Neural Network Enhanced Polyconvexification of Isotropic Energy Densities in Computational Mechanics

TL;DR

The paper addresses the challenge of computing parameter-dependent polyconvex envelopes for isotropic energy densities in computational mechanics, where nonconvex damage models impede robust simulation. It proposes a neural-network framework that encodes polyconvexity by operating on the minors of signed singular values via ICNN-based architectures (FICNN and PICNN) and enforces symmetry and upper-bound properties through penalty terms and data augmentation. The approach is validated on mathematical benchmarks (Kohn–Strang–Dolzmann family and generalisations) and engineering isotropic damage models, achieving 2–4% envelope accuracy with orders-of-magnitude speedups compared to traditional SVPC-based methods, and enabling real-time, parameter-dependent polyconvexification in large-scale problems. The splitting technique further reduces parameter dimensionality, improving training feasibility and robustness, with evidence of good extrapolation to unseen parameter values. The work paves the way for real-time, structure-preserving relaxation in computational mechanics and hints at future extensions for derivatives and determinant constraints.

Abstract

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). Polyconvex underestimation is weakly enforced by penalisation during training, as are the symmetries of the function. As a guiding example, we focus on a well-known isotropic damage problem, reformulated in terms of signed singular values, and apply a splitting approach to reduce the dimensionality of the parameter space, thereby making training more tractable. Numerical experiments show that the networks achieve sufficient 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.

Figures (14)

• Figure 1: A fully input convex neural network (FICNN).
• Figure 2: A partially input convex neural network (PICNN), coloured weights need to be non-negative to ensure convexity of the network in $\hat{m}$.
• Figure 3: Left: Network architecture for Kohn--Strang--Dolzmann example. Right: Learning curves based on the loss function $\mathcal{L}$ from \ref{['eq:loss']} for a single network initialisation.
• Figure 4: Comparison of the predicted polyconvex envelope $\Phi^{{\operatorname{pc}}}_{\rm pred}$ (averaged over ten network realisations) and the analytical polyconvex envelope $\Phi^{{\operatorname{pc}}}$ for the Kohn--Strang--Dolzmann function along two cross-sections. The standard deviation of the ten predictions is marked by $\sigma$.
• Figure 5: Left: Training data for the generalised Kohn--Strang--Dolzmann example. One-dimensional cross-section along the diagonal $(\nu_1, \nu_1$)-axis of the analytical polyconvex envelopes. Envelopes are evaluated at the training points in parameter space, i.e. for $(\lambda, \alpha) \in \{1, 1.2, \dotsc, 1.8, 2 \}^2$. Right: Learning curves based on the loss function $\mathcal{L}$ from \ref{['eq:loss']} for a single network initialisation.

Theorems & Definitions (6)

• Proposition 4.1
• Proposition 4.2
• Remark 4.3
• Remark 4.4: Symmetry a posteriori
• Remark 5.1
• Remark 6.1