优，但延迟高约60倍；本文不追求超越该变压器的原始精度，而是主导≤500万参数、<200ms CPU的轻量化帕累托前沿。
原文摘要 · Abstract (English)
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
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。