# Weak neural variational inference for solving Bayesian inverse problems without forward models: applications in elastography

arXiv: https://arxiv.org/abs/2407.20697 · HTML: https://mlsanity.com/paper/2407.20697/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 80 references that are in the map.
- First posted on arXiv 30 Jul 2024.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Bayesian Deep Convolutional Encoder-Decoder Networks for Surrogate Modeling and Uncertainty Quantification](https://mlsanity.com/paper/1801.06879/) (2018)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [Deep convolutional encoder-decoder networks for uncertainty quantification of dynamic multiphase flow in heterogeneous media](https://mlsanity.com/paper/1807.00882/) (2018)
- [Inference via low-dimensional couplings](https://mlsanity.com/paper/1703.06131/) (2017)
- [Solving inverse problems using conditional invertible neural networks](https://mlsanity.com/paper/2007.15849/) (2020)
- [Numerical Solution of Inverse Problems by Weak Adversarial Networks](https://mlsanity.com/paper/2002.11340/) (2020)
- [A probabilistic generative model for semi-supervised training of coarse-grained surrogates and enforcing physical constraints through virtual observables](https://mlsanity.com/paper/2006.01789/) (2020)
- [Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems](https://mlsanity.com/paper/1912.12976/) (2019)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Variational Relevance Vector Machines](https://mlsanity.com/paper/1301.3838/) (2013)
- [Fully probabilistic deep models for forward and inverse problems in parametric PDEs](https://mlsanity.com/paper/2208.04856/) (2022)
- [Bayesian Physics-Informed Extreme Learning Machine for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2205.06948/) (2022)
- [Accelerating PDE-constrained Inverse Solutions with Deep Learning and Reduced Order Models](https://mlsanity.com/paper/1912.08864/) (2019)
- [Sparse Variational Bayesian Approximations for Nonlinear Inverse Problems: applications in nonlinear elastography](https://mlsanity.com/paper/1412.0473/) (2014)
- [Beyond black-boxes in Bayesian inverse problems and model validation: applications in solid mechanics of elastography](https://mlsanity.com/paper/1803.00930/) (2018)
- [ANOVA Gaussian process modeling for high-dimensional stochastic computational models](https://mlsanity.com/paper/1911.05580/) (2019)

## Built on by (most cited first, 5 of 5)

- [DGenNO: A Novel Physics-aware Neural Operator for Solving Forward and Inverse PDE Problems based on Deep, Generative Probabilistic Modeling](https://mlsanity.com/paper/2502.06250/) (2025)
- [A unified physics-informed generative operator framework for general inverse problems](https://mlsanity.com/paper/2511.03241/) (2025)
- [The Illusion of Fit: Spatially Resolved Assessment of Constitutive Model Validity in Elastography and Physics-Based Inverse Problems](https://mlsanity.com/paper/2502.07415/) (2025)
- [Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data](https://mlsanity.com/paper/2505.11308/) (2025)
- [GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media](https://mlsanity.com/paper/2602.14642/) (2026)

## Abstract (arXiv)

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).