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We introduce the concept of machine-learning minimal-residual (ML-MRes) finite element discretizations of partial differential equations (PDEs), which resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The methods are obtained within a machine-learning framework during which the parameters defining the method are tuned against available training data. In particular, we use a provably stable parametric Petrov–Galerkin method that is equivalent to a minimal-residual formulation using a weighted norm. While the trial space is a standard finite element space, the test space has parameters that are tuned in an off-line stage. Finding the optimal test space therefore amounts to obtaining a goal-oriented discretization that is completely tailored towards the quantity of interest. We use an artificial neural network to define the parametric family of test spaces. Using numerical examples for the Laplacian and advection equation in one and two dimensions, we demonstrate that the ML-MRes finite element method has superior approximation of quantities of interest even on very coarse meshes. [ABSTRACT FROM AUTHOR]

There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyze a framework for the neural optimization of discrete weak formulations , suitable for finite element methods. The main idea of the framework is to include a neural-network function acting as a control variable in the weak form. Finding the neural control that (quasi-) minimizes a suitable cost (or loss) functional, then yields a numerical approximation with desirable attributes. In particular, the framework allows in a natural way the incorporation of known data of the exact solution, or the incorporation of stabilization mechanisms (e.g., to remove spurious oscillations). The main result of our analysis pertains to the well-posedness and convergence of the associated constrained-optimization problem. In particular, we prove under certain conditions, that the discrete weak forms are stable, and that quasi-minimizing neural controls exist, which converge quasi-optimally. We specialize the analysis results to Galerkin, least squares and minimal-residual formulations, where the neural-network dependence appears in the form of suitable weights. Elementary numerical experiments support our findings and demonstrate the potential of the framework. [ABSTRACT FROM AUTHOR]

Abstract: The numerical solution of fluid–structure-interaction problems poses a paradox in that most of the computational resources are consumed by the subsystem of least practical interest, viz., the fluid. Goal-oriented adaptive discretization methods provide a paradigm to bypass this paradox. Based on the solution of a dual problem, the contribution of local residuals to the error in a specific goal functional is estimated, and only the regions that yield a dominant contribution are refined. In the present work, we address a fundamental complication in the application of goal-oriented adaptivity to fluid–structure-interaction problems, namely, that the treatment of the interface conditions has nontrivial consequences for the properties of the dual problem. In the context of a linearized model problem, we consider two equivalent discretizations differing only on the formulation of the interface coupling terms. By means of an adjoint consistency analysis, we show that only one of these discretizations is adjoint consistent. Numerical experiments convey that the two discretizations behave very differently for the dual problem, and that the adjoint-consistent discretization yields more reliable error estimates. Based on the adjoint-consistent discretization, we finally present some h- and hp-adaptive results, confirming that tremendous savings in computational cost can be realized through the use of goal-oriented refinement strategies. The numerical experiments illustrate that the goal-oriented approach effectively equilibrates the error contributions of the fluid and structure subsystems, which is imperative for efficiently resolving the coupled fluid–structure-interaction problem, and which cannot be accomplished by uniform or residual-based refinement strategies. [ABSTRACT FROM AUTHOR]

Abstract Goal-oriented adaptivity is a powerful tool to accurately approximate physically relevant solution features for partial differential equations. In time dependent problems, we seek to represent the error in the quantity of interest as an integral over the whole space–time domain. A full space–time variational formulation allows such representation. Most authors employ implicit time marching schemes to perform goal-oriented adaptivity as it is known that they can be reinterpreted as Galerkin methods. In this work, we consider variational forms for explicit methods in time. We derive an appropriate error representation and propose a goal-oriented adaptive algorithm in space. For that, we derive the forward Euler method in time employing a discontinuous-in-time Petrov–Galerkin formulation. In terms of time domain adaptivity, we impose the Courant–Friedrichs–Lewy condition to ensure the stability of the method. We provide some numerical results in 1D space + time for the diffusion and advection–diffusion equations to show the performance of the proposed explicit-in-time goal-oriented adaptive algorithm. [ABSTRACT FROM AUTHOR]