AI 中文总结
针对从欠分辨率血流噪声测量重建速度和压力场这一不适定逆问题，提出贝叶斯有限元回归框架，无需离线训练数据，能重建含量化不确定性的场，在特定患者几何结构上表现优于三立方插值和PINN，更准确恢复壁面剪应力。
AI 中文摘要
从欠分辨率的血流噪声测量中重建准确的速度和压力场是一个不适定的逆问题，因为存在未知的进出口边界条件。我们提出了一种贝叶斯有限元回归框架，可在无离线训练数据的情况下，从噪声速度观测中重建具有量化不确定性的三维稳定速度和压力场。我们用泰勒-胡德有限元基函数表示速度和压力场，根据最大熵原理构建节点自由度上的物理先验。结合噪声模型指定的似然性，得到后验概率，其最大后验估计给出速度和压力重建。通过求解大规模稀疏非线性最小二乘问题计算最大后验估计，在此过程中解析消除压力、精确施加无滑移壁且无需正向/伴随求解或自动微分计算梯度。后验概率的拉普拉斯近似量化了重建中的不确定性，并将其传播到临床相关的感兴趣量。在特定患者的脑动脉瘤、主动脉瘤和主动脉缩窄几何结构上，该方法比三立方插值更准确地重建速度和压力，与PINN相当，同时比两者更准确地恢复感兴趣区域的壁面剪应力。
英文摘要
Reconstructing accurate velocity and pressure fields from under-resolved noisy measurements of blood flow is an ill-posed inverse problem due to unknown inlet and outlet boundary conditions. We present a Bayesian finite element regression framework that reconstructs steady three-dimensional velocity and pressure fields, with quantified uncertainty, from noisy velocity observations without offline training data. We represent velocity and pressure fields in Taylor-Hood finite element basis functions, and construct physics-informed priors on the nodal degrees of freedom from maximum-entropy principles. Combined with a likelihood specified by a noise-model, this yields a posterior whose maximum-a-posteriori estimate (MAP) gives velocity and pressure reconstructions. The MAP estimate is computed by solving a large-scale sparse nonlinear least-squares problem where pressure is eliminated analytically, no-slip walls are enforced exactly, and gradient is computed without forward/adjoint solves or automatic differentiation. A Laplace approximation of the posterior quantifies the uncertainties in our reconstructions and propagates them to clinically relevant quantities of interest including, pressure drop, flow rates, and wall shear stress. On patient-specific cerebral aneurysm, aortic aneurysm, and aortic coarctation geometries, the method reconstructs velocity and pressure more accurately than tricubic interpolation and comparably to a PINN, while recovering region-of-interest wall shear stress more accurately than both.
