A Multi-batch L-BFGS Method for Machine Learning, Proceedings of the 30th International Conference on Neural Information Processing Systems. Curran Associates Inc., USA, NIPS'16, pp.1063-1071, 2016. ,
A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing, vol.317, pp.28-41, 2018. ,
Julia: A fresh approach to numerical computing, SIAM Rev, vol.59, pp.65-98, 2017. ,
Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models, Math. Program, vol.163, pp.359-368, 2017. ,
, Pattern recognition and machine learning, Information Science and Statistics, 2006.
, Numerical Methods for Least Squares Problems, vol.51, 1996.
Optimization methods for large-scale machine learning, SIAM Rev, vol.60, pp.223-311, 2018. ,
General highly accurate algebraic coarsening, Electron. Trans. Numer. Anal, vol.10, pp.1-20, 2000. ,
, A Multigrid Tutorial, 2000.
On high-order multilevel optimization strategies, 2019. ,
URL : https://hal.archives-ouvertes.fr/hal-02943218
On the solution of systems of the form A T Ax = A T b + c, preprint, 2019. ,
Adaptive cubic regularisation methods for unconstrained optimization. part I: motivation, convergence and numerical results, Math. Program, vol.127, pp.245-295, 2011. ,
AMG strategies for PDE systems with applications in industrial semiconductor simulation, 2005. ,
How large a shift is needed in the shifted Helmholtz preconditioner for its effective inversion by multigrid?, SIAM. J. Sci. Comput, vol.39, pp.438-478, 2017. ,
URL : https://hal.archives-ouvertes.fr/hal-01578444
A constrained-optimization approach to training neural networks for smooth function approximation and system identification, IEEE International Joint Conference on Neural Networks (IEEE World Congress on Computational Intelligence, pp.2353-2359, 2008. ,
Neural-network-based approximations for solving partial differential equations, Commun. Numer. Methods. Eng, vol.10, pp.195-201, 1994. ,
The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems, 2017. ,
Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods, Numerical Analysis of Multiscale Problems, pp.325-363, 2012. ,
A class of iterative solvers for the Helmholtz equation: factorizations, sweeping preconditioners, source transfer, single layer potentials, polarized traces, and optimized Schwarz methods, SIAM Rev, vol.61, pp.3-76, 2019. ,
Recursive trust-region methods for multiscale nonlinear optimization, SIAM. J. Optim, vol.19, pp.414-444, 2008. ,
On the convergence of recursive trust-region methods for multiscale nonlinear optimization and applications to nonlinear mechanics, SIAM. J. Numer. Anal, vol.47, pp.3044-3069, 2009. ,
Learning across scales-Multiscale methods for convolution neural networks, Thirty-Second AAAI Conference on Artificial Intelligence, 2018. ,
, Multi-grid Methods and Applications, vol.4, 1985.
Solving high-dimensional partial differential equations using deep learning, Proc. Nat. Acad. Sci, vol.115, pp.8505-8510, 2018. ,
, Neural Networks: a Comprehensive Foundation, 1994.
, Theory of the backpropagation neural network, International 1989 Joint Conference on Neural Networks, vol.1, pp.593-605, 1989.
Deep learning: an introduction for applied mathematicians, SIAM Rev, vol.61, pp.860-891, 2019. ,
Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations, 2018. ,
A proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients, 2018. ,
Multigrid Neural Architectures, Proceedings of the 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR, pp.4067-4075, 2017. ,
A first-order multigrid method for bound-constrained convex optimization, Optim. Method. Softw, vol.31, pp.622-644, 2016. ,
Imagenet classification with deep convolutional neural networks, NIPS'12 Proceedings of the 25th International Conference on Neural Information Processing Systems, vol.1, pp.1097-1105, 2012. ,
Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Netw, vol.9, pp.987-1000, 1998. ,
Neural algorithm for solving differential equations, J. Comput. Phys, vol.91, pp.110-131, 1990. ,
Model problems for the multigrid optimization of systems governed by differential equations, SIAM. J. Sci. Comput, vol.26, pp.1811-1837, 2005. ,
Using inexact gradients in a multilevel optimization algorithm, Comput. Optim. Appl, vol.56, pp.39-61, 2013. ,
Solving PDEs in complex geometries: a diffuse domain approach, Commun. Math. Sci, vol.7, pp.81-107, 2009. ,
Learning PDEs from data, Proceedings of the 35th International Conference on Machine Learning, pp.3208-3216, 2018. ,
Neural network time series forecasting of finite-element mesh adaptation, Neurocomputing, vol.63, pp.447-463, 2005. ,
A machine learning framework for data driven acceleration of computations of differential equations, Seminar for Applied Mathematics, 2018. ,
Artificial neural networks in hardware: A survey of two decades of progress, Neurocomputing, vol.74, pp.239-255, 2010. ,
A multigrid approach to discretized optimization problems, Optimization Methods and Software, vol.14, pp.99-116, 2000. ,
Properties of a class of multilevel optimization algorithms for equality constrained problems, Optimization Methods and Software, vol.29, pp.137-159, 2014. ,
Hidden physics models: machine learning of nonlinear partial differential equations, J. Comput. Phys, vol.357, pp.125-141, 2018. ,
Physics informed deep learning (part I): datadriven solutions of nonlinear partial differential equations, 2017. ,
Physics informed deep learning (part II): datadriven discovery of nonlinear partial differential equations, 2017. ,
Numerical Gaussian processes for time-dependent and nonlinear partial differential equations, SIAM. J. Sci. Comput, vol.40, pp.172-198, 2018. ,
Finite-element neural networks for solving differential equations, IEEE Transactions on Neural Networks, vol.16, pp.1381-1392, 2005. ,
Solving partial differential equations using artificial neural networks, 2013. ,
Data-driven discovery of partial differential equations, Sci. Adv, vol.3, 2017. ,
Algebraic multigrid, pp.73-130, 1987. ,
Engineering fast multilevel support vector machines, 2017. ,
Learning partial differential equations via data discovery and sparse optimization, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, p.473, 2017. ,
Communication in the presence of noise, Proc. IEEE, vol.86, pp.447-457, 1998. ,
Multilayer perceptron neural networks with novel unsupervised training method for numerical solution of the partial differential equations, Appl. Soft. Comput, vol.9, pp.20-29, 2009. ,
Neural network representation of finite element method, Neural. Netw, vol.7, pp.389-395, 1994. ,
, , 2000.
A line search multigrid method for large-scale nonlinear optimization, SIAM J. Optim, vol.20, pp.1478-1503, 2009. ,