Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations
2026-06-02 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study a way to improve fast approximations of PDE (partial differential equation) solutions called neural operator surrogates, which often miss fine details due to spectral bias. They combine sparse accurate sensor measurements with these surrogates using a diffusion-based method called FreqNO-DPS, which corrects the bias by weighting frequencies according to their accuracy. Their approach avoids extra computational complexity and performs better than using sensors or surrogates alone, especially in retaining high-frequency details. They also provide theoretical guarantees and a diagnostic tool to verify their method's assumptions.
Neural operator surrogatePartial differential equationsSpectral biasDiffusion posterior samplingSparse sensor measurementsScore-based diffusion priorFrequency-dependent weightingElastic wavefield predictionSurrogate modelingCoherence diagnostic
Authors
Niccolò Perrone, Fanny Lehmann, Stefania Fresca, Filippo Gatti
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.