The Deep Operator Network (DeepONet) structure has shown great potential in approximating complex solution operators with low generalization errors. Recently, a sequential DeepONet (S-DeepONet) was proposed to use sequential learning models in the branch of DeepONet to predict final solutions given time-dependent inputs. In this novel work, the S-DeepONet architecture is further extended by modifying the information combination mechanism between the branch and trunk networks to simultaneously predict vector solutions with multiple components at multiple time steps of the evolution history. Two example problems, one on transient fluid flow and the other on path-dependent plastic loading were shown to demonstrate the capabilities of the model to handle different physics problems. The use of a trained S-DeepONet model in inverse parameter identification via the genetic algorithm is shown to demonstrate the application of the model. In almost all cases, the trained model achieved an $R^2$ value of above 0.99 and relative $L_2$ error of less than 10\% with only 3200 training data points, indicating superior accuracy. The vector S-DeepONet model, having only 0.4\% more parameters than a scalar model, can predict two output components simultaneously at an accuracy similar to the two independently trained scalar models with a 20.8\% faster training time. The S-DeepONet inference is at least three orders of magnitude faster than direct numerical simulations, and inverse parameter identifications using the trained model is highly efficient and accurate.

翻译：深度算子网络（DeepONet）结构在逼近低泛化误差的复杂解算子方面展现出巨大潜力。近期提出的序贯深度算子网络（S-DeepONet）通过在DeepONet分支中引入序贯学习模型，实现给定时间依赖输入的最终解预测。本研究进一步扩展S-DeepONet架构，通过改进分支网络与主干网络间的信息组合机制，同步预测含多个时间步演化的向量解的多分量。通过瞬态流体流动与路径相关塑性加载两个实例问题，验证了模型处理不同物理问题的能力。基于遗传算法的逆参数辨识实验展示了训练后S-DeepONet模型的应用潜力。几乎所有案例中，仅利用3200个训练数据点，训练模型即实现$R^2$值超过0.99且相对$L_2$误差小于10%，表明其卓越精度。相比标量模型，向量S-DeepONet模型仅增加0.4%参数，即可同时预测两个输出分量，且精度与独立训练的两个标量模型相当，训练时间缩短20.8%。S-DeepONet推理速度比直接数值模拟快至少三个数量级，基于该模型的逆参数辨识兼具高效率与高精度。