Physicsconstrained deep learning for highdimensional surrogate modeling and uncertainty quantification without labeled data
Abstract
Surrogate modeling and uncertainty quantification tasks for PDE systems are most often considered as supervised learning problems where input and output data pairs are used for training. The construction of such emulators is by definition a small data problem which poses challenges to deep learning approaches that have been developed to operate in the big data regime. Even in cases where such models have been shown to have good predictive capability in high dimensions, they fail to address constraints in the data implied by the PDE model. This paper provides a methodology that incorporates the governing equations of the physical model in the loss/likelihood functions. The resulting physicsconstrained, deep learning models are trained without any labeled data (e.g. employing only input data) and provide comparable predictive responses with datadriven models while obeying the constraints of the problem at hand. This work employs a convolutional encoderdecoder neural network approach as well as a conditional flowbased generative model for the solution of PDEs, surrogate model construction, and uncertainty quantification tasks. The methodology is posed as a minimization problem of the reverse KullbackLeibler (KL) divergence between the model predictive density and the reference conditional density, where the later is defined as the BoltzmannGibbs distribution at a given inverse temperature with the underlying potential relating to the PDE system of interest. The generalization capability of these models to outofdistribution input is considered. Quantification and interpretation of the predictive uncertainty is provided for a number of problems.
 Publication:

Journal of Computational Physics
 Pub Date:
 October 2019
 DOI:
 10.1016/j.jcp.2019.05.024
 arXiv:
 arXiv:1901.06314
 Bibcode:
 2019JCoPh.394...56Z
 Keywords:

 Physicsconstrained;
 Normalizing flow;
 Conditional generative model;
 Reverse KL divergence;
 Surrogate modeling;
 Uncertainty quantification;
 Physics  Computational Physics;
 Computer Science  Computer Vision and Pattern Recognition;
 Computer Science  Machine Learning;
 Statistics  Machine Learning
 EPrint:
 51 pages, 18 figures, submitted to Journal of Computational Physics