OnlyDense: Reduced-Order Modeling for Lagrangian simulation

2026-06-08

AI Summary

This work addresses the high computational cost of Lagrangian particle methods—such as Smoothed Particle Hydrodynamics (SPH) and the Material Point Method (MPM)—in multiscale dynamic simulations. The authors model the state of particle systems as function trajectories in a Hilbert space and propose a linear subspace reduced-order method based on explicit neural basis functions. By eschewing graph structures or nonlinear latent manifolds, this approach unifies classical projection-based model reduction with deep learning frameworks while guaranteeing permutation invariance with respect to particle count. Using only 32 neural basis functions, the method achieves highly accurate dynamic reconstruction and prediction in million-particle SPH simulations, attaining R² > 0.99 and substantially improving computational efficiency.

Abstract

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.

Problem

• Lagrangian simulation
• reduced-order modeling
• computational cost
• multi-scale phenomena
• particle systems

Innovation

• reduced-order modeling
• neural basis functions
• Hilbert space representation
• Lagrangian simulation
• scalable dynamics learning