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

2026-02-03

AI Summary

This work addresses the challenge of accurately modeling high-frequency Helmholtz equations with neural operators, which is hindered by extreme input–output sensitivity and spectral bias in conventional deterministic approaches. To overcome these limitations, the study introduces, for the first time, a score-based conditional diffusion operator that learns the full distribution of solutions within a probabilistic framework, rather than a single deterministic mapping. By integrating neural operator architectures with stability-aware considerations specific to the Helmholtz operator, the proposed method consistently achieves the lowest L², H¹, and energy norm errors across multiple frequencies. It significantly outperforms existing data-driven techniques and reliably quantifies uncertainty in the solution field induced by perturbations in sound speed.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Helmholtz equation
• high-frequency waves
• operator learning
• spectral bias
• input-output sensitivity

Innovation

• probabilistic neural operator
• diffusion models
• high-frequency Helmholtz equation
• uncertainty quantification
• score-based generative modeling