# Efficient Bayesian Physics Informed Neural Networks for Inverse Problems via Ensemble Kalman Inversion

arXiv: https://arxiv.org/abs/2303.07392 · HTML: https://mlsanity.com/paper/2303.07392/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 51 references that are in the map.
- First posted on arXiv 13 Mar 2023.

## Builds on (most cited first, 19 of 19)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Scaling Hamiltonian Monte Carlo Inference for Bayesian Neural Networks with Symmetric Splitting](https://mlsanity.com/paper/2010.06772/) (2020)
- [Challenges in Markov chain Monte Carlo for Bayesian neural networks](https://mlsanity.com/paper/1910.06539/) (2019)
- [Ensemble Kalman Inversion: A Derivative-Free Technique For Machine Learning Tasks](https://mlsanity.com/paper/1808.03620/) (2018)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [A regularizing iterative ensemble Kalman method for PDE-constrained inverse problems](https://mlsanity.com/paper/1505.03876/) (2015)
- [Reduced-Order Autodifferentiable Ensemble Kalman Filters](https://mlsanity.com/paper/2301.11961/) (2023)
- [Tikhonov Regularization Within Ensemble Kalman Inversion](https://mlsanity.com/paper/1901.10382/) (2019)
- [Novel Deep neural networks for solving Bayesian statistical inverse](https://mlsanity.com/paper/2102.03974/) (2021)
- [Never look back - A modified EnKF method and its application to the training of neural networks without back propagation](https://mlsanity.com/paper/1805.08034/) (2018)
- [Convergence Acceleration of Ensemble Kalman Inversion in Nonlinear Settings](https://mlsanity.com/paper/1911.02424/) (2019)
- [Parameterizations for Ensemble Kalman Inversion](https://mlsanity.com/paper/1709.01781/) (2017)
- [Ensemble Kalman Inversion for General Likelihoods](https://mlsanity.com/paper/2110.03034/) (2021)
- [A Review of Innovation-Based Methods to Jointly Estimate Model and Observation Error Covariance Matrices in Ensemble Data Assimilation](https://mlsanity.com/paper/1807.11221/) (2018)
- [A Novel Neural Network Training Framework with Data Assimilation](https://mlsanity.com/paper/2010.02626/) (2020)

## Built on by (most cited first, 6 of 6)

- [PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representations](https://mlsanity.com/paper/2412.05994/) (2024)
- [E-PINNs: Epistemic Physics-Informed Neural Networks](https://mlsanity.com/paper/2503.19333/) (2025)
- [Enhanced BPINN Training Convergence in Solving General and Multi-scale Elliptic PDEs with Noise](https://mlsanity.com/paper/2408.09340/) (2024)
- [Bayesian BiLO: Bilevel Local Operator Learning for Efficient Uncertainty Quantification of Bayesian PDE Inverse Problems with Low-Rank Adaptation](https://mlsanity.com/paper/2507.17019/) (2025)
- [PILD: Physics-Informed Learning via Diffusion](https://mlsanity.com/paper/2601.21284/) (2026)
- [DBPnet: Damper Characteristics-Based Bayesian Physics-Informed Neural Network for Wheel Load Estimation](https://mlsanity.com/paper/2605.24860/) (2026)

## Abstract (arXiv)

Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for high-dimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient inference algorithm for B-PINNs that uses Ensemble Kalman Inversion (EKI) for high-dimensional inference tasks. We find that our proposed method can achieve inference results with informative uncertainty estimates comparable to Hamiltonian Monte Carlo (HMC)-based B-PINNs with a much reduced computational cost. These findings suggest that our proposed approach has great potential for uncertainty quantification in physics-informed machine learning for practical applications.