56,517 research outputs found
Model Reduction and Neural Networks for Parametric PDEs
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 reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. Numerically we demonstrate the effectiveness of the method on a class of parametric elliptic PDE problems, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare our method with existing algorithms from the literature
Model Reduction and Neural Networks for Parametric PDEs
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 reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. Numerically we demonstrate the effectiveness of the method on a class of parametric elliptic PDE problems, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare our method with existing algorithms from the literature
On the possible Computational Power of the Human Mind
The aim of this paper is to address the question: Can an artificial neural
network (ANN) model be used as a possible characterization of the power of the
human mind? We will discuss what might be the relationship between such a model
and its natural counterpart. A possible characterization of the different power
capabilities of the mind is suggested in terms of the information contained (in
its computational complexity) or achievable by it. Such characterization takes
advantage of recent results based on natural neural networks (NNN) and the
computational power of arbitrary artificial neural networks (ANN). The possible
acceptance of neural networks as the model of the human mind's operation makes
the aforementioned quite relevant.Comment: Complexity, Science and Society Conference, 2005, University of
Liverpool, UK. 23 page
Differentiable Programming Tensor Networks
Differentiable programming is a fresh programming paradigm which composes
parameterized algorithmic components and trains them using automatic
differentiation (AD). The concept emerges from deep learning but is not only
limited to training neural networks. We present theory and practice of
programming tensor network algorithms in a fully differentiable way. By
formulating the tensor network algorithm as a computation graph, one can
compute higher order derivatives of the program accurately and efficiently
using AD. We present essential techniques to differentiate through the tensor
networks contractions, including stable AD for tensor decomposition and
efficient backpropagation through fixed point iterations. As a demonstration,
we compute the specific heat of the Ising model directly by taking the second
order derivative of the free energy obtained in the tensor renormalization
group calculation. Next, we perform gradient based variational optimization of
infinite projected entangled pair states for quantum antiferromagnetic
Heisenberg model and obtain start-of-the-art variational energy and
magnetization with moderate efforts. Differentiable programming removes
laborious human efforts in deriving and implementing analytical gradients for
tensor network programs, which opens the door to more innovations in tensor
network algorithms and applications.Comment: Typos corrected, discussion and refs added; revised version accepted
for publication in PRX. Source code available at
https://github.com/wangleiphy/tensorgra
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