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Register a new userNonlinear mixed-dimension model for embedded tubular networks with application to root water uptake
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Added by
Timo Koch
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We present a numerical scheme for the solution of nonlinear mixed-dimensional PDEs describing coupled processes in embedded tubular network system in exchange with a bulk domain. Such problems arise in various biological and technical applications such as in the modeling of root-water uptake, heat exchangers, or geothermal wells. The nonlinearity appears in form of solution-dependent parameters such as pressure-dependent permeability or temperature-dependent thermal conductivity. We derive and analyse a numerical scheme based on distributing the bulk-network coupling source term by a smoothing kernel with local support. By the use of local analytical solutions, interface unknowns and fluxes at the bulk-network interface can be accurately reconstructed from coarsely resolved numerical solutions in the bulk domain. Numerical examples give confidence in the robustness of the method and show the results in comparison to previously published methods. The new method outperforms these existing methods in accuracy and efficiency. In a root water uptake scenario, we accurately estimate the transpiration rate using only a few thousand 3D mesh cells and a structured cube grid whereas other state-of-the-art numerical schemes require millions of cells and local grid refinement to reach comparable accuracy.
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Embedding representations power machine intelligence in many applications, including recommendation systems, but they are space intensive -- potentially occupying hundreds of gigabytes in large-scale settings. To help manage this outsized memory consumption, we explore mixed dimension embeddings, an embedding layer architecture in which a particular embedding vectors dimension scales with its query frequency. Through theoretical analysis and systematic experiments, we demonstrate that using mixed dimensions can drastically reduce the memory usage, while maintaining and even improving the ML performance. Empirically, we show that the proposed mixed dimension layers improve accuracy by 0.1% using half as many parameters or maintain it using 16X fewer parameters for click-through rate prediction task on the Criteo Kaggle dataset.
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