Title: A Probabilistic Framework For Solving High-Frequency Helmholtz Equation Via Diffusion Models
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
Keywords: Probabilistic Learning; Helmholtz Equation; Diffusion Model; High Frequency Wave; Operator Learning; Uncertainty Quantification

Abstract: Deterministic neural operators perform well on many PDEs but can struggle with the approximation of high-frequency wave phenomena, where strong input-to-output sensitivity makes operator learning challenging and spectral bias blurs oscillations. We argue for a probabilistic approach, and use a conditional diffusion operator as a concrete tool to investigate it. Our study couples theory with practice: a theoretical sensitivity analysis explains why high frequency amplifies prediction errors, and suggests an evaluation protocol (including an energy-form metric) to test whether learned surrogates preserve \emph{stable} quantities while capturing uncertainty. Across a range of regimes, the probabilistic neural operator is found to produce robust, full-domain predictions, better preserves energy at high frequency, and provides calibrated uncertainty that reflects input sensitivity, whereas deterministic approaches tend to oversmooth. These results position probabilistic operator learning as a principled and effective approach for solving complex PDEs such as Helmholtz in the challenging high-frequency regime.

TL;DR: We present a probabilistic framework based on conditional diffusion models that approximates high-frequency Helmholtz solutions with higher accuracy and calibrated uncertainty, outperforming deterministic neural operators.