PICProp: Physics-Informed Confidence Propagation for Uncertainty Quantification
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Abstract (arXiv)
Standard approaches for uncertainty quantification in deep learning and physics-informed learning have persistent limitations. Indicatively, strong assumptions regarding the data likelihood are required, the performance highly depends on the selection of priors, and the posterior can be sampled only approximately, which leads to poor approximations because of the associated computational cost. This paper introduces and studies confidence interval (CI) estimation for deterministic partial differential equations as a novel problem. That is, to propagate confidence, in the form of CIs, from data locations to the entire domain with probabilistic guarantees. We propose a method, termed Physics-Informed Confidence Propagation (PICProp), based on bi-level optimization to compute a valid CI without making heavy assumptions. We provide a theorem regarding the validity of our method, and computational experiments, where the focus is on physics-informed learning.
Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 24 of its 38 references that are in the map. First posted on arXiv 10 Oct 2023.
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24 of 38 references.
References · 24
- 2015Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learningcited by 4,448
- 2016Simple and Scalable Predictive Uncertainty Estimation using Deep Ensemblescited by 3,926
- 2020A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challengescited by 819
- 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 335
- 2019Optimizing Millions of Hyperparameters by Implicit Differentiationcited by 304
- 2020Bilevel Optimization: Convergence Analysis and Enhanced Designcited by 222
- 2022Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisonscited by 170
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- 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 147
- 2018Physics-Informed Generative Adversarial Networks for Stochastic Differential Equationscited by 135
- 2018Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 120
- 2020Bayesian Deep Ensembles via the Neural Tangent Kernelcited by 90
12 more
- 2023Tackling the Curse of Dimensionality with Physics-Informed Neural Networkscited by 79
- 2020Multi-fidelity Bayesian Neural Networks: Algorithms and Applicationscited by 59
- 2018The promises and pitfalls of Stochastic Gradient Langevin Dynamicscited by 56
- 2021Meta-learning PINN loss functionscited by 51
- 2022NeuralUQ: A comprehensive library for uncertainty quantification in neural differential equations and operatorscited by 33
- 2021From Predictions to Decisions: The Importance of Joint Predictive Distributionscited by 29
- 2020Confidence sequences for sampling without replacementcited by 23
- 2021The Neural Testbed: Evaluating Joint Predictionscited by 22
- 2020Beyond Marginal Uncertainty: How Accurately can Bayesian Regression Models Estimate Posterior Predictive Correlations?cited by 21
- 2021Wasserstein Generative Adversarial Uncertainty Quantification in Physics-Informed Neural Networkscited by 16
- 2015Multivariate Chebyshev Inequality with Estimated Mean and Variancecited by 5
- 2019On the Expressiveness of Approximate Inference in Bayesian Neural Networkscited by 5
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