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Section 3.3: Diagram of our ansatz u=d∂Ω·uNN+g~\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u = d_{\partial \Omega }\cdot u_{NN} + {\tilde{g}}$$\end{document} for the two dimensional Poisson problem. Here we used the abbreviations ui:=u(xi,yi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_i := u(x_i, y_i)$$\end{document} and fi:=f(xi,yi)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_i := f(x_i,y_i)$$\end{document} for points xi=(xi,yi)∈Ω¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{x}_i = (x_i,y_i) \in {\bar{\Omega }}$$\end{document}

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In this work, we are concerned with neural network guided goal-oriented a posteriori error estimation and adaptivity using the dual weighted residual method. The primal problem is solved using classical Galerkin finite elements. The adjoint problem is solved in strong form with a feedforward neural network using two or three hidden layers. The main...

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... These include the test function choice [7], forward solve [8], adjoint solve [9], derivative recovery procedure [10], error estimation [11], metric/monitor function/sizing field construction step [12,13,14], and the entire mesh adaptation loop [6,15,16]. ...
... A similar 'focused' approach is also used in [9], which emulates the adjoint solve procedure. This is done on the base mesh and the data-driven adjoint solution is projected into an enriched space, where error indicators are assembled and thereby used to drive mesh adaptation. ...
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... Using ansatz functions of the above form has become increasingly popular and it has been observed that it simplifies the training process and produces more accurate solutions, see for instance Berg and Nyström (2018); Roth et al. (2021); Lyu et al. (2020); Chen et al. (2020). It is also possible to encode Neumann or Robin boundary conditions in a similar way, we refer the reader to Lyu et al. (2020). ...
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