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Data-driven constitutive modeling in continuum mechanics assumes that abundant material data are available and can effectively replace the constitutive law. To this end, Kirchdoerfer and Ortiz proposed an approach, which is often referred to as the distance-minimizing method \cite{kirchdoerferOrtiz2016}. Generally, in the distance-minimizing paradigm, a data-driven boundary value problem is formulated as a double-minimization problem with an associated metric. Such a metric can be defined as a quadratic functional form on the strain-stress phase space \cite{kirchdoerferOrtiz2016, Nguyen2018, Nguyen2020}. To weigh the contributions of the strain and the stress appropriately in the metric, a priori chosen tensor-valued hyperparameters are employed. The role of these hyperparameters remains poorly understood to date. In particular, it has been unclear whether and how they could be chosen to optimize the numerical properties of the distance-minimizing method. Herein, we demonstrate that choosing these hyperparameters equal to the tangent of the constitutive manifold underlying the available material data can substantially reduce the computational cost and improve the accuracy of the distance-minimizing method. As the tangent of the constitutive manifold is typically not known in a data-driven setting, and as its calculated values may change during an iterative solution process, we propose an adaptive strategy that continuously updates the hyperparameters based on an approximate tangent of the hidden constitutive manifold \cite{Nguyen2022}. The optimal convergence rate for the distance-minimizing method can be derived for a simplified problem of truss structure.

Data-driven computational mechanics; Optimization problem; Fixed-point method; Hyperparameters