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 the current work, the S-DeepONet architecture is 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, which is the first in the literature using DeepONets. 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 a 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：时序深度算子网络（Deep Operator Network, DeepONet）结构在逼近复杂解算子方面展现出低泛化误差的巨大潜力。近期提出的时序DeepONet（S-DeepONet）通过在DeepONet的分支网络中引入时序学习模型，能够根据时间相关输入预测最终解。本研究对S-DeepONet架构进行扩展，通过改进分支网络与主干网络之间的信息融合机制，首次在文献中利用DeepONet同时预测演化历程中多个时间步的多分量矢量解。通过两个示例问题——瞬态流体流动与路径相关的塑性加载——验证了该模型处理不同物理问题的能力。此外，展示了利用遗传算法通过训练后的S-DeepONet模型进行逆参数识别的应用。在几乎所有测试案例中，仅使用3200个训练数据点，训练模型即可达到$R^2$值高于0.99且相对$L_2$误差低于10%的优异精度。包含仅比标量模型多0.4%参数的矢量S-DeepONet模型，能以与两个独立训练的标量模型相近的精度同时预测两个输出分量，训练时间减少20.8%。S-DeepONet的推理速度比直接数值模拟快至少三个数量级，且基于训练模型的逆参数识别过程兼具高效率与高精度。