用概率模型解决高频波动方程，预测更准还带不确定性分析。
A Probabilistic Framework for Solving High-Frequency Helmholtz Equations via Diffusion Models
- 用基于得分的扩散模型构建条件概率算子，应对高频波的敏感性问题。
- 在多种频率下误差最低，$L^2$、$H^1$和能量范数均优于现有方法。
- 首次实现输入声速不确定性的传播建模，适合高可靠性需求场景。
确定性神经算子在多数偏微分方程中表现良好，但在高频波现象逼近中面临挑战：强输入-输出敏感性导致算子学习困难，谱偏差又模糊了振荡。本文主张采用概率方法逼近高频波，并提出基于得分的条件扩散算子框架。通过分析赫姆霍兹算子的稳定性，我们在广泛频率范围内进行数值实验，对比了多种主流数据驱动与机器学习方法。结果表明，该概率神经算子在$ L^2 $、$ H^1 $和能量范数下始终取得最低误差。更重要的是，与其他确定性方法不同，本框架能有效捕捉输入声速场不确定性向解场的传播。研究证明，概率算子学习为求解赫姆霍兹等复杂偏微分方程在高频区域提供了原理清晰且高效的新路径。
原文摘要 · Abstract (English)
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.
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