Model Reduction and Neural Networks for Parametric PDEs [PDF]
SMAI Journal of Computational Mathematics, 2020 We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model ...
K. Bhattacharya +3 more
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Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs [PDF]
Advances in Computational Mathematics, 2021 Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov ...
Tim De Ryck, Siddhartha Mishra
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A metalearning approach for Physics-Informed Neural Networks (PINNs): Application to parameterized PDEs [PDF]
Journal of Computational Physics, 2021 Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world.
Michael Penwarden +3 more
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PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs [PDF]
Computer Methods in Applied Mechanics and Engineering, 2019 Physics-informed neural networks (PINNs) encode physical conservation laws and prior physical knowledge into the neural networks, ensuring the correct physics is represented accurately while alleviating the need for supervised learning to a great degree.
Xuhui Meng +3 more
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A Comprehensive Deep Learning-Based Approach to Reduced Order Modeling of Nonlinear Time-Dependent Parametrized PDEs [PDF]
Journal of Scientific Computing, 2020 Conventional reduced order modeling techniques such as the reduced basis (RB) method (relying, e.g., on proper orthogonal decomposition (POD)) may incur in severe limitations when dealing with nonlinear time-dependent parametrized PDEs, as these are ...
S. Fresca, L. Dede’, A. Manzoni
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Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning [PDF]
Nonlinearity, 2020 In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations (PDEs) in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning.
W. E, Jiequn Han, Arnulf Jentzen
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Greedy training algorithms for neural networks and applications to PDEs [PDF]
Journal of Computational Physics, 2021 Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or empirically ...
Jonathan W. Siegel +4 more
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PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network [PDF]
Journal of Computational Physics, 2018 Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In
Zichao Long, Yiping Lu, Bin Dong
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Physics Informed Neural Networks (PINNs)for approximating nonlinear dispersive PDEs [PDF]
Journal of Computational Mathematics, 2021 We propose a novel algorithm, based on physics-informed neural networks (PINNs) to efficiently approximate solutions of nonlinear dispersive PDEs such as the KdV-Kawahara, Camassa-Holm andBenjamin-Ono equations.
Genming Bai +3 more
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Solving high-dimensional Hamilton–Jacobi–Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space [PDF]
Partial Differential Equations and Applications, 2020 Optimal control of diffusion processes is intimately connected to the problem of solving certain Hamilton–Jacobi–Bellman equations. Building on recent machine learning inspired approaches towards high-dimensional PDEs, we investigate the potential of ...
N. Nüsken, Lorenz Richter
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