# Exploring Efficient Quantification of Modeling Uncertainties with Differentiable Physics-Informed Machine Learning Architectures

arXiv: https://arxiv.org/abs/2506.18247 · HTML: https://mlsanity.com/paper/2506.18247/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 43 references that are in the map.
- First posted on arXiv 23 Jun 2025.

## Builds on (most cited first, 14 of 14)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Bayesian Neural Networks: An Introduction and Survey](https://mlsanity.com/paper/2006.12024/) (2020)
- [Physics-Informed Gaussian Process Regression Generalizes Linear PDE Solvers](https://mlsanity.com/paper/2212.12474/) (2022)
- [Modern Monte Carlo Methods for Efficient Uncertainty Quantification and Propagation: A Survey](https://mlsanity.com/paper/2011.00680/) (2020)
- [Uncertainty Decomposition in Bayesian Neural Networks with Latent Variables](https://mlsanity.com/paper/1706.08495/) (2017)
- [Physics-informed Neural Networks with Periodic Activation Functions for Solute Transport in Heterogeneous Porous Media](https://mlsanity.com/paper/2212.08965/) (2022)
- [Data-driven method for real-time prediction and uncertainty quantification of fatigue failure under stochastic loading using artificial neural networks and Gaussian process regression](https://mlsanity.com/paper/2103.08349/) (2021)
- [Physics-Informed Machine Learning Towards A Real-Time Spacecraft Thermal Simulator](https://mlsanity.com/paper/2407.06099/) (2024)
- [Physics-Infused Machine Learning Based Prediction of VTOL Aerodynamics with Sparse Datasets](https://mlsanity.com/paper/2307.03286/) (2023)

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

- [Faster Thermal Profiling of a Lunar Rover with Machine Learning Adapted Finite Difference Model](https://mlsanity.com/paper/2605.27651/) (2026)

## Abstract (arXiv)

Quantifying and propagating modeling uncertainties is crucial for reliability analysis, robust optimization, and other model-based algorithmic processes in engineering design and control. Now, physics-informed machine learning (PIML) methods have emerged in recent years as a new alternative to traditional computational modeling and surrogate modeling methods, offering a balance between computing efficiency, modeling accuracy, and interpretability. However, their ability to predict and propagate modeling uncertainties remains mostly unexplored. In this paper, a promising class of auto-differentiable hybrid PIML architectures that combine partial physics and neural networks or ANNs (for input transformation or adaptive parameter estimation) is integrated with Bayesian Neural networks (replacing the ANNs); this is done with the goal to explore whether BNNs can successfully provision uncertainty propagation capabilities in the PIML architectures as well, further supported by the auto-differentiability of these architectures. A two-stage training process is used to alleviate the challenges traditionally encountered in training probabilistic ML models. The resulting BNN-integrated PIML architecture is evaluated on an analytical benchmark problem and flight experiments data for a fixed-wing RC aircraft, with prediction performance observed to be slightly worse or at par with purely data-driven ML and original PIML models. Moreover, Monte Carlo sampling of probabilistic BNN weights was found to be most effective in propagating uncertainty in the BNN-integrated PIML architectures.