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

TL;DR

The paper tackles the trade-off between the structured generalization of linear model reduction and the expressive power of nonlinear neural models for deformable object simulation. It introduces a symmetric convex-inspired decoder built from Input Convex Neural Networks (ICNNs) augmented with an odd-function constraint, enabling a nonlinear yet structured mapping from a reduced latent space to full-space displacements. By applying convexity to an intermediate latent stage and enforcing antisymmetry, the approach achieves stable extrapolation, better generalization to unseen loadings, and improved robustness under sparse data and limited cubature points, while remaining suitable for real-time interaction. Empirical results across unseen loading directions, magnitude variations, and collision scenarios demonstrate stronger generalization, faster convergence in many cases, and compact reduced spaces compared to traditional linear or vanilla nonlinear baselines. This framework offers a practical route to reliable, real-time reduced-order physics with potential extensions to energy-based learning and multi-stable systems.

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.

Figures (13)

• Figure 1: We regularize nonlinear model reduction through a convex-inspired functional design. The convex structure discourages oscillations in the displacement field, while an odd-symmetric construction captures the intrinsic symmetry of physical deformation. Together, these constraints yield smoothly varying displacements in the latent reduced space, even for coordinates outside the training set. We visualize the displacement norm and the corresponding full-space deformations mapped from points in the latent reduced space.
• Figure 2: Our convex-inspired structure builds upon the input-convex neural network (ICNN), in which the output is, by construction, convex with respect to the input. This convexity is ensured through non-negative weights and non-decreasing activation functions.
• Figure 3: We demonstrate the advantage of input convex neural networks (ICNNs) on the convex function $z = x^2 + y^2$. Both an ICNN and a standard MLP are trained on samples within a disk of radius 2 (dashed blue circle). The left plot shows the ground truth. The middle plot shows the ICNN prediction, which accurately captures the convexity and generalizes smoothly outside the training region. The right plot shows the MLP prediction, which fits well inside the training region but produces non-convex distortions outside.
• Figure 4: Overview of our construction. During training, the full-space deformation is first projected to a reduced space via an encoder. To reconstruct the deformation, we compute a nonlinear mapping from the reduced-space coordinates to an intermediate variable using an input-convex functional. This variable, of slightly higher dimensionality, is then augmented with an odd-symmetric construction to capture the intrinsic symmetry of the physical system. Finally, the augmented variable is linearly mapped back to the full-space displacement through the last linear layer.
• Figure 5: Ablation study comparing convexity regularization in the full space and the reduced space. Enforcing convexity in the reduced space achieves faster convergence and lower loss, benefiting from PCA-based initialization. In contrast, initialization is not feasible in the full-space formulation due to stricter non-negativity constraints on the weight matrices.