Polyconvexity for Cosserat nonlinear elasticity and nonlinear couple-stress theory

What the math gives to ML

The paper provides a lifted representation of nonlinear deformation in which a rotation field and relative stretch are optimized separately: \(\overline U=\overline R^T D\varphi\). Its transferable asset is a polyconvex treatment of stretch using \((\overline U,\operatorname{Cof}\overline U,\det\overline U)\), together with an \(\mathrm{SO}(3)\)-valued curvature penalty that controls spatial variation of rotations. A neural analogue is a Jacobian-based deformation or flow module whose local orientation and distortion losses are evaluated in a learned rotation frame, using an input-convex energy over stretch minors rather than ad hoc Frobenius penalties. This could improve stability when training coordinate networks, implicit neural representations, or invertible deformation models near folding and ill-conditioned Jacobians.