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Multiscale phenomena exhibit complex structure-function relationships, andpredicting their macroscopic behavior requires deducing differential equationsat different scales. The complexity of these equations and the number ofessential parameters make developing effective, predictive models challenging.To overcome this, researchers explore leveraging advanced numerical techniquesfrom artificial intelligence and machine learning. Here, we focus on afundamental aspect in multiscale phenomena, i.e the recognition of thehierarchical role of variables. By adopting a Pareto front interpretation, weaim to deduce simple and accurate relations for material modeling, startingfrom experimental multiscale analyses. From a physical point of view, the aimis to deduce information at higher scales from lower scales data, possiblyrespecting their hierarchical order. A crucial aspect of the proposed approachis the deduction of causality relations among the different variables to becompared with the available theoretical notions and possibly newinterpretations resulting by the data modelling. This result in a stepwiseapproximation going from data modelling to theoretical equations and back todata modelling. To demonstrate the key advantages of our multiscale numericalapproach, compared to classical, non-physically based data modellingtechniques, we consider the explicit example of spider silk, known for itsexceptional properties and bioinspiration potential. Indeed, it presents acomplex behavior resulting from mesostructures formed by the aggregation ofamino acids at the molecular scale. We argue that, due to the generality of ourresults, our approach may represent a proof of concept in many fields wheremultiscale, hierarchical differential equations regulate the observedphenomenon.

Hierarchical physically based machine learning in material science: the case study of spider silk