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000616787 1001_ $$0P:(DE-H253)PIP1013212$$aWestphal, Alexander$$b0$$eCorresponding author$$udesy
000616787 1112_ $$aFH SciComp Workshop$$cHamburg$$d2024-07-01 - 2024-07-02$$wGermany
000616787 245__ $$aFrom testing cosmological inflation models to solving PDEs - dNNsolve: an efficient NN-based PDE solver
000616787 260__ $$c2024
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000616787 502__ $$cDESY Hamburg
000616787 520__ $$aNeural Networks (NNs) can be used to solve Ordinary and Partial Differential Equations (ODEs and PDEs) by redefining the question as an optimization problem. The objective function to be optimized is the sum of the squares of the PDE to be solved and of the initial/boundary conditions. A feed forward NN is trained to minimise this loss function evaluated on a set of collocation points sampled from the domain where the problem is defined. A compact and smooth solution, that only depends on the weights of the trained NN, is then obtained. This approach is often referred to as PINN, from Physics Informed Neural Network. Despite the success of the PINN approach in solving various classes of PDEs, an implementation of this idea that is capable of solving a large class of ODEs and PDEs with good accuracy and without the need to finely tune the hyperparameters of the network, is not available yet. In this paper, we introduce a new implementation of this concept - called dNNsolve - that makes use of dual Neural Networks with different activation functions to solve ODEs/PDEs. We show that dNNsolve is capable of solving a broad range of ODEs/PDEs in 1, 2 and 3 spacetime dimensions, without the need of hyperparameter fine-tuning.
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000616787 8564_ $$uhttps://indico.desy.de/event/44550/contributions/171599/
000616787 8564_ $$uhttps://bib-pubdb1.desy.de/record/616787/files/FHH-SciComp_Westphal_2024.pdf$$yRestricted
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000616787 9141_ $$y2024
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