SIMR-NO：一种具有频谱信息的多分辨率神经算子用于湍流流场超分辨率
SIMR-NO: A Spectrally-Informed Multi-Resolution Neural Operator for Turbulent Flow Super-Resolution
AI总结：
本文提出SIMR-NO，一种多分辨率神经算子，通过频谱信息和多尺度结构恢复湍流流场的高分辨率。在Kolmogorov强迫的二维湍流中，从极粗的观测重建高分辨率流场，取得更低的误差和更准确的频谱复现。
摘要：
从严重欠解的观测中重建高分辨率湍流流场是计算流体力学和科学机器学习中的基本反问题。经典插值方法无法恢复缺失的细尺度结构，而现有深度学习方法依赖卷积架构，缺乏用于大尺度放大时物理忠实重建所需的频谱和多尺度归纳偏置。我们引入了具有频谱信息的多分辨率神经算子（SIMR-NO），一种分层算子学习框架，该框架在中间空间分辨率上分解了不正则的逆映射，结合确定性插值先验与每个阶段的频谱门控傅里叶残差修正，并结合局部细化模块以恢复超出截断傅里叶基的细尺度空间特征。所提出的方法在Kolmogorov强迫的二维湍流中进行了评估，其中从极粗的8×8观测（代表16×的下采样因子）重建128×128的涡旋场。在201个独立测试实例中，SIMR-NO的平均相对ℓ2误差为26.04%，在所有方法中具有最低的误差方差，比FNO减少31.7%的误差，比EDSR减少26.0%的误差，比LapSRN减少9.3%的误差。除了点准确度外，SIMR-NO是唯一能够忠实复现地面真实能量和涡旋谱的算法，展示了湍流流场的物理一致超分辨率。
英文摘要：
Reconstructing high-resolution turbulent flow fields from severely under-resolved observations is a fundamental inverse problem in computational fluid dynamics and scientific machine learning. Classical interpolation methods fail to recover missing fine-scale structures, while existing deep learning approaches rely on convolutional architectures that lack the spectral and multiscale inductive biases necessary for physically faithful reconstruction at large upscaling factors. We introduce the Spectrally-Informed Multi-Resolution Neural Operator (SIMR-NO), a hierarchical operator learning framework that factorizes the ill-posed inverse mapping across intermediate spatial resolutions, combines deterministic interpolation priors with spectrally gated Fourier residual corrections at each stage, and incorporates local refinement modules to recover fine-scale spatial features beyond the truncated Fourier basis. The proposed method is evaluated on Kolmogorov-forced two-dimensional turbulence, where $128\times128$ vorticity fields are reconstructed from extremely coarse $8\times8$ observations representing a $16\times$ downsampling factor. Across 201 independent test realizations, SIMR-NO achieves a mean relative $\ell_2$ error of $26.04\%$ with the lowest error variance among all methods, reducing reconstruction error by $31.7\%$ over FNO, $26.0\%$ over EDSR, and $9.3\%$ over LapSRN. Beyond pointwise accuracy, SIMR-NO is the only method that faithfully reproduces the ground-truth energy and enstrophy spectra across the full resolved wavenumber range, demonstrating physically consistent super-resolution of turbulent flow fields.
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