Physics-guided mixture density networks for uncertainty quantification

Jie Chen, Yang Yu, Yongming Liu

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


This paper proposes a Physics-guided Mixture Density Network (PgMDN) model for uncertainty quantification of regression-type analysis. It integrates a Mixture Density Network for probabilistic modeling and physics knowledge as regularizations. This model can handle arbitrary distribution of data (e.g., strongly non-Gaussian, multi-mode, and truncated distributions). The physics knowledge from parameters and their partial derivatives is used as equality/ inequality constraints. The training of physics-guided machine learning is formulated as a constrained optimization problem. The constrained optimization problem is transformed to an unconstrained one using a dynamic penalty function algorithm. Thus, the commonly used backpropagation algorithm can be used to train the neural network. With the physics constraints, the required training data size can be reduced, and the overfitting problem can be mitigated. This paper demonstrates the application of the PgMDN using a numerical example, an engineering problem for fatigue stress-life curve estimation, an engineering problem for natural frequency prediction of bridges, and an engineering problem for fatigue life prediction of corroded steel reinforcing bars. Some discussions are given to illustrate the effectiveness of incorporating the physics knowledge when data are sparse, the improvement of the dynamic penalty function method compared with the static method, and the benefits achieved from the distribution mixture compared with a single Gaussian distribution.

Original languageEnglish (US)
Article number108823
JournalReliability Engineering and System Safety
StatePublished - Dec 2022


  • Constrained optimization
  • Neural network
  • Physics-guided machine learning
  • Probabilistic
  • Uncertainty quantification

ASJC Scopus subject areas

  • Safety, Risk, Reliability and Quality
  • Industrial and Manufacturing Engineering


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