A deep learning approach to Reduced Order Modelling of parameter dependent partial differential equations
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Publication:5058646
DOI10.1090/mcom/3781zbMath1504.65255arXiv2103.06183OpenAlexW3135895479MaRDI QIDQ5058646
Paolo Zunino, Andrea Manzoni, Unnamed Author
Publication date: 21 December 2022
Published in: Mathematics of Computation (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/2103.06183
Artificial neural networks and deep learning (68T07) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs (35B30) Theoretical approximation in context of PDEs (35A35) PDEs with randomness, stochastic partial differential equations (35R60)
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Uses Software
Cites Work
- Solution manifolds and submanifolds of parametrized equations and their discretization errors
- A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs
- Bayesian deep convolutional encoder-decoder networks for surrogate modeling and uncertainty quantification
- An `empirical interpolation' method: Application to efficient reduced-basis discretization of partial differential equations
- Proper orthogonal decomposition methods for partial differential equations
- Optimal nonlinear approximation
- Reduced order modeling for nonlinear structural analysis using Gaussian process regression
- Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs
- Numerical solution of the parametric diffusion equation by deep neural networks
- Model reduction and neural networks for parametric PDEs
- Approximation rates for neural networks with encodable weights in smoothness spaces
- POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition
- A theoretical analysis of deep neural networks and parametric PDEs
- Nonlinear approximation and (deep) ReLU networks
- High-order approximation rates for shallow neural networks with cosine and \(\mathrm{ReLU}^k\) activation functions
- Physics-informed machine learning for reduced-order modeling of nonlinear problems
- Optimal approximation of piecewise smooth functions using deep ReLU neural networks
- Non-intrusive reduced order modeling of unsteady flows using artificial neural networks with application to a combustion problem
- Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
- \(L^ 2\) estimates for the finite element method for the Dirichlet problem with singular data
- Decay of the Kolmogorov \(N\)-width for wave problems
- Error bounds for approximations with deep ReLU networks
- Efficient model reduction of parametrized systems by matrix discrete empirical interpolation
- A nonlinear manifold-based reduced order model for multiscale analysis of heterogeneous hyperelastic materials
- An extension of Tietze's theorem
- Optimal stable nonlinear approximation
- Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
- A Certified Reduced Basis Method for the Fokker–Planck Equation of Dilute Polymeric Fluids: FENE Dumbbells in Extensional Flow
- Convergence Rates for Greedy Algorithms in Reduced Basis Methods
- Reducing the Dimensionality of Data with Neural Networks
- A Goal-Oriented Reduced Basis Methods-Accelerated Generalized Polynomial Chaos Algorithm
- Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ
- Geometric Group Theory
- Reduced-order modelling of parameter-dependent, linear and nonlinear dynamic partial differential equation models
- Drei Sätze über die n-dimensionale euklidische Sphäre
- Enhancing Accuracy of Deep Learning Algorithms by Training with Low-Discrepancy Sequences
- Optimal Approximation with Sparsely Connected Deep Neural Networks
- Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs
- Error estimates for DeepONets: a deep learning framework in infinite dimensions
- Parameter identification for elliptic boundary value problems: an abstract framework and applications
- Error bounds for approximations with deep ReLU neural networks in Ws,p norms
- On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
- Reduced Basis Methods for Partial Differential Equations
- A Stochastic Collocation Method for Elliptic Partial Differential Equations with Random Input Data
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