Recurrent neural networks are a powerful means to cope with time series. We show how autoregressive linear, i.e., linearly activated recurrent neural networks (LRNNs) can approximate any time-dependent function f(t) given by a number of function values. The approximation can effectively be learned by simply solving a linear equation system; no backpropagation or similar methods are needed. Furthermore, and this is probably the main contribution of this article, the size of an LRNN can be reduced significantly in one step after inspecting the spectrum of the network transition matrix, i.e., its eigenvalues, by taking only the most relevant components. Therefore, in contrast to other approaches, we do not only learn network weights but also the network architecture. LRNNs have interesting properties: They end up in ellipse trajectories in the long run and allow the prediction of further values and compact representations of functions. We demonstrate this by several experiments, among them multiple superimposed oscillators (MSO), robotic soccer, and predicting stock prices. LRNNs outperform the previous state-of-the-art for the MSO task with a minimal number of units.

In RoboCup, several To assist automated learning of team behavior, we provide a large dataset generated using 10different leagues exist to emphasize specific research problems by using different kinds of the top participants in RoboCup 2016 or 2017. of robots and rules. There are different soccer While it is possible to use the simulator for robot leagues in the RoboCup with different types and learning, we also generate additional data that is sizes of hardware and software: small size, middle not normally available from playing other teams size, standard platform league, humanoid, 2D directly: We modified the simulator to record and 3D simulation (Kitano et al., 1997). In the data from each robots local perspective, that is, soccer simulation leagues (Akiyama et al., 2015), with the restricted views that depend on each the emphasis is on multi-robot team work with robots situation and actions, and also include partial and noisy information, in real-time.