This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush-Kuhn-Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer-Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

翻译：本文探讨了物理信息神经网络（PINNs）在小变形弹性接触力学正反问题求解中的能力。我们采用混合变量形式的PINNs，并结合输出变换以硬约束形式强制施加狄利克雷和诺伊曼边界条件。接触问题中的不等式约束，即Karush-Kuhn-Tucker（KKT）型条件，则通过将其纳入网络训练过程中的损失函数，作为软约束施加。针对KKT约束的损失函数贡献项构建，我们调研了现有应用于弹塑性问题的方法，并探索了一种非线性互补问题（NCP）函数——即Fischer-Burmeister函数，该函数在优化方面具有优越特性。基于赫兹接触问题，我们展示了PINNs不仅可作为纯偏微分方程（PDE）求解器、数据增强的正向模型、参数识别的反问题求解器，还可作为快速评估的替代模型。此外，我们证明了选择恰当超参数（如损失权重）以及采用Adam与L-BFGS-B优化器组合的重要性，旨在提升求解精度与训练效率。