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000456491 0247_ $$2datacite_doi$$a10.3204/PUBDB-2021-01487
000456491 037__ $$aPUBDB-2021-01487
000456491 041__ $$aEnglish
000456491 088__ $$2arXiv$$aarXiv:2103.08662
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000456491 1001_ $$0P:(DE-H253)PIP1087239$$aGuidetti, V.$$b0$$udesy
000456491 245__ $$adNNsolve: an efficient NN-based PDE solver
000456491 260__ $$c2021
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000456491 500__ $$a16 pages + 9 pages of appendices, 7 figures, LaTeX, code to be released soon
000456491 520__ $$aNeural Networks (NNs) can be used to solve Ordinary and Partial Differential Equations (ODEs and PDEs) by redefining the question as an optimization problem. The objective function to be optimized is the sum of the squares of the PDE to be solved and of the initial/boundary conditions. A feed forward NN is trained to minimise this loss function evaluated on a set of collocation points sampled from the domain where the problem is defined. A compact and smooth solution, that only depends on the weights of the trained NN, is then obtained. This approach is often referred to as PINN, from Physics Informed Neural Network~\cite{raissi2017physics_1, raissi2017physics_2}. Despite the success of the PINN approach in solving various classes of PDEs, an implementation of this idea that is capable of solving a large class of ODEs and PDEs with good accuracy and without the need to finely tune the hyperparameters of the network, is not available yet. In this paper, we introduce a new implementation of this concept - called dNNsolve - that makes use of dual Neural Networks to solve ODEs/PDEs. These include: i) sine and sigmoidal activation functions, that provide a more efficient basis to capture both secular and periodic patterns in the solutions; ii) a newly designed architecture, that makes it easy for the the NN to approximate the solution using the basis functions mentioned above. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions, without the need of hyperparameter fine-tuning.
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000456491 7001_ $$0P:(DE-H253)PIP1091885$$aMuia, F.$$b1$$udesy
000456491 7001_ $$0P:(DE-H253)PIP1086156$$aWelling, Yvette Maria$$b2$$eCorresponding author
000456491 7001_ $$0P:(DE-H253)PIP1013212$$aWestphal, Alexander$$b3
000456491 773__ $$p1-25
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