基于随机投影神经网络的时间并行求解方法
Parallel-in-Time Solutions with Random Projection Neural Networks
AI总结：
本文将经典Parareal时间并行方法的粗传播器替换为随机投影神经网络，理论分析了收敛性，在多个方程算例中验证了该方法在保证精度的同时可提升拟合效率。
摘要：
本文研究了用于求解常微分方程的基础时间并行方法之一Parareal（并行时间积分法），并通过采用神经网络作为粗传播器对其进行扩展。我们对所提算法的收敛特性提供了理论分析，并在包括Lorenz方程和Burgers'方程在内的多个算例上验证了其有效性。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）——一种两层神经网络，其第一层权重为随机抽取而非优化得到。如SIR系统算例所示，与标准前馈网络相比，这一限制大幅提升了RPNN权重的拟合效率，且不会对精度造成负面影响。
英文摘要：
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.
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