OnlyDense: Reduced-Order Modeling for Lagrangian simulation
Authors: Tu Do, Shannon Ryan, Santu Rana
Organizations: A2I2, Deakin University, Geelong, Victoria, Australia
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 R2 score above 0.99 with as few as 32 basis functions.
Pure Lagrangian neural simulators offer geometric flexibility and exact advection, making them well-suited for modeling moving domains and free surfaces. However, the absence of a fixed global reference frame introduces two severe limitations: a spatial bottleneck, in which model capacity is wasted on uniform regions because the dense particle neighborhoods required for stable gradients are applied indiscriminately, and rapid temporal drift, caused by purely local message passing that lacks a global anchor. Inspired by classical hybrid numerical solvers, we propose a Hybrid Lagrangian-Eulerian neural simulator that augments Lagrangian dynamics with an Eulerian representation. To address the spatial bottleneck, we introduce adaptive downsampling that eliminates kinematic redundancy, preserving micro-scale details on particles while aggregating compressed features onto Eulerian nodes to resolve large-scale dynamics. To counter temporal drift, we employ a cross-attention mechanism that queries these Eulerian features, using the fixed grid as a stable spatial anchor to correct trajectory deviations at every timestep. Comprehensive experiments show that this hierarchical, cross-attended design substantially suppresses error accumulation, establishing a new state-of-the-art for accuracy and rollout stability in Lagrangian fluid simulation.
A unified simulator that can model diverse physical phenomena without solver-specific redesign is a long-standing goal across simulation science. We present a learning-based particle simulator built on a single transformer architecture to model cloth, elastic solds, Newtonian and non-Newtonian fluids, granular materials, and molecular dynamics. Our model follows a prediction-correction design on a shared Lagrangian particle representation. An explicit predictor first advances particles under the known external forces, producing an intermediate state that captures externally driven motion but not inter-particle interactions. A learned corrector then predicts the residual position and velocity updates through three stages: a particle tokenizer that encodes local particle-particle, particle-boundary, and topology-guided interactions; a super-token encoder that hierarchically merges particle tokens into a compact set of super tokens via alternating self-attention and token merging; and a super-token decoder that lifts these super tokens back to particle resolution through cross-attention to predict per-particle position and velocity corrections. Progressive token merging reduces the attention cost at successive encoder layers by halving the token count at each level, and the decoder communicates through the compact super-token set rather than full particle-to-particle attention. Across the six dynamics categories, the same architecture generalizes to unseen materials, boundary configurations, initial conditions, and external forces. We further demonstrate downstream interactive control, inverse design, and learning from real-world manipulation data, reducing the need for per-phenomenon solver engineering.
We present a novel formulation for mesh-free, reduced-order simulation of deformable hyperelastic objects. Existing work in reduced-order elastodynamic simulation represents the input geometry by either meshes, which can be difficult to obtain due to challenges in scanning and triangulating complex shapes, or by neural fields that require per-shape optimization. We propose to adopt a Reproducing Kernel Particle Method (RKPM) representation, which enables the construction of reduced-order skinning weights by solving a generalized eigensystem on the Hessian matrix of the elastic energy. We demonstrate that this formulation not only leads to a 40x training speedup compared with the per-shape optimization of neural fields, but also achieves lower simulation error when evaluated against the converged results of finite element method. We show our simulation results on a wide variety of objects in different representations including meshes and Gaussian splats, as well as the application of our method in the downstream task of robot simulation.