Numerical solution of the parametric diffusion equation by deep neural networks
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Publication:2049099
DOI10.1007/s10915-021-01532-wzbMath1473.35331arXiv2004.12131OpenAlexW3166995168MaRDI QIDQ2049099
Moritz Geist, Philipp Petersen, Mones Raslan, Gitta Kutyniok, Reinhold Schneider
Publication date: 24 August 2021
Published in: Journal of Scientific Computing (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/2004.12131
Learning and adaptive systems in artificial intelligence (68T05) Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs (65N30) Rate of convergence, degree of approximation (41A25) Approximation by other special function classes (41A30) Elliptic equations and elliptic systems (35J99)
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Uses Software
Cites Work
- Unnamed Item
- Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
- Data driven approximation of parametrized PDEs by reduced basis and neural networks
- Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Application to transport and continuum mechanics.
- Non-intrusive reduced order modeling of nonlinear problems using neural networks
- Inferring solutions of differential equations using noisy multi-fidelity data
- Machine learning of linear differential equations using Gaussian processes
- Provable approximation properties for deep neural networks
- The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
- Deep UQ: learning deep neural network surrogate models for high dimensional uncertainty quantification
- DGM: a deep learning algorithm for solving partial differential equations
- Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs
- Optimal approximation of piecewise smooth functions using deep ReLU neural networks
- Theory of deep convolutional neural networks: downsampling
- An artificial neural network framework for reduced order modeling of transient flows
- Data-driven discovery of PDEs in complex datasets
- A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations
- Machine learning for fast and reliable solution of time-dependent differential equations
- Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders
- Approximation of infinitely differentiable multivariate functions is intractable
- Error bounds for approximations with deep ReLU networks
- Variational Monte Carlo -- bridging concepts of machine learning and high-dimensional partial differential equations
- An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications
- Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations
- On the mathematical foundations of learning
- An Introduction to Computational Stochastic PDEs
- Certified Reduced Basis Methods for Parametrized Partial Differential Equations
- Compressive sensing Petrov-Galerkin approximation of high-dimensional parametric operator equations
- Kolmogorov widths and low-rank approximations of parametric elliptic PDEs
- Learning Theory
- Universal approximation bounds for superpositions of a sigmoidal function
- Parametric PDEs: sparse or low-rank approximations?
- Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations
- Deep learning in high dimension: Neural network expression rates for generalized polynomial chaos expansions in UQ
- ANALYTIC REGULARITY AND POLYNOMIAL APPROXIMATION OF STOCHASTIC, PARAMETRIC ELLIPTIC MULTISCALE PDEs
- Solving high-dimensional partial differential equations using deep learning
- Optimal Approximation with Sparsely Connected Deep Neural Networks
- Deep ReLU networks and high-order finite element methods
- DeepXDE: A Deep Learning Library for Solving Differential Equations
- A mixed ℓ1 regularization approach for sparse simultaneous approximation of parameterized PDEs
- Equivalence of approximation by convolutional neural networks and fully-connected networks
- Approximation of high-dimensional parametric PDEs
- Reduced Basis Methods for Partial Differential Equations
- On the Theory of Dynamic Programming
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