Baking physics knowledge into deep learning

physics-constrained neural network for surrogate fluid modeling without labels

Significance 

Numerical simulations on fluid dynamics problems primarily rely on spatially or/and temporally discretization of the governing equation using polynomials into a finite-dimensional algebraic system. Due to the multi-scale nature of the physics and sensitivity from meshing a complicated geometry, such process can be computationally prohibitive for most real-time applications and many-query analyses. Therefore, developing a cost-effective surrogate model is of great practical significance. Deep learning (DL) has shown new promises for surrogate modeling due to its capability of handling strong nonlinearity and high dimensionality. However, the off-the-shelf DL architectures, success of which heavily relies on the large amount of training data and interpolatory nature of the problem, fail to operate when the data becomes sparse. Worse off, data is often insufficient in most parametric fluid dynamics problems since each data point in the parameter space requires an expensive numerical simulation based on the first principle, e.g., Navier–Stokes equations. To address this, a research group led by Professor Jian-Xun Wang from the University of Notre Dame in the United States developed a new physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data. Their work is currently published in the research journal, Computer Methods in Applied Mechanics and Engineering.

In their approach, the research team devised a structured deep neural network architecture to enforce the initial and boundary conditions. More so, the governing partial differential equations i.e., Navier–Stokes equations, were also incorporated into the loss of the DNN to drive the training. Numerical experiments were then conducted on a number of internal flows relevant to hemodynamics applications. Overall, forward propagation of uncertainties in fluid properties and domain geometry was studied as well.

The authors reported that the obtained comparisons illustrated the excellent agreement between the physics-constrained DNN surrogate models and CFD simulations. In addition, without the use of any labeled data in training, the DNN was able to accurately parameterize the velocity/pressure solutions with varying viscosity and geometries, which can be used to efficiently propagate uncertainties with enormous Monte Carlo simulation samples.

In summary, the objective of the study was to develop a physics-constrained, data-free DNN for surrogate modeling of incompressible flows. Going by the presented results, excellent agreement was obtained on the flow field and forward-propagated uncertainties between the DL surrogate approximations and the first-principles numerical simulations. In a statement to Advances in Engineering, Professor Jian-Xun Wang, the lead author highlighted that their results demonstrated the merit of the proposed method. In fact, their work suggests a great promise in developing DNN for surrogate fluid models without the need for CFD simulation data.

Baking physics knowledge into deep learning: physics-constrained neural network for surrogate fluid modeling without labels - Advances in Engineering

About the author

Luning Sun is a Ph.D. student in the department of aerospace and mechanical engineering, University of Notre Dame. He got his Master degree in Civil Engineering from University of Notre Dame and Bachelor degree from Tianjin University, China. His research interest lies in the scope of integrating physical laws and machine learning techniques.

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About the author

Han Gao is currently a Ph.D. student in the department of aerospace and mechanical engineering, University of Notre Dame. His primary research focuses on PDE-constrained inverse problem, physics-informed machine learning, uncertainty quantification, and scientific computing.

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About the author

Shaowu Pan is a Ph.D. student in the department of aerospace engineering at University of Michigan. His research interest lies in computational turbulence modeling and scientific machine learning.

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About the author

Dr. Jian-Xun Wang is an assistant professor of Aerospace and Mechanical Engineering at the University of Notre Dame. Dr. Wang received a Ph.D. in Aerospace Engineering from Virginia Tech in 2017 and was a postdoctoral scholar at the University of California, Berkeley before he joined Notre Dame in 2018.

His research focuses on developing data-driven/data-augmented computational modeling, which broadly revolves physics-informed machine/deep learning, Bayesian data assimilation, and uncertainty quantification.

Reference

Luning Sun, Han Gao, Shaowu Pan, Jian-Xun Wang. Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Computer Methods in Applied Mechanics and Engineering, issue 361 (2020) 112732.

Go To Computer Methods in Applied Mechanics and Engineering

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