This paper considers one of the fundamental parallel-in-time methods for the solution of ordinary differential equations, Parareal, and extends it by adopting a neural network as a coarse propagator. We provide a theoretical analysis of the convergence properties of the proposed algorithm and show its effectiveness for several examples, including Lorenz and Burgers' equations. In our numerical simulations, we further specialize the underpinning neural architecture to Random Projection Neural Networks (RPNNs), a 2-layer neural network where the first layer weights are drawn at random rather than optimized. This restriction substantially increases the efficiency of fitting RPNN's weights in comparison to a standard feedforward network without negatively impacting the accuracy, as demonstrated in the SIR system example.

翻译：本文研究了求解常微分方程的基本并行时间方法之一——Parareal算法，并通过采用神经网络作为粗粒度传播器对其进行了扩展。我们提供了所提出算法收敛性的理论分析，并展示了其在多个示例（包括洛伦兹方程和伯格斯方程）中的有效性。在数值模拟中，我们进一步将底层神经架构具体化为随机投影神经网络（RPNNs），这是一种两层神经网络，其第一层权重采用随机生成而非优化方式确定。如SIR系统示例所示，与标准前馈网络相比，这种限制显著提高了RPNN权重拟合的效率，且未对精度产生负面影响。