A Framework for Physics-Informed Deep Learning Over Freeform Domains

被引:4
作者
Mezzadri, Francesco [1 ]
Gasick, Joshua [2 ]
Qian, Xiaoping [2 ]
机构
[1] Univ Modena & Reggio Emilia, Dept Engn Enzo Ferrari, Via P Vivarelli 10-1,Bldg 26, I-41125 Modena, Italy
[2] Univ Wisconsin, Dept Mech Engn, 1513 Univ Ave, Madison, WI 53706 USA
基金
美国国家科学基金会;
关键词
Physics -informed deep learning; Neural network; NURBS; Computer -aided design; Partial differential equations; OPTIMIZATION; NETWORKS;
D O I
10.1016/j.cad.2023.103520
中图分类号
TP31 [计算机软件];
学科分类号
081202 ; 0835 ;
摘要
Deep learning is a popular approach for approximating the solutions to partial differential equations (PDEs) over different material parameters and boundary conditions. However, no work has yet been reported on learning PDE solutions over changing shapes of the underlying domain. We present a framework to train neural networks (NN) and physics-informed neural networks (PINNs) to learn the solutions to PDEs defined over varying freeform domains. This is made possible through our adoption of a parametric non-uniform rational B-Spline (NURBS) representation of the underlying physical shape. Distinct physical domains are mapped to a common parametric space via NURBS parameterization. In our approach, we formulate NNs and PINNs that learn the solutions to PDEs as a function of the shape of the domain itself through shape parameters. Under this formulation, the loss function is based on an unchanging parametric domain that maps to a variable physical domain. Residual computation in PINNs is made possible through the Jacobian of the mapping. Numerical results show that our networks can be trained to predict the solutions to a PDE defined over an entire set of shapes. We focus on the linear elasticity PDE and show how we can build a surrogate model that is able to predict displacements and stresses over a variety of freeform domains. To assess the efficacy of all networks in this work, data efficiency, network accuracy, and the capacity of networks to extrapolate are considered and compared between NNs and PINNs. The comparison includes cases where little training data is available. Transfer learning and applications to shape optimization are analyzed as well. A selection of the used codes and datasets is provided at https://github.com/fmezzadri/shape_parameterized.git. (c) 2023 Elsevier Ltd. All rights reserved.
引用
收藏
页数:23
相关论文
共 31 条
  • [1] Abadi M, 2016, PROCEEDINGS OF OSDI'16: 12TH USENIX SYMPOSIUM ON OPERATING SYSTEMS DESIGN AND IMPLEMENTATION, P265
  • [2] Baydin AG, 2018, J MACH LEARN RES, V18
  • [3] Bergstra J., 2011, ADV NEURAL INFORM PR, P2546
  • [4] WaveY-Net: Physics-Augmented Deep Learning for High-Speed Electromagnetic Simulation and Optimization
    Chen, Mingkun
    Lupoiu, Robert
    Mao, Chenkai
    Huang, Der-Han
    Jiang, Jiaqi
    Lalanne, Philippe
    Fan, Jonathan A.
    [J]. HIGH CONTRAST METASTRUCTURES XI, 2022, 12011
  • [5] Chen Z, 2018, PR MACH LEARN RES, V80
  • [6] Neural architecture search via standard machine learning methodologies
    Franchini, Giorgia
    Ruggiero, Valeria
    Porta, Federica
    Zanni, Luca
    [J]. MATHEMATICS IN ENGINEERING, 2023, 5 (01): : 1 - 21
  • [7] Fuchi KW, 2020, PROCEEDINGS OF THE ASME 2020 CONFERENCE ON SMART MATERIALS, ADAPTIVE STRUCTURES AND INTELLIGENT SYSTEMS (SMASIS2020)
  • [8] Gurney K., 1997, An introduction to neural networks
  • [9] Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training
    Haghighat, Ehsan
    Amini, Danial
    Juanes, Ruben
    [J]. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 2022, 397
  • [10] A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics
    Haghighat, Ehsan
    Raissi, Maziar
    Moure, Adrian
    Gomez, Hector
    Juanes, Ruben
    [J]. COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 2021, 379