Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries

doi:10.1016/j.jcp.2022.111510

Journal of Computational Physics

Volume 468, 1 November 2022, 111510

https://doi.org/10.1016/j.jcp.2022.111510 Get rights and content

Highlights

•
PIPN is a novel physics-informed deep learning framework.
•
PIPN solves forward and inverse time-independent problems on several irregular domains by training only once.
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PIPN overcomes the shortcoming of regular PINNs that need to be retrained for any single domain with a new geometry.
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Applications of PIPN are shown for incompressible flows and thermal fields.

Abstract

We present a novel physics-informed deep learning framework for solving steady-state incompressible flow on multiple sets of irregular geometries by incorporating two main elements: using a point-cloud based neural network to capture geometric features of computational domains, and using the mean squared residuals of the governing partial differential equations, boundary conditions, and sparse observations as the loss function of the network to capture the physics. While the solution of the continuity and Navier-Stokes equations is a function of the geometry of the computational domain, current versions of physics-informed neural networks have no mechanism to express this functionally in their outputs, and thus are restricted to obtain the solutions only for one computational domain with each training procedure. Using the proposed framework, three new facilities become available. First, the governing equations are solvable on a set of computational domains containing irregular geometries with high variations with respect to each other but requiring training only once. Second, after training the introduced framework on the set, it is now able to predict the solutions on domains with unseen geometries from seen and unseen categories as well. The former and the latter both lead to savings in computational costs. Finally, all the advantages of the point-cloud based neural network for irregular geometries, already used for supervised learning, are transferred to the proposed physics-informed framework. The effectiveness of our framework is shown through the method of manufactured solutions and thermally-driven flow for forward and inverse problems.

Section snippets

Introduction and motivation

To design a neural network for predicting the solution of coupled partial differential equations (PDEs) governing physical phenomena of interest, generally speaking, there are two common deep learning methodologies: “supervised models” and “physics-informed models.” In supervised models (see e.g., Refs. [59], [73], [65], [33], [35]), a neural network is trained over a set of labeled data as pairs of input and outputs of the network. The loss function in this approach is usually the mean squared

Governing equations of interest

The governing equations of conservation of mass, momentum, and energy for an incompressible steady flow of a Newtonian fluid in two dimensional spaces are respectively given by $\nabla \cdot u = 0 in V,$ $ρ (u \cdot \nabla) u - μ Δ u + \nabla p = f in V,$ $ρ (u \cdot \nabla) T - \frac{κ}{c_{p}} Δ T = 0 in V,$ where $u$ is the velocity vector with components u and v in the x and y directions, respectively. p indicates the pressure of the fluid and T shows the temperature of the fluid. $f$ is the external body force. We denote the x and y components of $f$ respectively by $f^{x}$ and $f^{y}$ . The

Methodology

In this subsection, we express the key ideas of PIPN in a general and abstract manner. After that, details of PIPN are explained in Sect. 3.2 and Sect. 3.3. Fundamentally, the PIPN skeleton is the combination of two major components: a neural network designed based on the PointNet [49] mechanism, and an associated loss function accorded to the physics-informed models.

As discussed in Sect. 1, the goal is to first train PIPN for obtaining the solution of PDEs (Eqs. (1)–(3)) over a set of

Method of manufactured solutions in non-trivial geometries

The concern of this subsection is to assess the efficiency of PIPN for a forward problem. We set up a machine-learning experiment by means of the method of manufactured solutions. The method of manufactured solutions is a widely-used scheme for the purpose of code verification (see e.g., Refs. [61], [68], [4]) and algorithm examination (see e.g., Refs. [10], [69], [67]). To utilize this strategy, we consider a divergence free velocity field given by $u (x, y) = \cos (x) \sin (y),$ $v (x, y) = - \cos (y) \sin (x),$

Summary and future studies

Since the late 2018s, artificial neural networks based on physics-informed models have become popular among the community of computational mathematics and mechanics, mainly due to their abilities to solve PDEs of interest without labeled data for forward problems and with sparse data (observations) for inverse problems. However, recent physics-informed neural networks have been limited to only solve a set of PDEs on a single computational domain with a fixed geometry, fundamentally due to the

CRediT authorship contribution statement

Ali Kashefi: Conceptualization, Methodology, Software, Visualization, Writing – original draft, Writing – review & editing. Tapan Mukerji: Conceptualization, Funding acquisition, Project administration, Writing – review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

We acknowledge funding by Shell-Stanford Collaborative Project on Digital Rock Physics 2.0 for supporting this study. We would also like to thank Steve Graham, the Dean of the School of Earth, Energy, and Environmental Sciences at Stanford University for funding. Additionally, we wish to thank the Stanford Research Computing Center for providing computational resources for this research project. Moreover, A. Kashefi would like to thank Davis Rempe in the department of Computer Science at

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View full text

Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries

Highlights

Abstract

Section snippets

Introduction and motivation

Governing equations of interest

Methodology

Method of manufactured solutions in non-trivial geometries

Summary and future studies

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgement

J. Pet. Sci. Eng.

J. Comput. Phys.

Comput. Fluids

J. Comput. Phys.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

Comput. Methods Appl. Mech. Eng.

J. Comput. Phys.

Comput. Methods Appl. Mech. Eng.

J. Comput. Phys.

J. Comput. Phys.

Int. J. Heat Mass Transf.

J. Comput. Phys.

J. Comput. Phys.

Comput. Methods Appl. Mech. Eng.

J. Comput. Phys.

Comput. Methods Appl. Mech. Eng.

J. Comput. Phys.

Comput. Methods Appl. Mech. Eng.

J. Comput. Phys.

J. Comput. Phys.

Comput. Fluids

J. Comput. Phys.

Comput. Fluids

Int. J. Heat Mass Transf.

Appl. Math. Lett.

J. Comput. Phys.

TensorFlow: large-scale machine learning on heterogeneous systems

Prediction of aerodynamic flow fields using convolutional neural networks

Comput. Mech.

Physics-informed neural networks (PINNs) for fluid mechanics: a review

Acta Mech. Sin.

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J. Heat Transf.

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Numerical solution of the Navier-Stokes equations

Math. Comput.

Scientific machine learning through physics-informed neural networks: where we are and what's next

Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels

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