ConvPDE-UQ: convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains

DOI10.1016/j.jcp.2019.05.026zbMath1457.65245OpenAlexW2946866513WikidataQ127858349 ScholiaQ127858349MaRDI QIDQ2222287

Karthik Ramani, Guang Lin, Nick Winovich

Publication date: 26 January 2021

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

Full work available at URL: https://doi.org/10.1016/j.jcp.2019.05.026

zbMATH Keywords

partial differential equationsconfidence intervalmachine learninguncertainty quantificationdeep learningconvolutional encoder-decoder networks

Mathematics Subject Classification ID

Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs (65N75) Artificial neural networks and deep learning (68T07) Learning and adaptive systems in artificial intelligence (68T05) Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation (35J05) Neural nets applied to problems in time-dependent statistical mechanics (82C32) Fundamental solutions, Green's function methods, etc. for boundary value problems involving PDEs (65N80)

Related Items (21)

ConvPDE-UQ ⋮ A modified batch intrinsic plasticity method for pre-training the random coefficients of extreme learning machines ⋮ Learning time-dependent PDEs with a linear and nonlinear separate convolutional neural network ⋮ On computing the hyperparameter of extreme learning machines: algorithm and application to computational PDEs, and comparison with classical and high-order finite elements ⋮ A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems ⋮ Fully probabilistic deep models for forward and inverse problems in parametric PDEs ⋮ Accuracy and architecture studies of residual neural network method for ordinary differential equations ⋮ Addressing discontinuous root-finding for subsequent differentiability in machine learning, inverse problems, and control ⋮ Label-free learning of elliptic partial differential equation solvers with generalizability across boundary value problems ⋮ A deep neural network-based method for solving obstacle problems ⋮ Numerical computation of partial differential equations by hidden-layer concatenated extreme learning machine ⋮ Bi-fidelity modeling of uncertain and partially unknown systems using DeepONets ⋮ Energy-dissipative evolutionary deep operator neural networks ⋮ Physics-informed ConvNet: learning physical field from a shallow neural network ⋮ DeepXDE: A Deep Learning Library for Solving Differential Equations ⋮ Output-based adaptive aerodynamic simulations using convolutional neural networks ⋮ Local extreme learning machines and domain decomposition for solving linear and nonlinear partial differential equations ⋮ Deep neural networks for waves assisted by the Wiener–Hopf method ⋮ PPINN: parareal physics-informed neural network for time-dependent PDEs ⋮ The Random Feature Model for Input-Output Maps between Banach Spaces ⋮ PhyCRNet: physics-informed convolutional-recurrent network for solving spatiotemporal PDEs

Uses Software

• FIAT
• DOLFIN
• FEniCS
• SyFi
• Unicorn
• UFL
• UFC
• FFC
• Adam
• U-Net
• DGM

Cites Work

• Unnamed Item
• Bayesian solution uncertainty quantification for differential equations
• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• Automated solution of differential equations by the finite element method. The FEniCS book
• Numerical solution for high order differential equations using a hybrid neural network-optimization method
• Multilayer feedforward networks are universal approximators
• Bayesian deep convolutional encoder-decoder networks for surrogate modeling and uncertainty quantification
• Sparsity-promoting elastic net method with rotations for high-dimensional nonlinear inverse problem
• Deep UQ: learning deep neural network surrogate models for high dimensional uncertainty quantification
• DGM: a deep learning algorithm for solving partial differential equations
• Optimal observations-based retrieval of topography in 2D shallow water equations using PC-EnKF
• Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• Unicorn: parallel adaptive finite element simulation of turbulent flow and fluid-structure interaction for deforming domains and complex geometry
• Automating the finite element method
• Neural algorithm for solving differential equations
• Framework for Massively Parallel Adaptive Finite Element Computational Fluid Dynamics on Tetrahedral Meshes
• On the efficiency of symbolic computations combined with code generation for finite element methods
• Optimizations for quadrature representations of finite element tensors through automated code generation
• DOLFIN
• Efficient Assembly of $H(\mathrm{div})$ and $H(\mathrm{curl})$ Conforming Finite Elements
• Efficient compilation of a class of variational forms
• Automated Code Generation for Discontinuous Galerkin Methods
• On the unimodality of the likelihood for the Cauchy distribution
• Neural‐network‐based approximations for solving partial differential equations
• Calibration of reduced-order model for a coupled Burgers equations based on PC-EnKF
• Geometric Optimization of the Evaluation of Finite Element Matrices
• Algorithm 839
• Optimizing the Evaluation of Finite Element Matrices
• Topological Optimization of the Evaluation of Finite Element Matrices
• Unified form language
• A table of integrals of the Error functions
• Approximation by superpositions of a sigmoidal function

This page was built for publication: ConvPDE-UQ: convolutional neural networks with quantified uncertainty for heterogeneous elliptic partial differential equations on varied domains