Neural Network Enhanced Polyconvexification of Isotropic Energy Densities in Computational Mechanics

We present a neural network approach for fast evaluation of parameter-dependent polyconvex envelopes, which are crucial in computational mechanics. Our method uses a neural network architecture that inherently encodes polyconvexity in the main variable by combining a feature extraction layer that computes the minors function on the signed singular value characterisation of isotropic energy densities with a Partially Input Convex Neural Network (PICNN). The envelope inequality is weakly enforced by penalisation during training, as are the symmetries of the function. As a guiding example, we focus on a pseudo time incremental variational damage problem, which is parameter-dependent on previous time-step iterates, the deformation gradient and the internal variable. This problem is reformulated in terms of signed singular values and a splitting approach is applied to reduce the dimension of the parameter space, thereby making training more tractable. Numerical experiments show that the networks achieve favourable accuracy for engineering applications while providing high compression and significant speed-up over traditional polyconvexification schemes. Most importantly, the network adapts to varying physical or material parameters, enabling real-time polyconvexification in large-scale computational mechanics scenarios.