OnlyDense: Reduced-Order Modeling for Lagrangian simulation
2026-06-08 • Machine Learning
Machine LearningArtificial Intelligence
AI summaryⓘ
The authors developed a new way to model large systems of particles, like those in fluid or material simulations, that is faster and more scalable than traditional methods. Instead of tracking each particle individually, they represent the whole system as a combination of learned basic patterns (basis functions) that evolve over time in a mathematical space called Hilbert space. This approach lets them predict complex behaviors accurately using far fewer parameters, making large simulations more efficient. Their tests showed they could reconstruct and predict particle dynamics very well, even with over a million particles involved.
Lagrangian simulationSmooth Particle Hydrodynamics (SPH)Material Point Method (MPM)Hilbert spaceBasis functionsReduced-order modelingProper Orthogonal DecompositionDeep learningDynamic systemsTrajectory modeling
Authors
Tu Do, Shannon Ryan, Santu Rana
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.