A Convex-Inspired Neural Construction for Structured and Generalizable Nonlinear Model Reduction

Abstract: Real-time simulation of deformable objects relies on model reduction to achieve interactive performance while maintaining physical fidelity. Traditional linear methods, such as principal component analysis (PCA), provide structured and predictable behavior thanks to their linear formulation, but are limited in expressiveness. Nonlinear model reduction, typically implemented with neural networks, offers richer representations and higher compression; however, without structural constraints, the learned mappings often fail to generalize beyond the training distribution, leading to unstable or implausible deformations. We present a symmetric, convex-inspired neural formulation that bridges the gap between linear and nonlinear model reduction. Our approach adopts an input-convex neural network (ICNN) augmented with symmetry constraints to impose structure on the nonlinear decoder. This design retains the flexibility of neural mappings while embedding physical consistency, yielding coherent and stable displacements even under unseen conditions. We evaluate our method on challenging deformation scenarios involving forces of different magnitudes, inverse directions, and sparsely sampled training data. Our approach demonstrates superior generalization while maintaining compact reduced spaces, and supports real-time interactive applications.