做物理模拟或PDE求解的团队终于有了解决高频细节丢失的实用方案——FreqNO-DPS用扩散模型校正神经算子的谱偏差，在稀疏观测下也能恢复全频带精度，建议做科学计算或工程仿真的直接试。
神经算子代理（NO）能比数值求解器快数个数量级地近似偏微分方程解，但存在谱偏差：高频成分被系统性地衰减，限制了其在需要精细尺度结构场景下的可靠性。稀疏传感器测量虽能提供无谱畸变的逐点精度，但仅覆盖小部分区域。本文提出FreqNO-DPS框架，将NO预测作为扩散后验采样中的辅助观测，结合基于分数的扩散先验和稀疏观测条件，并通过闭式谱整形引导分数避免重新引入偏差。在3D弹性波场预测中，5%和2%传感器覆盖率下，该方法在所有频带达到近零谱偏差，而单独使用代理或传感器均显示高频衰减。该框架仅需成对代理/参考数据，无需问题特定结构。
Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations
Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters. Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion but covering only a small fraction of the domain. We address this by treating NO predictions as auxiliary observations in a diffusion posterior sampling framework. Our method, FreqNO-DPS (https://github.com/niccoloperrone/FreqNO-DPS), combines an unconditional score-based diffusion prior, trained on high-fidelity simulations, with diffusion posterior sampling (DPS) conditioned on sparse observations and guided by a frozen neural operator. Naive integration reintroduces the surrogate's spectral bias; we resolve this with a closed-form, spectrally shaped guidance score that weights the surrogate by its frequency-dependent accuracy and needs no denoiser backpropagation. A distribution-free analysis bounds the approximation error across the frequency-diffusion-time plane and shows the guidance's frequency dependence is preserved regardless of distributional assumptions. On 3D elastic wavefield prediction at 5% and 2% sensor coverage, the method reaches near-zero spectral bias across all bands, where both the surrogate and sensor-only DPS show systematic high-frequency attenuation. Isotropic guidance, the natural baseline, improves pointwise accuracy but carries the bias into the posterior nearly intact, confirming that frequency-dependent calibration is essential, not merely beneficial. The framework needs only paired surrogate/reference data and exploits no problem-specific structure beyond the residual's approximate spectral diagonality, verifiable for new surrogates via the coherence diagnostic we provide.