Neural networks model complex elastic energies with guaranteed symmetry and convexity
Universal neural network approximation of symmetric polyconvex energies and envelopes
Computational Engineering, Finance, and ScienceMachine Learning
Summary
Modeling materials that stretch and deform can be tricky because their internal energy must follow special mathematical rules to behave realistically. The authors developed a new kind of neural network that always respects these rules about symmetry and smoothness in how materials respond to strain. This approach makes it easier and faster to approximate complicated energy functions that describe material behavior accurately. They proved mathematically that their network can represent any such energy and showed examples where it predicts stresses and energy shapes well.
What this means in practice
- •For elasticity modelers: Create neural network models that guarantee physically realistic material energy behavior for simulating elastic deformations.
- •For computer graphics engineers: Use symmetric polyconvex neural networks to generate stable and realistic deformation animations based on learned energy functions.
Authors
Florian Gammer
Abstract
Symmetric polyconvexity provides a sufficient condition for weak lower semicontinuity in small-strain elasticity and a tractable lower bound for the relaxation of non-(quasi)convex energies. Since classical algorithms for computing such functions can be computationally expensive, neural network approximations have recently emerged as an alternative. A central difficulty with establishing such neural network based approaches for symmetric polyconvexity is the additional monotonicity property. As we show, the corresponding weight restrictions on ReLU-based input-convex networks do not yield universal approximation of the lifted representers. Here, we introduce symmetric polyconvex neural networks (SPCNNs) for constitutive modeling and approximation of symmetric polyconvex envelopes. The deep architecture combines convexity with the monotonicity required of a representer in the space of minors. We prove a universal approximation theorem for symmetric polyconvex energies, without requiring isotropy, and quantify the number of affine pieces attainable by the lifted network and derive reduced architectures for isotropic energies. Numerical examples illustrate stress-based regression and approximation of the symmetric polyconvex envelope of a double-well potential. Symmetric polyconvexity is guaranteed by construction, while the accuracy of the fitted energies and envelopes is assessed numerically.