Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries
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 where 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. is the external body force. We denote the x and y components of respectively by and . 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
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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