发表机构
Department of Civil and Systems Engineering, Johns Hopkins University; Department of Aerospace Engineering and Mechanics, University of Minnesota（约翰霍普金斯大学土木与系统工程系; 明尼苏达大学航空航天工程与力学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究将算子引导高斯过程回归扩展到复值亥姆霍兹问题，通过转化复算子用实值GP条件推断。在基准问题和脑弹性成像中表现良好，能返回波场后验，确定了精度上限与未校准的不确定性，校准不确定性是后续核心步骤。
AI 中文摘要
亥姆霍兹方程支配时谐波传播，在耗散介质中，复模量使波数平方κ²为复数。从稀疏、有噪声的数据推断此类场需要能量化自身不确定性的求解器。物理引导的高斯过程（GP）回归通过返回解的后验来提供此功能，但算子条件公式几乎仅针对实值场开发。我们通过将复算子转化为等效耦合实块，将算子引导的GP回归扩展到复值亥姆霍兹问题，从而能用标准实值GP条件进行推断。该构造允许一系列先验，从适当的对角先验到共区域化和多尺度变体，并基于偏微分方程残差和边界迹进行条件设定。在一到三维的基准问题上，该求解器在内部约束预算小得多的情况下与有限差分和神经网络基线具有竞争力。与那些确定性基线不同，它返回复波场的后验而不是点估计。应用于体内脑磁共振弹性成像时，适当的多尺度先验将剪切卷曲场重建与测量的相关性提高到0.77，高于目标值0.75。增益来自多尺度核而非实虚耦合。我们还确定了由模型失配设定的低频精度上限以及尚未校准的后验不确定性。因此，校准不确定性成为耗散介质中概率波场推断的核心下一步。
英文摘要
The Helmholtz equation governs time-harmonic wave propagation, and in dissipative media a complex modulus renders its squared wavenumber $κ^2$ complex. Inferring such fields from sparse, noisy data calls for solvers that also quantify their own uncertainty. Physics-informed Gaussian-process (GP) regression supplies this by returning a posterior over the solution, yet operator-conditioned formulations have been developed almost exclusively for real-valued fields. We extend operator-informed GP regression to complex-valued Helmholtz problems by realifying the complex operator into an equivalent coupled real block, which enables inference with standard real-valued GP conditioning. The construction admits a family of priors, from a proper diagonal prior to coregionalized and multiscale variants, and conditions on PDE residuals and boundary traces. On benchmark problems in one to three dimensions, the solver is competitive with finite-difference and neural-network baselines at a far smaller interior-constraint budget. Unlike those deterministic baselines, it returns a posterior over the complex wavefield rather than a point estimate. Applied to \textit{in vivo} brain magnetic resonance elastography, a proper multiscale prior reconstructs the shear curl field to a correlation of $0.77$ with measurement, above a $0.75$ target. The gain arises from the multiscale kernel rather than from real--imaginary coupling. We further identify a low-frequency accuracy ceiling set by model mismatch and a posterior uncertainty that is not yet calibrated. Calibrated uncertainty therefore emerges as the central next step for probabilistic wavefield inference in dissipative media.
Comments26 pages, 7 figures
