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).

中文翻译：

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在没有前向模型的情况下求解贝叶斯逆问题的弱神经变分推理：在弹性成像中的应用


在本文中，我们介绍了一种新颖的数据驱动方法，用于解决基于偏微分方程 （PDE） 的高维贝叶斯逆问题，称为弱神经变分推理 （WNVI）。该方法使用从物理模型得出的虚拟观测值来补充实际测量。特别是，加权残差被用作主导偏微分方程的探针，以便制定和解决贝叶斯逆问题，而无需制定或求解正向模型。该公式将物理模型的状态变量视为潜在变量，使用随机变分推理 （SVI） 以及通常的未知数进行推断。采用的近似后验使用神经网络来近似从状态变量到未知数的逆映射。我们在生物医学环境中说明了所提出的方法，我们从噪声、组织变形数据中推断出空间变化的材料特性。我们证明，WNVI 不仅比依赖于将（非）线性正向问题作为黑盒反复求解的传统方法一样准确和高效，而且它还可以处理病态正向问题（例如，边界条件不足）。