mlsanityData to 14 Oct 2026

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

arXiv 2505.06459 ↗First posted 9 May 2025cs.LG cs.AI physics.comp-phcited by 3

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Abstract (arXiv)

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.

Key facts. Cited by 3 papers in the map (data through 14 Oct 2026). Builds on 19 of its 57 references that are in the map. First posted on arXiv 9 May 2025.

Builds on

19 of 57 references.

References · 19

  1. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  2. 2015Weight Uncertainty in Neural Networkscited by 554
  3. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  4. 2018Pyro: Deep Universal Probabilistic Programmingcited by 380
  5. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
  6. 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
  7. 2021Efficient training of physics-informed neural networks via importance samplingcited by 110
  8. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  9. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  10. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 57
  11. 2021Uncertainty Toolbox: an Open-Source Library for Assessing, Visualizing, and Improving Uncertainty Quantificationcited by 57
  12. 2022Bayesian Physics-Informed Neural Networks for real-world nonlinear dynamical systemscited by 54
7 more
  1. 2023A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositionscited by 48
  2. 2023Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operatorscited by 28
  3. 2020Solving Differential Equations Using Neural Network Solution Bundlescited by 21
  4. 2020Energy-based error bound of physics-informed neural network solutions in elasticitycited by 15
  5. 2023Residual-based error bound for physics-informed neural networkscited by 9
  6. 2021Uncertainty Quantification in Neural Differential Equationscited by 3
  7. 2022Evaluating Error Bound for Physics-Informed Neural Networks on Linear Dynamical Systemscited by 3

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2025: 1; 2026: 2.

3 papers.

  1. 2025Bayesian Physics Informed Neural Networks for Reliable Transformer Prognosticscited by 3
  2. 2026Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equationscited by 0
  3. 2026Rigorous Error Certification for Neural PDE Solvers: From Empirical Residuals to Solution Guaranteescited by 0