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.

Integrating Fourier Neural Operators with Diffusion Models to improve Spectral Representation of Synthetic Earthquake Ground Motion Response

Nuclear reactor buildings must be designed to withstand the dynamic load induced by strong ground motion earthquakes. For this reason, their structural behavior must be assessed in multiple realistic ground shaking scenarios (e.g., the Maximum Credible Earthquake). However, earthquake catalogs and recorded seismograms may not always be available in the region of interest. Therefore, synthetic earthquake ground motion is progressively being employed, although with some due precautions: earthquake physics is sometimes not well enough understood to be accurately reproduced with numerical tools, and the underlying epistemic uncertainties lead to prohibitive computational costs related to model calibration. In this study, we propose an AI physics-based approach to generate synthetic ground motion, based on the combination of a neural operator that approximates the elastodynamics Green's operator in arbitrary source-geology setups, enhanced by a denoising diffusion probabilistic model. The diffusion model is trained to correct the ground motion time series generated by the neural operator. Our results show that such an approach promisingly enhances the realism of the generated synthetic seismograms, with frequency biases and Goodness-Of-Fit (GOF) scores being improved by the diffusion model. This indicates that the latter is capable to mitigate the mid-frequency spectral falloff observed in the time series generated by the neural operator. Our method showcases fast and cheap inference in different site and source conditions.