# Bayesian Inference for PDE-based Inverse Problems using the Optimization of a Discrete Loss

arXiv: https://arxiv.org/abs/2510.15664 · HTML: https://mlsanity.com/paper/2510.15664/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 59 references that are in the map.
- First posted on arXiv 17 Oct 2025.

## Builds on (most cited first, 24 of 24)

- [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)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems](https://mlsanity.com/paper/2107.12146/) (2021)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion](https://mlsanity.com/paper/1308.1313/) (2013)
- [Personalized Radiotherapy Design for Glioblastoma: Integrating Mathematical Tumor Models, Multimodal Scans and Bayesian Inference](https://mlsanity.com/paper/1807.00499/) (2018)
- [A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems](https://mlsanity.com/paper/1308.6221/) (2013)
- [A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems](https://mlsanity.com/paper/1505.03876/) (2015)
- [A penalty method for PDE-constrained optimization in inverse problems](https://mlsanity.com/paper/1504.02249/) (2015)
- [Physics-Regularized Multi-Modal Image Assimilation for Brain Tumor Localization](https://mlsanity.com/paper/2409.20409/) (2024)
- [Graph Transformers for inverse physics: reconstructing flows around arbitrary 2D airfoils](https://mlsanity.com/paper/2501.17081/) (2025)
- [A Conformal Prediction Framework for Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/2509.13717/) (2025)
- [Representing model inadequacy: A stochastic operator approach](https://mlsanity.com/paper/1604.01651/) (2016)
- [Flow reconstruction by multiresolution optimization of a discrete loss with automatic differentiation](https://mlsanity.com/paper/2303.04679/) (2023)
- [E-PINNs: Epistemic Physics-Informed Neural Networks](https://mlsanity.com/paper/2503.19333/) (2025)
- [Optimal Navigation in Microfluidics via the Optimization of a Discrete Loss](https://mlsanity.com/paper/2506.15902/) (2025)
- [The stress-free state of human erythrocytes: data driven inference of a transferable RBC model](https://mlsanity.com/paper/2303.03404/) (2023)
- [Contactless Precision Steering of Particles in a Fluid inside a Cube with Rotating Walls](https://mlsanity.com/paper/2506.15958/) (2025)

## Built on by (most cited first, 1 of 1)

- [A Mechanistic Analysis of Transformers for Dynamical Systems](https://mlsanity.com/paper/2512.21113/) (2025)

## Abstract (arXiv)

Inverse problems are crucial for many applications in science, engineering and medicine that involve data assimilation, design, and imaging. Their solution infers the parameters or latent states of a complex system from noisy data and partially observable processes. When measurements are an incomplete or indirect view of the system, additional knowledge is required to accurately solve the inverse problem. Adopting a physical model of the system in the form of partial differential equations (PDEs) is a potent method to close this gap. In particular, the method of optimizing a discrete loss (ODIL) has shown great potential in terms of robustness and computational cost. In this work, we introduce B-ODIL, a Bayesian extension of ODIL, that integrates the PDE loss of ODIL as prior knowledge and combines it with a likelihood describing the data. B-ODIL employs a Bayesian formulation of PDE-based inverse problems to infer solutions with quantified uncertainties. We demonstrate the capabilities of B-ODIL in a series of synthetic benchmarks involving PDEs in one, two, and three dimensions. We showcase the application of B-ODIL in estimating tumor concentration and its uncertainty in a patient's brain from MRI scans using a three-dimensional tumor growth model.