Data driven modeling of self-similar dynamics

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.Comment: 11 pages,5 figures,1 tabl

μ-Benzene-1,2,4,5-tetra­carboxyl­ato-κ4 O 1,O 2:O 4,O 5-bis­[diaqua(phen­an­thro­line-κ2 N,N′)nickel(II)] 0.67-hydrate

The asymmetric unit of the title compound, [Ni2(C10H2O8)(C12H8N2)2(H2O)4]·0.67H2O, contains one complete binuclear complex and one half-mol­ecule, the latter being completed by crystallographic inversion symmetry, and 0.67 of a solvent water molecule. Each Ni2+ cation is coordinated by a 1,10-phenanthroline ligand, a bidentate benzene-1,2,4,5-tetra­carboxyl­ate (btec) tetra-anion and two water mol­ecules to generate a distorted cis-NiN2O4 octa­hedral coordination geometry. The btec species bridges the metal ions. In the crystal, the clusters and uncoordinated water mol­ecules are linked by O—H⋯O hydrogen bonds and π–π inter­actions [shortest centroid–centroid separation = 3.596 (2) Å] to form a three-dimensional network