Areas of computational mechanics such as uncertainty quantification and optimization usually involve repeated evaluation of numerical models that represent the behavior of engineering systems. In the case of complex nonlinear systems however, these models tend to be expensive to evaluate, making surrogate models quite valuable. Artificial neural networks approximate systems very well by taking advantage of the inherent information of its given training data. In this context, this paper investigates the improvement of the training process by including sensitivity information, which are partial derivatives w.r.t. inputs, as outlined by Sobolev training. In computational mechanics, sensitivities can be applied to neural networks by expanding the training loss function with additional loss terms, thereby improving training convergence resulting in lower generalisation error. This improvement is shown in two examples of linear and non-linear material behavior. More specifically, the Sobolev designed loss function is expanded with residual weights adjusting the effect of each loss on the training step. Residual weighting is the given scaling to the different training data, which in this case are response and sensitivities. These residual weights are optimized by an adaptive scheme, whereby varying objective functions are explored, with some showing improvements in accuracy and precision of the general training convergence.

翻译：在不确定性量化与优化等计算力学领域，通常需要对表征工程系统行为的数值模型进行反复评估。然而，对于复杂的非线性系统，这类模型的评估往往计算成本高昂，这使得代理模型具有重要价值。人工神经网络能够充分利用其给定训练数据的内在信息，从而实现对系统的高度近似。在此背景下，本文研究了通过引入灵敏度信息（即关于输入变量的偏导数）来改进训练过程的方法，该方法遵循Sobolev训练的理论框架。在计算力学中，可通过在训练损失函数中增加额外的损失项来将灵敏度信息应用于神经网络，从而改善训练收敛性并降低泛化误差。这一改进效果通过线性和非线性材料行为的两个算例得到验证。具体而言，本文对Sobolev设计的损失函数进行了扩展，引入残差权重以调节各项损失在训练步骤中的影响。残差权重是对不同训练数据（本文中指系统响应与灵敏度数据）赋予的缩放系数。这些残差权重通过自适应方案进行优化，该方案探索了多种目标函数，其中部分函数在整体训练收敛的准确性与精确度方面展现出显著提升。