Geometry-Informed Neural Operator for Large-Scale 3D PDEs

We propose the geometry-informed neural operator (GINO), a highly efficient approach to learning the solution operator of large-scale partial differential equations with varying geometries. GINO uses a signed distance function and point-cloud representations of the input shape and neural operators based on graph and Fourier architectures to learn the solution operator. The graph neural operator handles irregular grids and transforms them into and from regular latent grids on which Fourier neural operator can be efficiently applied. GINO is discretization-convergent, meaning the trained model can be applied to arbitrary discretization of the continuous domain and it converges to the continuum operator as the discretization is refined. To empirically validate the performance of our method on large-scale simulation, we generate the industry-standard aerodynamics dataset of 3D vehicle geometries with Reynolds numbers as high as five million. For this large-scale 3D fluid simulation, numerical methods are expensive to compute surface pressure. We successfully trained GINO to predict the pressure on car surfaces using only five hundred data points. The cost-accuracy experiments show a $26,000 \times$ speed-up compared to optimized GPU-based computational fluid dynamics (CFD) simulators on computing the drag coefficient. When tested on new combinations of geometries and boundary conditions (inlet velocities), GINO obtains a one-fourth reduction in error rate compared to deep neural network approaches.

[1]  Nikola B. Kovachki,et al.  Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs , 2023, ArXiv.

[2]  Yue Yu,et al.  Domain Agnostic Fourier Neural Operators , 2023, ArXiv.

[3]  Hang Su,et al.  GNOT: A General Neural Operator Transformer for Operator Learning , 2023, ICML.

[4]  Suman V. Ravuri,et al.  GraphCast: Learning skillful medium-range global weather forecasting , 2022, ArXiv.

[5]  Yufei Huang,et al.  Non-equispaced Fourier Neural Solvers for PDEs , 2022, ArXiv.

[6]  K. Azizzadenesheli,et al.  Real-time High-resolution CO2 Geological Storage Prediction using Nested Fourier Neural Operators , 2022, Energy & Environmental Science.

[7]  K. Azizzadenesheli,et al.  Accelerating Time-Reversal Imaging with Neural Operators for Real-time Earthquake Locations , 2022, 2210.06636.

[8]  Jayesh K. Gupta,et al.  Clifford Neural Layers for PDE Modeling , 2022, ICLR.

[9]  Daniel Z. Huang,et al.  Fourier Neural Operator with Learned Deformations for PDEs on General Geometries , 2022, J. Mach. Learn. Res..

[10]  A. Farimani,et al.  Transformer for Partial Differential Equations' Operator Learning , 2022, Trans. Mach. Learn. Res..

[11]  K. Azizzadenesheli,et al.  U-NO: U-shaped Neural Operators , 2022, Trans. Mach. Learn. Res..

[12]  K. Azizzadenesheli,et al.  FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators , 2022, ArXiv.

[13]  Jessica B. Hamrick,et al.  Physical Design using Differentiable Learned Simulators , 2022, ArXiv.

[14]  George J. Pappas,et al.  Learning Operators with Coupled Attention , 2022, J. Mach. Learn. Res..

[15]  Aaron C. Walden,et al.  Application of a Detached Eddy Simulation Approach with Finite-Rate Chemistry to Mars-Relevant Retropropulsion Operating Environments , 2022, AIAA SCITECH 2022 Forum.

[16]  Nikola B. Kovachki,et al.  Physics-Informed Neural Operator for Learning Partial Differential Equations , 2021, ACM / IMS Journal of Data Science.

[17]  P. Fua,et al.  DEBOSH: Deep Bayesian Shape Optimization , 2021, ArXiv.

[18]  Nikola B. Kovachki,et al.  Neural Operator: Learning Maps Between Function Spaces , 2021, ArXiv.

[19]  Kamyar Azizzadenesheli,et al.  Seismic wave propagation and inversion with Neural Operators , 2021, The Seismic Record.

[20]  Baskar Ganapathysubramanian,et al.  Distributed Multigrid Neural Solvers on Megavoxel Domains , 2021, SC21: International Conference for High Performance Computing, Networking, Storage and Analysis.

[21]  R. Klein,et al.  Teaching the incompressible Navier–Stokes equations to fast neural surrogate models in three dimensions , 2020, Physics of Fluids.

[22]  Nikola B. Kovachki,et al.  Fourier Neural Operator for Parametric Partial Differential Equations , 2020, ICLR.

[23]  T. Pfaff,et al.  Learning Mesh-Based Simulation with Graph Networks , 2020, ICLR.

[24]  Gordon Wetzstein,et al.  Implicit Neural Representations with Periodic Activation Functions , 2020, NeurIPS.

[25]  Nikola B. Kovachki,et al.  Multipole Graph Neural Operator for Parametric Partial Differential Equations , 2020, NeurIPS.

[26]  Pascal Fua,et al.  MeshSDF: Differentiable Iso-Surface Extraction , 2020, NeurIPS.

[27]  Nikola B. Kovachki,et al.  Model Reduction and Neural Networks for Parametric PDEs , 2020, The SMAI journal of computational mathematics.

[28]  Kamyar Azizzadenesheli,et al.  Neural Operator: Graph Kernel Network for Partial Differential Equations , 2020, ICLR 2020.

[29]  Jure Leskovec,et al.  Learning to Simulate Complex Physics with Graph Networks , 2020, ICML.

[30]  Fred A. Hamprecht,et al.  Accurate and versatile 3D segmentation of plant tissues at cellular resolution , 2020, bioRxiv.

[31]  Leslie Pack Kaelbling,et al.  Graph Element Networks: adaptive, structured computation and memory , 2019, ICML.

[32]  Jan Eric Lenssen,et al.  Fast Graph Representation Learning with PyTorch Geometric , 2019, ArXiv.

[33]  Nobuyuki Umetani,et al.  Learning three-dimensional flow for interactive aerodynamic design , 2018, ACM Trans. Graph..

[34]  Jessica B. Hamrick,et al.  Relational inductive biases, deep learning, and graph networks , 2018, ArXiv.

[35]  Albert Cohen,et al.  Shape Holomorphy of the Stationary Navier-Stokes Equations , 2018, SIAM J. Math. Anal..

[36]  Vladlen Koltun,et al.  Open3D: A Modern Library for 3D Data Processing , 2018, ArXiv.

[37]  Jure Leskovec,et al.  Inductive Representation Learning on Large Graphs , 2017, NIPS.

[38]  Serge J. Belongie,et al.  Arbitrary Style Transfer in Real-Time with Adaptive Instance Normalization , 2017, 2017 IEEE International Conference on Computer Vision (ICCV).

[39]  Razvan Pascanu,et al.  Interaction Networks for Learning about Objects, Relations and Physics , 2016, NIPS.

[40]  Max Welling,et al.  Semi-Supervised Classification with Graph Convolutional Networks , 2016, ICLR.

[41]  Andrea Vedaldi,et al.  Instance Normalization: The Missing Ingredient for Fast Stylization , 2016, ArXiv.

[42]  Leonidas J. Guibas,et al.  ShapeNet: An Information-Rich 3D Model Repository , 2015, ArXiv.

[43]  O. C. Zienkiewicz,et al.  The Finite Element Method for Fluid Dynamics , 2005 .

[44]  F. Menter ZONAL TWO EQUATION k-w TURBULENCE MODELS FOR AERODYNAMIC FLOWS , 1993 .

[45]  Gunther Ramm,et al.  Some salient features of the time - averaged ground vehicle wake , 1984 .

[46]  J. Cooper SINGULAR INTEGRALS AND DIFFERENTIABILITY PROPERTIES OF FUNCTIONS , 1973 .

[47]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[48]  Aleksandar Jemcov,et al.  OpenFOAM: A C++ Library for Complex Physics Simulations , 2007 .

[49]  R. Temam Navier-Stokes Equations: Theory and Numerical Analysis , 1979 .

[50]  H. Whitney Analytic Extensions of Differentiable Functions Defined in Closed Sets , 1934 .