Speaker
Description
Recent developments have shown that machine learning methods, such as neural networks, can significantly benefit from incorporating physical knowledge [2]. However, a correct and non-restrictive implementation remains a major challenge [1,3,4]. Classic invariants of the right Cauchy-Green tensor, for instance, are a sufficient, yet overly restrictive choice for the combination of objectivity, isotropy and polyconvexity. This work thus aims to incorporate physical and mathematical constraints into neural networks without limiting the required solution space, specifically in the context of isotropic hyperelasticity. The two key components of the approach are an input convex network architecture and a parametrization based on the signed singular values of the deformation gradient. This enables to rigorously capture frame indifference and polyconvexity, together with other physical constraints. A highly beneficial feature of the proposed design is its compliance with the universal approximation theorem. More precisely, the architecture can approximate any isotropic polyconvex hyperelastic energy, provided it is sufficiently large. This is achieved by employing a necessary and sufficient criterion for polyconvexity under the assumption of objectivity and isotropy. The benefits and unique aspects of the new approach will be demonstrated and discussed through the approximation of a polyconvex energy, a non-polyconvex energy, and the construction of a polyconvex hull - an outcome that cannot be achieved using regular architectures.
[1] K. Linka et al., Constitutive artificial neural networks: A fast and general approach to predictive data-driven constitutive modeling by deep learning, J. Comput. Phys., 429 (2021), 110010.
[2] B. Moseley, Physics-informed machine learning: from concepts to real-world applications. Phd Thesis, University of Oxford (2022).
[3] L. Linden et al., Neural networks meet hyperelasticity: A guide to enforcing physics, J. Mech. Phys. Solids, 179 (2023), 105363.
[4] G.-L. Geuken et al., Incorporating sufficient physical information into artificial neural networks: A guaranteed improvement via physics-based Rao-Blackwellization. Comput. Method. in Appl. M., 423 (2024), 116848.
