SPCNNs approximate symmetric polyconvex energies by construction
Original title: Universal neural network approximation of symmetric polyconvex energies and envelopes
The authors prove universal approximation without isotropy and illustrate stress regression plus a double-well envelope.
Symmetric polyconvex neural networks combine convexity with the monotonicity needed for representers in the space of minors. The paper proves a universal approximation theorem for symmetric polyconvex energies without requiring isotropy and derives reduced architectures for isotropic cases. Numerical examples cover stress-based regression and the symmetric polyconvex envelope of a double-well potential; polyconvexity holds by construction.
- ReLU input-convex restrictions do not universally approximate lifted representers.
- The lifted network's number of affine pieces is quantified.
- Isotropy is not required for the approximation theorem.
- Envelope accuracy is assessed numerically, not in closed form.
From the original article
1 min read at arxiv.orgSymmetric 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.
By Florian Gammer · arxiv.org
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