Data driven modeling for self-similar dynamics

Multiscale modeling of complex systems is crucial for understanding their intricacies. In recent years, data- driven multiscale modeling has emerged as a promising approach to tackle challenges associated with complex systems. Still, at present,this field is more focused on the prediction or control problems in specific fields, and there is no suitable framework to help us promote the establishment of complex system modeling theory. 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. Our framework can discern whether the dynamics are self-similar to deterministic dynamics. For uncertain dynamics, it not only can judge whether it is self-similar or not, but also 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, providing valuable insights for the establishment of complex system modeling theory.