Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastography
This addresses computational challenges in biomedical applications like elastography, offering a more efficient and robust approach for inverse problems, though it builds on existing variational inference techniques.
The paper tackles high-dimensional Bayesian inverse problems in PDEs, such as inferring material properties from tissue deformation data, by introducing Weak Neural Variational Inference (WNVI), which avoids solving forward models and is shown to be as accurate and more efficient than traditional methods while handling ill-posed problems.
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).