Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations

2026-06-02

AI Summary

This work addresses the spectral bias inherent in neural operators when solving partial differential equations, which impedes accurate recovery of high-frequency details under sparse observations. The authors propose embedding neural operator predictions as auxiliary observations within a Diffusion Posterior Sampling (DPS) framework, synergistically combining an unconditional diffusion prior with observational constraints to correct spectral bias while preserving pointwise accuracy. A novel spectral-weighted guidance score mechanism is introduced, enabling frequency-adaptive bias correction without backpropagation and yielding a distribution-agnostic error bound. In 3D elastic wavefield reconstruction, the method achieves near-zero spectral bias across the full frequency spectrum using only 2%–5% sensor coverage, substantially outperforming both standalone neural operators and standard DPS approaches.

Abstract

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.

Problem

• spectral bias
• neural operators
• sparse observations
• diffusion models
• high-frequency recovery

Innovation

• spectral bias
• neural operator
• diffusion posterior sampling
• frequency-dependent guidance
• sparse observations