We resort to game theory in order to formulate Data-Driven methods for solid mechanics in which stress and strain players pursue different objectives. The objective of the stress player is to minimize the discrepancy to a material data set, whereas the objective of the strain player is to ensure the admissibility of the mechanical state, in the sense of compatibility and equilibrium. We show that, unlike the cooperative Data-Driven games proposed in the past, the new non-cooperative Data-Driven games identify an effective material law from the data and reduce to conventional displacement boundary-value problems, which facilitates their practical implementation. However, unlike supervised machine learning methods, the proposed non-cooperative Data-Driven games are unsupervised, ansatz-free and parameter-free. In particular, the effective material law is learned from the data directly, without recourse to regression to a parameterized class of functions such as neural networks. We present analysis that elucidates sufficient conditions for convergence of the Data-Driven solutions with respect to the data. We also present selected examples of implementation and application that demonstrate the range and versatility of the approach.

翻译：我们借助博弈论来构建固体力学中的数据驱动方法，其中应力与应变参与者追求不同目标：应力参与者的目标是最小化与材料数据集的偏差，而应变参与者的目标是确保力学状态在相容性与平衡意义上的可容许性。研究表明，与以往提出的合作型数据驱动博弈不同，新型非合作型数据驱动博弈可从数据中识别有效材料本构关系，并简化为常规位移边值问题，从而便于实际应用。然而，与监督式机器学习方法不同，所提出的非合作型数据驱动博弈是无监督、无假设且无参数的：特别地，有效材料本构关系直接从数据中学习，无需回归至参数化函数类（如神经网络）。我们通过分析阐明了数据驱动解关于数据收敛的充分条件，并给出实施与应用的典型算例，展示了该方法的适用范围与灵活性。