Abstract
Solving partial differential equations (PDE) is an indispensable part of many branches of science as many processes can be modelled in terms of PDEs. However, recent numerical solvers require manual discretization of the underlying equation as well as sophisticated, tailored code for distributed computing. Scanning the parameters of the underlying model significantly increases the runtime as the simulations have to be cold-started for each parameter configuration. Machine Learning based surrogate models denote promising ways for learning complex relationship among input, parameter and solution. However, recent generative neural networks require lots of training data, i.e. full simulation runs making them costly. In contrast, we examine the applicability of continuous, mesh-free neural solvers for partial differential equations, physics-informed neural networks (PINNs) solely requiring initial/boundary values and validation points for training but no simulation data. The induced curse of dimensionality is approached by learning a domain decomposition that steers the number of neurons per unit volume and significantly improves runtime. Distributed training on large-scale cluster systems also promises great utilization of large quantities of GPUs which we assess by a comprehensive evaluation study. Finally, we discuss the accuracy of GatedPINN with respect to analytical solutions- as well as state-of-the-art numerical solvers, such as spectral solvers.
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The PyTorch implementations of the benchmarking dataset as well as the neural solvers for 1D and 2D Schrodinger equation and pretrained models are available online: https://github.com/ComputationalRadiationPhysics/NeuralSolvers.
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Acknowledgement
This work was partially funded by the Center of Advanced Systems Understanding (CASUS) which is financed by Germany’s Federal Ministry of Education and Research (BMBF) and by the Saxon Ministry for Science, Culture and Tourism (SMWK) with tax funds on the basis of the budget approved by the Saxon State Parliament. The authors gratefully acknowledge the GWK support for funding this project by providing computing time through the Center for Information Services and HPC (ZIH) at TU Dresden on the HPC-DA.
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Stiller, P. et al. (2020). Large-Scale Neural Solvers for Partial Differential Equations. In: Nichols, J., Verastegui, B., Maccabe, A.‘., Hernandez, O., Parete-Koon, S., Ahearn, T. (eds) Driving Scientific and Engineering Discoveries Through the Convergence of HPC, Big Data and AI. SMC 2020. Communications in Computer and Information Science, vol 1315. Springer, Cham. https://doi.org/10.1007/978-3-030-63393-6_2
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DOI: https://doi.org/10.1007/978-3-030-63393-6_2
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