We present an approach for the data-driven modeling of nonlinear viscoelastic materials at small strains which is based on physics-augmented neural networks (NNs) and requires only stress and strain paths for training. The model is built on the concept of generalized standard materials and is therefore thermodynamically consistent by construction. It consists of a free energy and a dissipation potential, which can be either expressed by the components of their tensor arguments or by a suitable set of invariants. The two potentials are described by fully/partially input convex neural networks. For training of the NN model by paths of stress and strain, an efficient and flexible training method based on a recurrent cell, particularly a long short-term memory cell, is developed to automatically generate the internal variable(s) during the training process. The proposed method is benchmarked and thoroughly compared with existing approaches. These include a method that obtains the internal variable by integrating the evolution equation over the entire sequence, while the other method uses an an auxiliary feedforward neural network for the internal variable(s). Databases for training are generated by using a conventional nonlinear viscoelastic reference model, where 3D and 2D plane strain data with either ideal or noisy stresses are generated. The coordinate-based and the invariant-based formulation are compared and the advantages of the latter are demonstrated. Afterwards, the invariant-based model is calibrated by applying the three training methods using ideal or noisy stress data. All methods yield good results, but differ in computation time and usability for large data sets. The presented training method based on a recurrent cell turns out to be particularly robust and widely applicable and thus represents a promising approach for the calibration of other types of models as well.

翻译：我们提出一种基于物理增强神经网络（NNs）的小应变非线性粘弹性材料数据驱动建模方法，该方法仅需应力与应变路径作为训练数据。模型基于广义标准材料概念构建，因此具有热力学一致性。模型由自由能与耗散势函数组成，可通过张量参数分量或合适的张量不变量集表示。这两个势函数采用全凸/部分输入凸神经网络描述。为通过应力-应变路径训练神经网络模型，我们开发了基于循环单元（特别是长短期记忆单元）的高效灵活训练方法，该方法可在训练过程中自动生成内变量。所提方法与现有方法进行了基准对比与全面比较：其中一种方法通过对整个序列积分演化方程获取内变量，另一种方法则采用辅助前馈神经网络生成内变量。训练数据集通过传统非线性粘弹性参考模型生成，包含三维和二维平面应变数据（含理想应力与含噪应力）。对基于坐标和基于不变量的两种模型表述进行了比较，验证了后者的优势。随后，采用三种训练方法（使用理想或含噪应力数据）对基于不变量的模型进行参数标定。所有方法均取得良好结果，但在计算效率和大数据集适用性方面存在差异。基于循环单元的训练方法展现出卓越的鲁棒性和广泛适用性，为其他类型模型的参数标定提供了有前景的解决方案。