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

2025-05-09

AI Summary

Physics-informed neural networks (PINNs) lack inherent uncertainty quantification capabilities, limiting their reliable deployment in forward solving and cosmological parameter inference. This work proposes a two-stage Bayesian neural network framework that, for the first time, explicitly incorporates the theoretical error bound of PINNs into Bayesian inference. It constructs a heteroscedastic variance model jointly governed by physical constraints and data-driven learning, enabling end-to-end propagation of solution uncertainty from forward problems to inverse parameter estimation. Evaluated on multiple PDE benchmarks and a cosmological parameter inference task, the method significantly improves uncertainty calibration: calibration error decreases by 37%, and 95% credible interval coverage increases to 92%, outperforming state-of-the-art stochastic and Monte Carlo approaches.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Probabilistic Inference
• Intelligent Robots: State Estimation

Application Category

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• Social Networks and Social Media: Influence propagation, information diffusion, and the prediction on networks
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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.

Problem

• Quantify uncertainty in Physics-Informed Neural Networks (PINNs)
• Improve uncertainty estimation using error bounds and Bayesian methods)
• Apply enhanced PINNs to inverse problems in cosmology)

Innovation

• Uses Bayesian Neural Networks for uncertainty quantification
• Applies error bounds to improve variance estimation
• Utilizes uncertainties in cosmological parameter estimation