Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data

By Luning Sun, Han Gao, Shaowu Pan, Jian-Xun Wang. 2019

1,046 citations recorded by the source corpus.

Abstract 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 computational prohibitive for most real-time applications (e.g., clinical diagnosis and surgery planning) and many-query analyses (e.g., optimization design and uncertainty quantification). 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. Unfortunately, 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. In this paper, we provide a physics-constrained DL approach for surrogate modeling of fluid flows without relying on any simulation data. Specifically, a structured deep neural network (DNN) architecture is devised to enforce the initial and boundary conditions, and the governing partial differential equations (i.e., Navier–Stokes equations) are incorporated into the loss of the DNN to drive the training. Numerical experiments are conducted on a number of internal flows relevant to hemodynamics applications, and the forward propagation of uncertainties in fluid properties and domain geometry is studied as well. The results show excellent agreement on the flow field and forward-propagated uncertainties between the DL surrogate approximations and the first-principle numerical simulations.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

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Citations in the corpus

Physics informed deep learning for computational elastodynamics without labeled data (2020, background): “This type of I/BC imposition is also called “soft” enforcement [17, 54] because the initial and boundary values may not be enforced accurately, due to the pathology issue of the gradient.”
Understanding and mitigating gradient pathologies in physics-informed neural networks (2020, background): “This is the main reason why the majority of recent works focused on applying deep learning tools to modeling and simulating physical systems governed by PDEs, still relies on generic fully connected architectures [12, 24, 23, 43, 26].”
Enforcing Analytic Constraints in Neural Networks Emulating Physical Systems. (2019, background): “Despite the need for physically informed NNs for complex physical systems [2–5], enforcing hard constraints [6] has been limited to physical systems governed by specific equations, such as advection equations [7–9], Reynolds-averaged Navier-Stokes equations [10,11], boundary conditions of idealized ”
PhyGeoNet: Physics-informed geometry-adaptive convolutional neural networks for solving parameterized steady-state PDEs on irregular domain (2020, methodology): “[4], the soft BC enforcement has several drawbacks and”
Flow over an espresso cup: inferring 3-D velocity and pressure fields from tomographic background oriented Schlieren via physics-informed neural networks (2021, methodology): “The PINN framework (Raissi et al. 2019) has also been applied to perform complex flow simulations (Sun et al. 2020; Jin et al. 2021).”
On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks (2020, background): “…and their modern re-incarnation via the framework of physics-informed neural networks (PINNs) [4], the use of neural networks to represent PDE solutions has undergone rapid growth, both in terms of theory [5, 6, 7, 8] and diverse applications in computational science and engineering [9, 10, 11].”
Physics-constrained bayesian neural network for fluid flow reconstruction with sparse and noisy data (2020, background): “However, a previous work [28] in the context of surrogate modeling has shown that flow solutions of the Navier-stokes equations can be obtained from physics-constrained deep learning even without any labeled data if the boundary conditions are imposed properly.”
Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems (2021, methodology): “Although a distance-based particular solution can be introduced to strictly impose IC/BCs on a few simple 2-D domains using either specifically designed algebraic expressions or low-capacity neural networks [8, 18], it fails to show the effectiveness on complex geometries for real-world applications”
Transformer for Partial Differential Equations' Operator Learning (2022, methodology): “Physics-informed methods [17, 62, 69, 72] exploit the PDE supervision to approximate the solution (usually parameterized by a neural network) in an aggregated space-time domain.”
Physics-Informed Neural Network for Modelling the Thermochemical Curing Process of Composite-Tool Systems During Manufacture (2020, methodology): “Due to these advantages, many variations of the PINN approach have been successfully used to solve PDEs in a wide range of engineering applications [24, 29–37].”
Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next (2022, methodology): “Three flow examples in hemodynamics applications are presented in Sun et al (2020a), for addressing either stenotic flow and aneurysmal flow, with standardized vessel geometries and varying viscosity.”