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

2025-06-22

AI Summary

This study addresses the challenge of quantifying and propagating uncertainty in physics-informed machine learning (PIML), which limits reliability analysis and robust optimization. We propose a differentiable uncertainty-aware architecture that embeds Bayesian neural networks into a differentiable hybrid physics model, enabling end-to-end uncertainty propagation. A two-stage training strategy is introduced to enhance convergence stability. The method integrates physical constraints, automatic differentiation, Bayesian inference, and Monte Carlo sampling to jointly estimate predictive means and uncertainties in a fully differentiable manner. Evaluations on benchmark functions and real-world fixed-wing UAV flight data demonstrate that our approach achieves prediction accuracy comparable to state-of-the-art PIML models, attains >92% uncertainty coverage, and significantly improves propagation fidelity via Monte Carlo sampling. This provides a reliable, differentiable uncertainty quantification framework for model-driven engineering design and control.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Uncertainty Representations
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

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.

Problem

• Quantify modeling uncertainties in physics-informed machine learning
• Explore Bayesian Neural Networks for uncertainty propagation
• Evaluate hybrid PIML architectures on engineering applications

Innovation

• Differentiable hybrid PIML with Bayesian Neural Networks
• Two-stage training for probabilistic ML models
• Monte Carlo sampling for uncertainty propagation