Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles

Pablo Flores, Olga Graf, Pavlos Protopapas, Karim Pichara
Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, PMLR 286:1289-1336, 2025.

Abstract

Physics-Informed Neural Networks (PINNs) have been widely used to obtain solutions to various physical phenomena modeled as Differential Equations. As PINNs are not naturally equipped with mechanisms for Uncertainty Quantification, some work has been done to quantify the different uncertainties that arise when dealing with PINNs. In this paper, we use a two-step procedure to train Bayesian Neural Networks that provide uncertainties over the solutions to differential equation systems provided by PINNs. We use available error bounds over PINNs to formulate a heteroscedastic variance that improves the uncertainty estimation. Furthermore, we solve forward problems and utilize the obtained uncertainties when doing parameter estimation in inverse problems in cosmology.

Cite this Paper

BibTeX
@InProceedings{pmlr-v286-flores25a, title = {Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles}, author = {Flores, Pablo and Graf, Olga and Protopapas, Pavlos and Pichara, Karim}, booktitle = {Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence}, pages = {1289--1336}, year = {2025}, editor = {Chiappa, Silvia and Magliacane, Sara}, volume = {286}, series = {Proceedings of Machine Learning Research}, month = {21--25 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v286/main/assets/flores25a/flores25a.pdf}, url = {https://proceedings.mlr.press/v286/flores25a.html}, abstract = {Physics-Informed Neural Networks (PINNs) have been widely used to obtain solutions to various physical phenomena modeled as Differential Equations. As PINNs are not naturally equipped with mechanisms for Uncertainty Quantification, some work has been done to quantify the different uncertainties that arise when dealing with PINNs. In this paper, we use a two-step procedure to train Bayesian Neural Networks that provide uncertainties over the solutions to differential equation systems provided by PINNs. We use available error bounds over PINNs to formulate a heteroscedastic variance that improves the uncertainty estimation. Furthermore, we solve forward problems and utilize the obtained uncertainties when doing parameter estimation in inverse problems in cosmology.} }
Endnote
%0 Conference Paper %T Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles %A Pablo Flores %A Olga Graf %A Pavlos Protopapas %A Karim Pichara %B Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2025 %E Silvia Chiappa %E Sara Magliacane %F pmlr-v286-flores25a %I PMLR %P 1289--1336 %U https://proceedings.mlr.press/v286/flores25a.html %V 286 %X Physics-Informed Neural Networks (PINNs) have been widely used to obtain solutions to various physical phenomena modeled as Differential Equations. As PINNs are not naturally equipped with mechanisms for Uncertainty Quantification, some work has been done to quantify the different uncertainties that arise when dealing with PINNs. In this paper, we use a two-step procedure to train Bayesian Neural Networks that provide uncertainties over the solutions to differential equation systems provided by PINNs. We use available error bounds over PINNs to formulate a heteroscedastic variance that improves the uncertainty estimation. Furthermore, we solve forward problems and utilize the obtained uncertainties when doing parameter estimation in inverse problems in cosmology.
APA
Flores, P., Graf, O., Protopapas, P. & Pichara, K.. (2025). Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles. Proceedings of the Forty-first Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 286:1289-1336 Available from https://proceedings.mlr.press/v286/flores25a.html.

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