We describe a recurrent neural network (RNN) based architecture to learn the flow function of a causal, time-invariant and continuous-time control system from trajectory data. By restricting the class of control inputs to piecewise constant functions, we show that learning the flow function is equivalent to learning the input-to-state map of a discrete-time dynamical system. This motivates the use of an RNN together with encoder and decoder networks which map the state of the system to the hidden state of the RNN and back. We show that the proposed architecture is able to approximate the flow function by exploiting the system's causality and time-invariance. The output of the learned flow function model can be queried at any time instant. We experimentally validate the proposed method using models of the Van der Pol and FitzHugh Nagumo oscillators. In both cases, the results demonstrate that the architecture is able to closely reproduce the trajectories of these two systems. For the Van der Pol oscillator, we further show that the trained model generalises to the system's response with a prolonged prediction time horizon as well as control inputs outside the training distribution. For the FitzHugh-Nagumo oscillator, we show that the model accurately captures the input-dependent phenomena of excitability.

翻译：我们提出了一种基于循环神经网络（RNN）的架构，用于从轨迹数据中学习因果、时不变连续时间控制系统的流函数。通过将控制输入限制为分段常数函数，我们证明了学习流函数等价于学习离散时间动力系统的输入-状态映射。这一结论启发我们采用RNN，并结合将系统状态映射到RNN隐状态及其逆映射的编码器与解码器网络。我们证明了所提出的架构能够利用系统的因果性和时不变性来逼近流函数。该学习到的流函数模型可在任意时间点进行查询输出。我们利用Van der Pol和FitzHugh-Nagumo振荡器模型对提出的方法进行了实验验证。在两种情况下，结果均表明该架构能够精确复现这两个系统的轨迹。对于Van der Pol振荡器，我们进一步展示了训练后的模型可泛化至延长预测时间范围的系统响应以及训练分布之外的控制输入。对于FitzHugh-Nagumo振荡器，我们证明了该模型准确捕捉了依赖于输入的可兴奋性现象。