FreeForm: Reduced-Order Deformable Simulation from Particle-Based Skinning Eigenmodes

Tactile perception is key to dexterous manipulation, yet simulating high-resolution elastomer deformation remains computationally prohibitive. Finite element methods (FEM) deliver high fidelity but demand costly remeshing, while Material Point Methods (MPM) suffer from heavy particle-memory tradeoffs. We propose a {reduced-order neural simulation framework} that couples coarse-grained MPM dynamics with an implicit neural decoder to reconstruct sub-particle tactile details from compact latent states. The framework learns a continuous deformation manifold from paired high- and low-resolution simulations, enabling physically consistent, differentiable inference. Compared to the TacIPC, our method achieves over 65% faster simulation and {40% lower memory usage}, while maintaining better geometric fidelity. In tactile rendering and 3D surface reconstruction, our methods further improve accuracy by 25% and produce realistic depth images and surface mesh within a faster inference speed. These results demonstrate that the proposed reduced-order neural model enables high-detail, physically grounded tactile simulation with substantial efficiency gains for robotic interaction and optimization.

Physics-informed operator learning is an attractive candidate for surrogate modeling of microstructures, especially in multiscale finite-element simulations. Its practical use, however, is often limited by the high cost of loss evaluation. We address this bottleneck by combining the Equilibrium Neural Operator (EquiNO) with the QR-based discrete empirical interpolation method (Q-DEIM). EquiNO learns only the modal coefficients of reduced displacement-fluctuation and first Piola-Kirchhoff stress representations built from periodic and divergence-free bases, thereby enforcing periodicity and mechanical equilibrium by construction. Q-DEIM then identifies a small set of spatial points through a column-pivoted QR factorization of the stress basis and restricts constitutive evaluations during training to these points alone. This makes full-batch second-order optimization practical for three-dimensional representative volume elements (RVEs). Homogenized first Piola-Kirchhoff stresses are recovered directly from the offline-averaged reduced stress modes, without the need to reconstruct the full stress field at inference time. We validate the framework on two three-dimensional finite-strain hyperelastic RVEs. Q-DEIM reduces the per-step training cost by roughly three orders of magnitude relative to full-field loss evaluation, while reduced homogenization achieves speed-up factors of order $10^3$ to $10^4$ over direct full-field computations. Despite relying on only a small number of offline snapshot loading paths for basis construction, the method accurately interpolates and extrapolates both microscopic stress fields and homogenized stresses, with prediction quality improving systematically as more snapshots are added.

In science and engineering, Lagrangian simulation methods such as Smooth Particle Hydrodynamics (SPH) or Material Point Method (MPM) are often employed to study the behavior of dynamic systems. However, these methods can be prohibitively computationally expensive, particularly when simulating multi-scale spatial or temporal phenomena, e.g., void growth and coalescence within macro-scale geometries, structural failure of spacecraft components resulting from hypervelocity impact of space debris particles, etc. In contrast to graph-based methods, where the state of the system is understood as a discrete set of particles, we propose a learning framework for scalable representation and dynamics modeling of massive particle systems by treating the system state as a function and its evolution as a trajectory in Hilbert space. Rather than representing the state as a discrete set of particles or embedding it in a nonlinear latent manifold, we approximate the state space with a linear subspace spanned by learned neural basis functions. This parameterization enables direct projection to obtain latent coefficients and explicit access to the basis functions, avoiding optimization over a nonlinear latent space. The resulting representation admits a natural interpretation: latent variables correspond to coefficients in Hilbert space, and basis functions correspond to spatial modes, analogous to Proper Orthogonal Decomposition. The framework thus unifies classical projection-based reduced-order modeling with modern deep learning, while remaining invariant to the number of discretization points. Experiments on large-scale SPH simulations with over one million particles, including dynamic events with extreme deformation and fragmentation, demonstrate that the proposed method accurately reconstructs and predicts dynamics, achieving an R$^2$ score above $0.99$ with as few as $32$ basis functions.