Multiscale modeling of complex systems is crucial for understanding their intricacies. Data-driven multiscale modeling has emerged as a promising approach to tackle challenges associated with complex systems. On the other hand, self-similarity is prevalent in complex systems, hinting that large-scale complex systems can be modeled at a reduced cost. In this paper, we introduce a multiscale neural network framework that incorporates self-similarity as prior knowledge, facilitating the modeling of self-similar dynamical systems. For deterministic dynamics, our framework can discern whether the dynamics are self-similar. For uncertain dynamics, it can compare and determine which parameter set is closer to self-similarity. The framework allows us to extract scale-invariant kernels from the dynamics for modeling at any scale. Moreover, our method can identify the power law exponents in self-similar systems. Preliminary tests on the Ising model yielded critical exponents consistent with theoretical expectations, providing valuable insights for addressing critical phase transitions in non-equilibrium systems.

翻译：复杂系统的多尺度建模对于理解其复杂性至关重要。数据驱动的多尺度建模已成为应对复杂系统相关挑战的一种有前景的方法。另一方面，自相似性在复杂系统中普遍存在，这表明大规模复杂系统可以以较低成本进行建模。本文提出了一种将自相似性作为先验知识的多尺度神经网络框架，从而促进自相似动力系统的建模。对于确定性动力学，该框架能够判断动力学是否具有自相似性；对于不确定性动力学，它能够比较并确定哪组参数更接近自相似性。该框架允许我们从动力学中提取尺度不变核，用于任意尺度的建模。此外，该方法还能识别自相似系统中的幂律指数。在伊辛模型上的初步测试得到了与理论预期一致的临界指数，为非平衡系统中的临界相变问题提供了有价值的洞见。