Neural Control of Discrete Weak Formulations: Galerkin, Least-Squares and Minimal-Residual Methods with Quasi-Optimal Weights

Abstract

There is tremendous potential in using neural networks to optimize numerical methods. In this paper, we introduce and analyse 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.

35 pages, 9 figures

References in corpus (8)

• Machine learning materials physics: Integrable deep neural networks enable scale bridging by learning free energy functions
• Variational Physics Informed Neural Networks: the role of quadratures and test functions
• Least-Squares ReLU Neural Network (LSNN) Method For Linear Advection-Reaction Equation
• Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems
• Solving PDEs by Variational Physics-Informed Neural Networks: an a posteriori error analysis
• A priori and a posteriori error estimates for the Deep Ritz method applied to the Laplace and Stokes problem
• A Machine Learning approach to enhance the SUPG stabilization method for advection-dominated differential problems
• Error Estimates for the Deep Ritz Method with Boundary Penalty

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• Learning quantities of interest from parametric PDEs: An efficient neural-weighted Minimal Residual approach
• Neural network methods for Neumann series problems of Perron-Frobenius operators