Hybrid FEM-NN models: combining artificial neural networks with the finite element method

DOI10.1016/j.jcp.2021.110651OpenAlexW3120780149MaRDI QIDQ2133536

Simon W. Funke, Sebastian K. Mitusch, Miroslav Kuchta

Publication date: 29 April 2022

Published in: Journal of Computational Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/2101.00962

zbMATH Keywords

finite element methodpartial differential equationsmachine learningdata-driven scientific computinglearning unknown physics

Mathematics Subject Classification ID

Mathematical programming (90Cxx) Artificial intelligence (68Txx) Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems (65Mxx)

Related Items

Machine Learning Surrogate Modeling for Meshless Methods: Leveraging Universal Approximation, Integrated finite element neural network (I-FENN) for non-local continuum damage mechanics, Finite element interpolated neural networks for solving forward and inverse problems, A conservative hybrid deep learning method for Maxwell-Ampère-Nernst-Planck equations, DNN-MG: a hybrid neural network/finite element method with applications to 3D simulations of the Navier-Stokes equations, Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation

Uses Software

• L-BFGS-B
• L-BFGS
• DOLFIN
• tn
• LBFGS-B
• SciPy
• dcc
• UFL
• Python
• PyTorch
• PDE-Net
• dolfin-adjoint
• XPINNs
• ADCME
• DeepONet
• PINNsNTK
• hp-VPINNs

Cites Work

• Unnamed Item
• Discovering governing equations from data by sparse identification of nonlinear dynamical systems
• Learning constitutive relations from indirect observations using deep neural networks
• On the limited memory BFGS method for large scale optimization
• Singly diagonally implicit Runge-Kutta methods with an explicit first stage
• A numerical study of inverse heat conduction problems
• When and why PINNs fail to train: a neural tangent kernel perspective
• Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems
• PDE-Net 2.0: learning PDEs from data with a numeric-symbolic hybrid deep network
• Adaptive activation functions accelerate convergence in deep and physics-informed neural networks
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• DOLFIN
• Newton-Type Minimization via the Lanczos Method
• The Art of Differentiating Computer Programs
• Algorithm 778: L-BFGS-B
• Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations
• Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks
• Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations
• Automated Adjoints of Coupled PDE-ODE Systems
• Turbulence Modeling in the Age of Data
• Unified form language