In this paper, we introduce a novel, data-driven approach for solving high-dimensional Bayesian inverse problems based on partial differential equations (PDEs), called Weak Neural Variational Inference (WNVI). The method complements real measurements with virtual observations derived from the physical model. In particular, weighted residuals are employed as probes to the governing PDE in order to formulate and solve a Bayesian inverse problem without ever formulating nor solving a forward model. The formulation treats the state variables of the physical model as latent variables, inferred using Stochastic Variational Inference (SVI), along with the usual unknowns. The approximate posterior employed uses neural networks to approximate the inverse mapping from state variables to the unknowns. We illustrate the proposed method in a biomedical setting where we infer spatially varying material properties from noisy tissue deformation data. We demonstrate that WNVI is not only as accurate and more efficient than traditional methods that rely on repeatedly solving the (non)linear forward problem as a black-box, but it can also handle ill-posed forward problems (e.g., with insufficient boundary conditions).

翻译：本文提出了一种新颖的、数据驱动的求解基于偏微分方程的高维贝叶斯反问题的方法，称为弱神经变分推断。该方法通过物理模型导出的虚拟观测数据对实际测量进行补充。具体而言，我们采用加权残差作为控制偏微分方程的探针，从而在不构建也不求解前向模型的情况下，构建并求解贝叶斯反问题。该公式将物理模型的状态变量视为潜变量，与常规未知量一同通过随机变分推断进行推断。所采用的近似后验分布使用神经网络来近似从状态变量到未知量的逆映射。我们在生物医学场景中展示了所提方法，从含噪声的组织变形数据推断空间变化的材料属性。实验证明，WNVI不仅与传统依赖重复求解（非）线性前向问题（作为黑箱）的方法同样精确且更高效，还能处理不适定的前向问题（例如边界条件不足的情况）。