A Probabilistic Framework for Solving High-Frequency Helmholtz Equations via Diffusion Models

TL;DR

This paper introduces a probabilistic neural operator framework using diffusion models to accurately and robustly solve high-frequency Helmholtz equations, capturing uncertainties and outperforming deterministic methods.

Contribution

It develops a novel probabilistic approach with a score-based diffusion operator for high-frequency wave PDEs, demonstrating improved accuracy and uncertainty quantification.

Findings

Achieves lowest errors in multiple norms compared to other methods.

Robustly captures uncertainties in input parameters.

Outperforms deterministic approaches in high-frequency regimes.

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 adopting a probabilistic approach for approximating waves in high-frequency regime, and develop our probabilistic framework using a score-based conditional diffusion operator. After demonstrating a stability analysis of the Helmholtz operator, we present our numerical experiments across a wide range of frequencies, benchmarked against other popular data-driven and machine learning approaches for waves. We show that our probabilistic neural operator consistently produces robust predictions with the lowest errors in $L^2$, $H^1$, and energy norms. Moreover, unlike all the other tested deterministic approaches, our framework remarkably captures uncertainties in the input sound speed map propagated to the solution field. We envision that our results position probabilistic operator learning as a principled and effective approach for solving complex PDEs such as Helmholtz in the challenging high-frequency regime.