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：针对未知动力系统，有效学习其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，以及如何选择这些关键样本？本文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，以实现对系统演化的逐步精确学习。核心挑战在于：由于真实系统状态未知，网络建模误差无法获知，而这正是关键采样所必需的。为解决该问题，我们提出多步互逆预测网络，其中前向与反向演化网络分别学习时间维度的正向与反向演化行为。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。这使得我们能够从高网络建模误差区域动态选择关键样本。此外，通过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，从而以少量样本实现系统演化算子的鲁棒学习。大量实验结果表明，所提方法能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量显著降低数百倍。