Title: Critical Sampling for Robust Evolution Behavior Learning of Unknown Dynamical Systems. Abstract: We study the following new and important problem: 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. Specifically, starting from a small initial set of samples, we adaptively discover and collect 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 of the ground-truth system state, which is however needed for critical sampling. To address this challenge, we introduce a multi-step reciprocal prediction network where a forward evolution network and a backward evolution network are designed to learn and predict 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. More importantly, this multi-step reciprocal prediction error can be directly computed from the current system state without knowing the ground-truth or data statistics. This allows us to perform dynamic selection of critical samples from regions with high network modeling errors and develop an adaptive sampling-learning method for dynamical systems. To achieve accurate and robust learning from this small set of critical samples, we introduce a joint spatial-temporal evolution network 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 numbers of samples needed for effective learning and accurate prediction of evolution behaviors of unknown dynamical systems by up to hundreds of times, especially for high-dimensional dynamical systems.