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：针对一个未知动力系统，为其有效学习控制规律并准确预测未来演化行为，所需的最少样本数量是多少，以及如何选取这些关键样本？在本文中，我们基于一种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，以实现对系统演化日益精确的学习。此处的一个核心挑战在于，由于真实系统状态未知，我们无法获知网络建模误差，而该误差恰是关键采样所需。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正向与反向时间方向上的时间演化行为。有趣的是，我们发现期望的网络建模误差与多步互逆预测误差高度相关，后者可直接根据当前系统状态计算得出。这使我们能够针对动力系统从网络建模误差较高的区域动态选取关键样本。此外，我们引入了一种联合时空演化网络，将空间动力学建模融入时间演化预测，以实现仅用少量样本即可对系统演化算子进行鲁棒学习。大量实验结果表明，我们的方法能够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量大幅减少，最高可达数百倍。