Complex systems often show macroscopic coherent behavior due to the interactions of microscopic agents like molecules, cells, or individuals in a population with their environment. However, simulating such systems poses several computational challenges during simulation as the underlying dynamics vary and span wide spatiotemporal scales of interest. To capture the fast-evolving features, finer time steps are required while ensuring that the simulation time is long enough to capture the slow-scale behavior, making the analyses computationally unmanageable. This paper showcases how deep learning techniques can be used to develop a precise time-stepping approach for multiscale systems using the joint discovery of coordinates and flow maps. While the former allows us to represent the multiscale dynamics on a representative basis, the latter enables the iterative time-stepping estimation of the reduced variables. The resulting framework achieves state-of-the-art predictive accuracy while incurring lesser computational costs. We demonstrate this ability of the proposed scheme on the large-scale Fitzhugh Nagumo neuron model and the 1D Kuramoto-Sivashinsky equation in the chaotic regime.

翻译：复杂系统通常由于微观主体（如分子、细胞或群体中的个体）与其环境的相互作用而表现出宏观相干行为。然而，模拟此类系统在仿真过程中面临若干计算挑战，因为其底层动力学存在差异且跨越了广泛的时空尺度。为捕捉快速演化的特征，需要采用更精细的时间步长，同时确保仿真时间足够长以捕获慢尺度行为，这使得分析在计算上难以处理。本文展示了如何利用深度学习技术，通过坐标与流映射的联合发现，为多尺度系统开发一种精确的时间步进方法。前者使我们能够在代表性基上表示多尺度动力学，而后者则能对约简变量进行迭代的时间步进估计。所得框架在实现最先进预测精度的同时，计算成本更低。我们在大规模Fitzhugh-Nagumo神经元模型和混沌状态下的1D Kuramoto-Sivashinsky方程上验证了所提方案的这一能力。