On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
From MaRDI portal
Publication:5162370
DOI10.4208/cicp.OA-2020-0193zbMath1473.65349arXiv2004.01806OpenAlexW3102139197WikidataQ114021249 ScholiaQ114021249MaRDI QIDQ5162370
Yeonjong Shin, Jérôme Darbon, George Em. Karniadakis
Publication date: 2 November 2021
Published in: Communications in Computational Physics (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/2004.01806
convergenceelliptic and parabolic PDEsphysics informed neural networksSchauder approachHölder regularization
Artificial neural networks and deep learning (68T07) Boundary value problems for second-order elliptic equations (35J25) Initial-boundary value problems for second-order parabolic equations (35K20) Stability and convergence of numerical methods for boundary value problems involving PDEs (65N12) Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs (65M12) Numerical methods for partial differential equations, initial value and time-dependent initial-boundary value problems (65M99) Numerical methods for partial differential equations, boundary value problems (65N99)
Related Items
Identification and prediction of time-varying parameters of COVID-19 model: a data-driven deep learning approach ⋮ Convergence of Physics-Informed Neural Networks Applied to Linear Second-Order Elliptic Interface Problems ⋮ When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization? ⋮ A two-stage physics-informed neural network method based on conserved quantities and applications in localized wave solutions ⋮ When and why PINNs fail to train: a neural tangent kernel perspective ⋮ Modeling, analysis and physics informed neural network approaches for studying the dynamics of COVID-19 involving human-human and human-pathogen interaction ⋮ Neural networks enforcing physical symmetries in nonlinear dynamical lattices: the case example of the Ablowitz-Ladik model ⋮ CAN-PINN: a fast physics-informed neural network based on coupled-automatic-numerical differentiation method ⋮ A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations ⋮ Learning generative neural networks with physics knowledge ⋮ A decision-making machine learning approach in Hermite spectral approximations of partial differential equations ⋮ Improved architectures and training algorithms for deep operator networks ⋮ High Order Deep Neural Network for Solving High Frequency Partial Differential Equations ⋮ Blood and breath alcohol concentration from transdermal alcohol biosensor data: estimation and uncertainty quantification via forward and inverse filtering for a covariate-dependent, physics-informed, hidden Markov model* ⋮ The deep parametric PDE method and applications to option pricing ⋮ A Deep Learning Method for Elliptic Hemivariational Inequalities ⋮ Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations ⋮ Scientific machine learning through physics-informed neural networks: where we are and what's next ⋮ Convergence Rate Analysis for Deep Ritz Method ⋮ RPINNs: rectified-physics informed neural networks for solving stationary partial differential equations ⋮ Fractional Chebyshev deep neural network (FCDNN) for solving differential models ⋮ Wasserstein generative adversarial uncertainty quantification in physics-informed neural networks ⋮ The deep learning Galerkin method for the general Stokes equations ⋮ A unified scalable framework for causal sweeping strategies for physics-informed neural networks (PINNs) and their temporal decompositions ⋮ Data-driven method to learn the most probable transition pathway and stochastic differential equation ⋮ Kolmogorov n-width and Lagrangian physics-informed neural networks: a causality-conforming manifold for convection-dominated PDEs ⋮ Deep learning methods for partial differential equations and related parameter identification problems ⋮ A priori generalization error analysis of two-layer neural networks for solving high dimensional Schrödinger eigenvalue problems ⋮ A deep first-order system least squares method for solving elliptic PDEs ⋮ Solving nonconvex energy minimization problems in martensitic phase transitions with a mesh-free deep learning approach ⋮ Physics-informed neural networks for approximating dynamic (hyperbolic) PDEs of second order in time: error analysis and algorithms ⋮ A cusp-capturing PINN for elliptic interface problems ⋮ A framework based on symbolic regression coupled with eXtended physics-informed neural networks for gray-box learning of equations of motion from data ⋮ Uncertainty quantification in scientific machine learning: methods, metrics, and comparisons ⋮ A metalearning approach for physics-informed neural networks (PINNs): application to parameterized PDEs ⋮ Physics-informed deep learning for melting heat transfer analysis with model-based transfer learning ⋮ Pre-training strategy for solving evolution equations based on physics-informed neural networks ⋮ Error estimates and physics informed augmentation of neural networks for thermally coupled incompressible Navier Stokes equations ⋮ Solving an inverse source problem by deep neural network method with convergence and error analysis ⋮ An Efficient Neural-Network and Finite-Difference Hybrid Method for Elliptic Interface Problems with Applications ⋮ Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance ⋮ Physics-informed neural networks for data-driven simulation: advantages, limitations, and opportunities ⋮ A novel physics-informed deep learning strategy with local time-updating discrete scheme for multi-dimensional forward and inverse consolidation problems ⋮ Isogeometric neural networks: a new deep learning approach for solving parameterized partial differential equations ⋮ A deep Fourier residual method for solving PDEs using neural networks ⋮ A deep double Ritz method (\(\mathrm{D^2RM}\)) for solving partial differential equations using neural networks ⋮ A Rate of Convergence of Weak Adversarial Neural Networks for the Second Order Parabolic PDEs ⋮ Learning Specialized Activation Functions for Physics-Informed Neural Networks ⋮ Boundary-safe PINNs extension: application to non-linear parabolic PDEs in counterparty credit risk ⋮ An artificial neural network approach to bifurcating phenomena in computational fluid dynamics ⋮ Physics-informed deep learning for three-dimensional transient heat transfer analysis of functionally graded materials ⋮ Solving free-surface problems for non-shallow water using boundary and initial conditions-free physics-informed neural network (bif-PINN) ⋮ Improved training of physics-informed neural networks for parabolic differential equations with sharply perturbed initial conditions ⋮ Finite basis physics-informed neural networks (FBPINNs): a scalable domain decomposition approach for solving differential equations ⋮ Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent? ⋮ Optimization of physics-informed neural networks for solving the nolinear Schrödinger equation ⋮ Control of partial differential equations via physics-informed neural networks ⋮ An overview on deep learning-based approximation methods for partial differential equations ⋮ Greedy training algorithms for neural networks and applications to PDEs ⋮ On the eigenvector bias of Fourier feature networks: from regression to solving multi-scale PDEs with physics-informed neural networks ⋮ Data-driven vector soliton solutions of coupled nonlinear Schrödinger equation using a deep learning algorithm ⋮ Physics-Informed Neural Networks with Hard Constraints for Inverse Design ⋮ Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks ⋮ Data-driven peakon and periodic peakon solutions and parameter discovery of some nonlinear dispersive equations via deep learning ⋮ INN: interfaced neural networks as an accessible meshless approach for solving interface PDE problems ⋮ A non-gradient method for solving elliptic partial differential equations with deep neural networks ⋮ Self-adaptive physics-informed neural networks ⋮ On stability and regularization for data-driven solution of parabolic inverse source problems
Uses Software
Cites Work
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures
- On the limited memory BFGS method for large scale optimization
- Applied functional analysis. Functional analysis, Sobolev spaces and elliptic differential equations
- Generalization bounds for function approximation from scattered noisy data
- Elliptic partial differential equations of second order
- Global optimization in Hilbert space
- The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
- DGM: a deep learning algorithm for solving partial differential equations
- Physics-informed neural networks for high-speed flows
- Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
- Variational Analysis in Sobolev and BV Spaces
- Neural‐network‐based approximations for solving partial differential equations
- Solving high-dimensional partial differential equations using deep learning
- Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations
- Consistency of Lipschitz Learning with Infinite Unlabeled Data and Finite Labeled Data
- DeepXDE: A Deep Learning Library for Solving Differential Equations
- Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks
- fPINNs: Fractional Physics-Informed Neural Networks
