Integration of Mechanical Constraints Into Deep Symbolic Optimization for Discovering Viscoelastic Models

The current issue of mathematically modelling the mechanical behavior of polymer materials produced using additive technologies, specifically thermoplastic polyurethane (TPU), is addressed in the paper. The widespread use of 3D printing to create flexible structures and biomedical implants necessitates accurate constitutive models that can describe complex nonlinear phenomena, such as hysteresis, stress relaxation, and strain rate dependence. Traditional phenomenological approaches, based, for example, on hyperelastic potentials (Mooney-Rivlin, Ogden), have significant limitations: the need for a priori selection of the model structure, such as fixing the number of relaxation branches, and the complexity of identifying a large number of parameters. This often leads to the problem of solution inconsistency, which complicates the prediction of material behavior under conditions of complex anisotropy caused by layer-by-layer printing. On the other hand, modern machine learning methods, such as neural networks, although providing high approximation accuracy, act as “black boxes” and do not provide physically interpretable dependencies suitable for further analysis. The goal of this work is to develop a method for the automated search for determining equations of viscoelasticity that combines the flexibility of data-driven approaches with physical meaningfulness. We propose the use of a modified deep symbolic optimization (DSO) framework with the integration of “mechanical content.” The algorithm searches for the analytical form of the elastic deformation potential and relaxation function. Applying the method to experimental data has yielded a compact determining equation that provides high approximation accuracy and accurately describes the relaxation behavior of 3D-printed materials. The results indicate the promise of symbolic regression for automating the creation of transparent (“white-box”) material models in finite element analysis.