Title: Mitigating Spectral Bias in Neural Operators Through Diffusion Posterior Sampling Using Sparse Data

Abstract:

While neural operator surrogates (NO) offer a significant speed advantage over traditional numerical solvers for approximating partial differential equation (PDE) solutions, they are hindered by a phenomenon known as spectral bias. This issue results in the systematic attenuation of high-frequency components, thereby compromising reliability in regions requiring fine-scale detail. Although sparse sensor measurements provide pointwise accuracy free from spectral distortion, they typically cover only a limited portion of the domain. To bridge this gap, we propose integrating NO predictions as auxiliary observations within a diffusion posterior sampling framework.

Our proposed method, FreqNO-DPS (https://github.com/niccoloperrone/FreqNO-DPS), merges an unconditional score-based diffusion prior—trained on high-fidelity simulations—with diffusion posterior sampling (DPS). This process is conditioned on sparse observations and steered by a frozen neural operator. Simply combining these elements naively would perpetuate the surrogate’s spectral bias. To prevent this, we introduce a closed-form guidance score that applies spectral shaping. This score weights the surrogate based on its frequency-dependent accuracy and eliminates the need for denoiser backpropagation.

We provide a distribution-free analysis that bounds the approximation error throughout the frequency-diffusion-time plane, demonstrating that the guidance’s frequency dependence remains intact irrespective of distributional assumptions. In tests involving 3D elastic wavefield prediction with sensor coverage rates of 5% and 2%, our approach achieved near-zero spectral bias across all frequency bands. In contrast, both the surrogate alone and standard DPS using only sensors exhibited systematic high-frequency attenuation. While isotropic guidance serves as a natural baseline and enhances pointwise accuracy, it fails to correct the bias, allowing it to persist in the posterior. This confirms that frequency-dependent calibration is crucial. The framework requires only paired surrogate and reference data, leveraging no problem-specific structures other than the approximate spectral diagonality of the residual—a property verifiable for new surrogates using the coherence diagnostic we introduce.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC