Deep Operator Network (DeepONet), a recently introduced deep learning operator network, approximates linear and nonlinear solution operators by taking parametric functions (infinite-dimensional objects) as inputs and mapping them to solution functions in contrast to classical neural networks that need re-training for every new set of parametric inputs. In this work, we have extended the classical formulation of DeepONets by introducing sequential learning models like the gated recurrent unit (GRU) and long short-term memory (LSTM) in the branch network to allow for accurate predictions of the solution contour plots under parametric and time-dependent loading histories. Two example problems, one on transient heat transfer and the other on path-dependent plastic loading, were shown to demonstrate the capabilities of the new architectures compared to the benchmark DeepONet model with a feed-forward neural network (FNN) in the branch. Despite being more computationally expensive, the GRU- and LSTM-DeepONets lowered the prediction error by half (0.06\% vs. 0.12\%) compared to FNN-DeepONet in the heat transfer problem, and by 2.5 times (0.85\% vs. 3\%) in the plasticity problem. In all cases, the proposed DeepONets achieved a prediction $R^2$ value of above 0.995, indicating superior accuracy. Results show that once trained, the proposed DeepONets can accurately predict the final full-field solution over the entire domain and are at least two orders of magnitude faster than direct finite element simulations, rendering it an accurate and robust surrogate model for rapid preliminary evaluations.

翻译：深度算子网络（DeepONet）是一种近期提出的深度学习算子网络，通过将参数化函数（无限维对象）作为输入并将其映射为解函数，从而逼近线性和非线性解算子。与经典神经网络需针对每组新参数输入重新训练不同，本文通过引入序列学习模型（如门控循环单元（GRU）和长短期记忆网络（LSTM））扩展了传统的DeepONet结构，使得分支网络能够准确预测参数化和时变载荷历程下的解等高线图。通过两个算例（瞬态热传导问题和路径相关塑性加载问题），展示了新架构相较于以前馈神经网络（FNN）为分支网络的基准DeepONet模型的性能优势。尽管计算成本更高，GRU-和LSTM-DeepONet在热传导问题中将预测误差降低了一半（0.06% vs. 0.12%），而在塑性问题中误差降低了2.5倍（0.85% vs. 3%）。在所有案例中，所提出的DeepONet模型的预测$R^2$值均超过0.995，表明其具有卓越的精度。结果表明，训练完成后，所提出的DeepONet模型能准确预测整个域内的最终全场解，且其计算速度比直接有限元模拟快至少两个数量级，从而成为快速初步评估中精确且鲁棒的替代模型。