PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain

Abstract

Recently, the advent of deep learning has spurred interest in the development of physics-informed neural networks (PINN) for efficiently solving partial differential equations (PDEs), particularly in a parametric setting. Among all different classes of deep neural networks, the convolutional neural network (CNN) has attracted increasing attention in the scientific machine learning community, since the parameter-sharing feature in CNN enables efficient learning for problems with large-scale spatiotemporal fields. However, one of the biggest challenges is that CNN only can handle regular geometries with image-like format (i.e., rectangular domains with uniform grids). In this paper, we propose a novel physics-constrained CNN learning architecture, aiming to learn solutions of parametric PDEs on irregular domains without any labeled data. In order to leverage powerful classic CNN backbones, elliptic coordinate mapping is introduced to enable coordinate transforms between the irregular physical domain and regular reference domain. The proposed method has been assessed by solving a number of PDEs on irregular domains, including heat equations and steady Navier-Stokes equations with parameterized boundary conditions and varying geometries. Moreover, the proposed method has also been compared against the state-of-the-art PINN with fully-connected neural network (FC-NN) formulation. The numerical results demonstrate the effectiveness of the proposed approach and exhibit notable superiority over the FC-NN based PINN in terms of efficiency and accuracy.

57 pages, 26 figures

References in corpus (15)

• NSFnets (Navier-Stokes Flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations
• B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data
• Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data
• hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition
• Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks
• Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures
• Deep Learning of Subsurface Flow via Theory-guided Neural Network
• Variational Physics-Informed Neural Networks For Solving Partial Differential Equations
• Understanding and mitigating gradient pathologies in physics-informed neural networks
• Embedding Hard Physical Constraints in Neural Network Coarse-Graining of 3D Turbulence
• Turbulence Enrichment using Physics-informed Generative Adversarial Networks
• VarNet: Variational Neural Networks for the Solution of Partial Differential Equations
• Three-dimensional convolutional neural network (3D-CNN) for heterogeneous material homogenization
• Learning and Meta-Learning of Stochastic Advection-Diffusion-Reaction Systems from Sparse Measurements
• TIME: A Transparent, Interpretable, Model-Adaptive and Explainable Neural Network for Dynamic Physical Processes

Cited by in corpus (67)

• Deep learning in nano-photonics: inverse design and beyond
• CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method
• Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations
• Physics-informed graph neural Galerkin networks: A unified framework for solving PDE-governed forward and inverse problems
• PhyCRNet: Physics-informed Convolutional-Recurrent Network for Solving Spatiotemporal PDEs
• Uncovering near-wall blood flow from sparse data with physics-informed neural networks
• Transfer learning based physics-informed neural networks for solving inverse problems in engineering structures under different loading scenarios
• Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels
• Super-Resolution Analysis via Machine Learning: A Survey for Fluid Flows
• Deep Learning in Deterministic Computational Mechanics
• Physics-Informed Neural Operator for Learning Partial Differential Equations
• Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries
• Convolutional neural networks for fluid flow analysis: toward effective metamodeling and low-dimensionalization
• Graph neural networks for laminar flow prediction around random 2D shapes
• Learning in Sinusoidal Spaces with Physics-Informed Neural Networks
• Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous media
• Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• Physics-informed neural networks (PINNs) for fluid mechanics: A review
• Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics
• Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks
• $\Delta$-PINNs: physics-informed neural networks on complex geometries
• MAgNET: A Graph U-Net Architecture for Mesh-Based Simulations
• Adaptive deep density approximation for Fokker-Planck equations
• Nonlinear proper orthogonal decomposition for convection-dominated flows
• An unsupervised latent/output physics-informed convolutional-LSTM network for solving partial differential equations using peridynamic differential operator
• ViscoelasticNet: A physics informed neural network framework for stress discovery and model selection
• Finite Volume Graph Network(FVGN): Predicting unsteady incompressible fluid dynamics with finite volume informed neural network
• Investigating and Mitigating Failure Modes in Physics-informed Neural Networks (PINNs)
• Quantum Convolutional Neural Networks for Multi-Channel Supervised Learning
• Adaptive physics-informed neural operator for coarse-grained non-equilibrium flows
• Variational operator learning: A unified paradigm marrying training neural operators and solving partial differential equations
• Physics-informed MeshGraphNets (PI-MGNs): Neural finite element solvers for non-stationary and nonlinear simulations on arbitrary meshes
• Multi-scale Time-stepping of Partial Differential Equations with Transformers
• Deep learning for surrogate modelling of 2D mantle convection
• Reconstructing three-dimensional bluff body wake from sectional flow fields with convolutional neural networks
• Hard Encoding of Physics for Learning Spatiotemporal Dynamics
• Adapting Physics-Informed Neural Networks to Improve ODE Optimization in Mosquito Population Dynamics
• Modelling parametric uncertainty in PDEs models via Physics-Informed Neural Networks
• Deep learning for nano-photonic materials -- The solution to everything!?
• An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Study
• A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs
• Physics Informed Neural Networks for Modeling of 3D Flow-Thermal Problems with Sparse Domain Data
• Physics-informed Convolutional Neural Networks for Temperature Field Prediction of Heat Source Layout without Labeled Data
• Meshfree Variational Physics Informed Neural Networks (MF-VPINN): an adaptive training strategy
• Finite-difference-informed graph network for solving steady-state incompressible flows on block-structured grids
• Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials
• LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry
• Multi-Fidelity Machine Learning Applied to Steady Fluid Flows
• Stochastic Modeling of Inhomogeneities in the Aortic Wall and Uncertainty Quantification using a Bayesian Encoder-Decoder Surrogate
• Discovering Physics-Informed Neural Networks Model for Solving Partial Differential Equations through Evolutionary Computation
• Convolutional Autoencoders, Clustering and POD for Low-dimensional Parametrization of Navier-Stokes Equations
• Approximating Families of Sharp Solutions to Fisher's Equation with Physics-Informed Neural Networks
• Preconditioned FEM-based Neural Networks for Solving Incompressible Fluid Flows and Related Inverse Problems
• Physics-Guided Discovery of Highly Nonlinear Parametric Partial Differential Equations
• Physics-informed Partitioned Coupled Neural Operator for Complex Networks
• A composable autoencoder-based iterative algorithm for accelerating numerical simulations
• Physics-Informed Neural Networks for Solving Multiscale Mode-Resolved Phonon Boltzmann Transport Equation
• Learnable-Differentiable Finite Volume Solver for Accelerated Simulation of Flows
• Studying First Passage Problems using Neural Networks: A Case Study in the Slit-Well Microfluidic Device
• Neural Networks with Inputs Based on Domain of Dependence and A Converging Sequence for Solving Conservation Laws, Part I: 1D Riemann Problems
• Physics-informed Dyna-Style Model-Based Deep Reinforcement Learning for Dynamic Control
• Physics-informed Shadowgraph Network: An End-to-end Density Field Reconstruction Method
• Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: an application to rubber calendering process
• Unfitted finite element interpolated neural networks
• CFDagent: A Language-Guided, Zero-Shot Multi-Agent System for Complex Flow Simulation
• SlotPi: Physics-informed Object-centric Reasoning Models