Given an unknown dynamical system, what is the minimum number of samples needed for effective learning of its governing laws and accurate prediction of its future evolution behavior, and how to select these critical samples? In this work, we propose to explore this problem based on a design approach. Starting from a small initial set of samples, we adaptively discover critical samples to achieve increasingly accurate learning of the system evolution. One central challenge here is that we do not know the network modeling error since the ground-truth system state is unknown, which is however needed for critical sampling. To address this challenge, we introduce a multi-step reciprocal prediction network where forward and backward evolution networks are designed to learn the temporal evolution behavior in the forward and backward time directions, respectively. Very interestingly, we find that the desired network modeling error is highly correlated with the multi-step reciprocal prediction error, which can be directly computed from the current system state. This allows us to perform a dynamic selection of critical samples from regions with high network modeling errors for dynamical systems. Additionally, a joint spatial-temporal evolution network is introduced which incorporates spatial dynamics modeling into the temporal evolution prediction for robust learning of the system evolution operator with few samples. Our extensive experimental results demonstrate that our proposed method is able to dramatically reduce the number of samples needed for effective learning and accurate prediction of evolution behaviors of unknown dynamical systems by up to hundreds of times.

翻译：给定一个未知动力系统，有效学习其控制法则并准确预测其未来演化行为所需的最小样本数是多少？如何选取这些关键样本？本文提出基于设计方法探索该问题。从少量初始样本出发，我们自适应地发现关键样本，以实现对系统演化越来越精确的学习。此处的核心挑战在于：由于真实系统状态未知，我们无法获知网络建模误差，而后者正是关键采样所需的关键信息。为应对这一挑战，我们引入一种多步互逆预测网络，其中前向与后向演化网络分别用于学习时间维度上前向与后向的演化行为。有趣的是，我们发现所需的网络建模误差与可直接从当前系统状态计算的多步互逆预测误差高度相关，从而能够对动力系统中网络建模误差较高的区域进行动态关键采样。此外，我们引入联合时空演化网络，将空间动力学建模融入时间演化预测，从而实现利用少量样本对系统演化算子的鲁棒学习。大量实验结果表明，所提方法可将有效学习与准确预测未知动力系统演化行为所需的样本数减少高达数百倍。