Neural-Manifold Primal-Dual Model Order Reduction and Hyper-Reduction for Nonlinear Parametric Variational Inequalities

Abstract

We propose a model order reduction framework for nonlinear parametrized variational inequalities arising in computational mechanics. The high-dimensional model is written in mixed primal-dual form with projection-based complementarity conditions, leading to nonlinear nonsmooth algebraic systems solved by a semi-smooth Newton method. On this basis, reduced models are constructed by proper orthogonal decomposition (POD) of both primal and dual solution snapshots. To address cases where low-dimensional linear spaces provide limited approximation efficiency, we introduce a neural-network-augmented reduced model. Two feedforward networks learn corrections in the truncated POD coordinates of the primal and dual variables from solutions of enlarged Galerkin ROMs. The learned reconstructions enter the coupled equilibrium and constraint equations directly, defining a nonlinear reduced system solved by approximate semi-smooth Newton iterations. The online cost associated with high-dimensional residual evaluations is reduced through hyper-reduction, using a sparse cubature approach based on greedy nonnegative least squares. The proposed methodology is assessed on two nonlinear variational inequalities with distinct sources of nonlinearity: a two-dimensional obstacle problem with a cubic nonlinearity in the state equation, and a three-dimensional frictional contact problem in which the Coulomb law induces a nonlinear projection in the constraint equation. Numerical results compare the high-dimensional model, the linear reduced model, the neural-network-augmented reduced model, and their hyper-reduced variants, demonstrating accurate approximations with substantial reductions in online computational cost. The performance study also evaluates feasibility over the complete constraint set and measures how errors in the reconstructed obstacle and contact forces affect the primal response.