Odd-DC: Generalizable Neural Model Reduction via Odd Difference-of-Convex Structure

Abstract

Model reduction is essential for real-time simulation of deformable objects. Linear techniques such as PCA provide structured and predictable behavior, but their limited expressiveness restricts accuracy under large or nonlinear deformations. Nonlinear model reduction with neural networks offers richer representations and higher compression; however, without structural constraints, the learned mapping from latent coordinates to displacements often generalizes poorly beyond the training distribution. We present an odd difference-of-convex (DC) neural formulation that bridges linear and nonlinear model reduction. Our goal is to obtain a latent space that behaves reliably under unseen load magnitudes and directions. To improve extrapolation in magnitude, we introduce convexity into the decoder to discourage oscillatory responses. Yet convexity alone cannot represent the odd symmetry required by many symmetric systems, which is crucial for generalization to inverse force directions. We therefore adopt a DC formulation that preserves the stabilizing effect of convexity while explicitly enforcing odd symmetry. Practically, we realize this structure using an input-convex neural network (ICNN) augmented with symmetry constraints. Across challenging deformation scenarios with varying magnitudes and reversed load directions, our method demonstrates stronger generalization than unconstrained nonlinear reductions while maintaining compact latent spaces and real-time performance. Our DC formulation extends to both mesh-based and neural-field reductions, demonstrating applicability across multiple classes of neural nonlinear model reduction.