Solving parametric PDE problems with artificial neural networks

DOI10.1017/S0956792520000182zbMath1501.65154arXiv1707.03351MaRDI QIDQ5014839

Yuehaw Khoo, Lexing Ying, Jian-feng Lu

Publication date: 8 December 2021

Published in: European Journal of Applied Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1707.03351

zbMATH Keywords

uncertainty quantification neural-network parametric PDE

Mathematics Subject Classification ID

Artificial neural networks and deep learning (68T07) PDEs with randomness, stochastic partial differential equations (35R60) Numerical methods for partial differential equations, boundary value problems (65N99)

Related Items

Stationary Density Estimation of Itô Diffusions Using Deep Learning, Deep Adaptive Basis Galerkin Method for High-Dimensional Evolution Equations With Oscillatory Solutions, A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations, A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder, Normalizing field flows: solving forward and inverse stochastic differential equations using physics-informed flow models, Computing the invariant distribution of randomly perturbed dynamical systems using deep learning, Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms, Learning deep implicit Fourier neural operators (IFNOs) with applications to heterogeneous material modeling, Neural Parametric Fokker--Planck Equation, A finite element based deep learning solver for parametric PDEs, DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method, A survey of unsupervised learning methods for high-dimensional uncertainty quantification in black-box-type problems, Nonlocal kernel network (NKN): a stable and resolution-independent deep neural network, Deep global model reduction learning in porous media flow simulation, Simultaneous neural network approximation for smooth functions, Probabilistic partition of unity networks for high‐dimensional regression problems, Deep capsule encoder–decoder network for surrogate modeling and uncertainty quantification, Solving nonconvex energy minimization problems in martensitic phase transitions with a mesh-free deep learning approach, Data-driven vortex solitons and parameter discovery of 2D generalized nonlinear Schrödinger equations with a \(\mathcal{PT}\)-symmetric optical lattice, Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation, Friedrichs Learning: Weak Solutions of Partial Differential Equations via Deep Learning, Neural Galerkin schemes with active learning for high-dimensional evolution equations, A mathematical perspective of machine learning, ADLGM: an efficient adaptive sampling deep learning Galerkin method, A conservative hybrid deep learning method for Maxwell-Ampère-Nernst-Planck equations, Structure-preserving recurrent neural networks for a class of Birkhoffian systems, A Review of Data‐Driven Discovery for Dynamic Systems, Deep convolutional Ritz method: parametric PDE surrogates without labeled data, A Variational Neural Network Approach for Glacier Modelling with Nonlinear Rheology, Bi-Orthogonal fPINN: A Physics-Informed Neural Network Method for Solving Time-Dependent Stochastic Fractional PDEs, Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing, opPINN: physics-informed neural network with operator learning to approximate solutions to the Fokker-Planck-Landau equation, Energy-dissipative evolutionary deep operator neural networks, Dual order-reduced Gaussian process emulators (DORGP) for quantifying high-dimensional uncertain crack growth using limited and noisy data, A Reduced Order Schwarz Method for Nonlinear Multiscale Elliptic Equations Based on Two-Layer Neural Networks, Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent?, Solving inverse problems with deep learning, On mathematical modeling in image reconstruction and beyond, Kernel methods are competitive for operator learning, Greedy training algorithms for neural networks and applications to PDEs, Solving many-electron Schrödinger equation using deep neural networks, Networks for nonlinear diffusion problems in imaging, General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning, Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations, A Multiscale Neural Network Based on Hierarchical Matrices, Solving inverse problems using data-driven models, A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations, A multiscale neural network based on hierarchical nested bases, Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations, Monte Carlo fPINNs: deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations, Learning phase field mean curvature flows with neural networks, Efficient Construction of Tensor Ring Representations from Sampling, Efficient coupled deep neural networks for the time-dependent coupled Stokes-Darcy problems, Multilevel Fine-Tuning: Closing Generalization Gaps in Approximation of Solution Maps under a Limited Budget for Training Data, CAS4DL: Christoffel adaptive sampling for function approximation via deep learning, Committor functions via tensor networks, Connections between deep learning and partial differential equations, Solving parametric partial differential equations with deep rectified quadratic unit neural networks, B-DeepONet: an enhanced Bayesian deeponet for solving noisy parametric PDEs using accelerated replica exchange SGLD, DNN expression rate analysis of high-dimensional PDEs: application to option pricing, A theoretical analysis of deep neural networks and parametric PDEs

Uses Software

Cites Work