# Error-Aware B-PINNs: Improving Uncertainty Quantification in Bayesian Physics-Informed Neural Networks

arXiv: https://arxiv.org/abs/2212.06965 · HTML: https://mlsanity.com/paper/2212.06965/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 22 references that are in the map.
- First posted on arXiv 14 Dec 2022.

## Builds on (most cited first, 15 of 15)

- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Scalable Bayesian Optimization Using Deep Neural Networks](https://mlsanity.com/paper/1502.05700/) (2015)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Adversarial Uncertainty Quantification in Physics-Informed Neural Networks](https://mlsanity.com/paper/1811.04026/) (2018)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Physical Symmetries Embedded in Neural Networks](https://mlsanity.com/paper/1904.08991/) (2019)
- [Learning Functional Priors and Posteriors from Data and Physics](https://mlsanity.com/paper/2106.05863/) (2021)
- [NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operators](https://mlsanity.com/paper/2208.11866/) (2022)
- [Solving differential equations with unknown constitutive relations as recurrent neural networks](https://mlsanity.com/paper/1710.02242/) (2017)
- [Uncertainty Quantification in Neural Differential Equations](https://mlsanity.com/paper/2111.04207/) (2021)
- [Evaluating Error Bound for Physics-Informed Neural Networks on Linear Dynamical Systems](https://mlsanity.com/paper/2207.01114/) (2022)

## Built on by (most cited first, 4 of 4)

- [Optimizing the Optimizer for Physics-Informed Neural Networks and Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2501.16371/) (2025)
- [One-Shot Transfer Learning for Nonlinear ODEs](https://mlsanity.com/paper/2311.14931/) (2023)
- [Damper-B-PINN: Damper Characteristics-Based Bayesian Physics-Informed Neural Network for Vehicle State Estimation](https://mlsanity.com/paper/2502.20772/) (2025)
- [Quantifying constraint hierarchies in Bayesian PINNs via per-constraint Hessian decomposition](https://mlsanity.com/paper/2510.03278/) (2025)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) are gaining popularity as a method for solving differential equations. While being more feasible in some contexts than the classical numerical techniques, PINNs still lack credibility. A remedy for that can be found in Uncertainty Quantification (UQ) which is just beginning to emerge in the context of PINNs. Assessing how well the trained PINN complies with imposed differential equation is the key to tackling uncertainty, yet there is lack of comprehensive methodology for this task. We propose a framework for UQ in Bayesian PINNs (B-PINNs) that incorporates the discrepancy between the B-PINN solution and the unknown true solution. We exploit recent results on error bounds for PINNs on linear dynamical systems and demonstrate the predictive uncertainty on a class of linear ODEs.