Deep Operator Network (DeepONet) is a neural network framework for learning nonlinear operators such as those from ordinary differential equations (ODEs) describing complex systems. Multiple-input deep neural operators (MIONet) extended DeepONet to allow multiple input functions in different Banach spaces. MIONet offers flexibility in training dataset grid spacing, without constraints on output location. However, it requires offline inputs and cannot handle varying sequence lengths in testing datasets, limiting its real-time application in dynamic complex systems. This work redesigns MIONet, integrating Long Short Term Memory (LSTM) to learn neural operators from time-dependent data. This approach overcomes data discretization constraints and harnesses LSTM's capability with variable-length, real-time data. Factors affecting learning performance, like algorithm extrapolation ability are presented. The framework is enhanced with uncertainty quantification through a novel Bayesian method, sampling from MIONet parameter distributions. Consequently, we develop the B-LSTM-MIONet, incorporating LSTM's temporal strengths with Bayesian robustness, resulting in a more precise and reliable model for noisy datasets.

翻译：深度算子网络（DeepONet）是一种用于学习非线性算子的神经网络框架，例如描述复杂系统的常微分方程（ODEs）中的算子。多输入深度神经算子（MIONet）扩展了DeepONet，允许在不同巴拿赫空间中包含多个输入函数。MIONet在训练数据集网格间距方面具有灵活性，且不限制输出位置。然而，它需要离线输入数据，无法处理测试数据集中变长的序列，限制了其在动态复杂系统中的实时应用。本研究重新设计了MIONet，集成长短期记忆网络（LSTM）以从时间依赖数据中学习神经算子。该方法克服了数据离散化的约束，并利用了LSTM处理变长实时数据的能力。本文还分析了影响学习性能的因素，如算法外推能力。通过一种新颖的贝叶斯方法，从MIONet参数分布中采样，增强了框架的不确定性量化能力。最终，我们开发了B-LSTM-MIONet，融合了LSTM的时间处理优势与贝叶斯鲁棒性，为噪声数据集提供了一个更精确、更可靠的模型。