Reduced-Order Neural Operators: Learning Lagrangian Dynamics on Highly Sparse Graphs

TL;DR

GIOROM tackles the costly problem of simulating Lagrangian dynamics on high-resolution domains by introducing a data-driven, discretization-invariant reduced-order framework. It learns a neural operator that maps sparse observations on graphs to pointwise field values and then uses a neural kernel to interpolate to arbitrary query points, eliminating the need for full PDE solvers or explicit projection-based decoders. A neural time-stepper on sparse graphs (phi_Theta) evolves a reduced latent representation by predicting accelerations from a window of past velocities, while a kernel-ROM interpolates the full-field deformation at query locations. The approach achieves substantial input-size reductions (6.6× to 32×) while maintaining high fidelity across fluids, granular media, and elastoplastic dynamics, and it demonstrates strong generalization across discretizations and sampling densities, outperforming conventional ROM baselines. This data-driven, discretization-invariant framework reduces computational cost and extends the applicability of ROMs to complex, multi-physics Lagrangian systems with minimal PDE priors.

Abstract

Simulating complex physical systems governed by Lagrangian dynamics often requires solving partial differential equations (PDEs) over high-resolution spatial domains, resulting in substantial computational costs. We present GIOROM (\textit{G}raph \textit{I}nf\textit{O}rmed \textit{R}educed \textit{O}rder \textit{M}odeling), a data-driven discretization invariant framework for accelerating Lagrangian simulations through reduced-order modeling (ROM). Previous discretization invariant ROM approaches rely on PDE time-steppers for spatiotemporally evolving low-dimensional reduced-order latent states. Instead, we leverage a data-driven graph-based neural approximation of the PDE solution operator. This operator estimates point-wise function values from a sparse set of input observations, reducing reliance on known governing equations of numerical solvers. Order reduction is achieved by embedding these point-wise estimates within the reduced-order latent space using a learned kernel parameterization. This latent representation enables the reconstruction of the solution at arbitrary spatial query points by evolving latent variables over local neighborhoods on the solution manifold, using the kernel. Empirically, GIOROM achieves a 6.6$\times$-32$\times$ reduction in input dimensionality while maintaining high-fidelity reconstructions across diverse Lagrangian regimes including fluid flows, granular media, and elastoplastic dynamics. The resulting framework enables learnable, data-driven and discretization-invariant order-reduction with reduced reliance on analytical PDE formulations. Our code is at \href{https://github.com/HrishikeshVish/GIOROM}{https://github.com/HrishikeshVish/GIOROM}

Figures (19)

• Figure 1: Overview of GIOROM. This figure illustrates the temporal evolution of Lagrangian systems using reduced-order modeling. (a) Prior discretization invariant methods evolve a global latent state $\hat{\mathbf{x}}$ using PDE time-steppers and reconstruct the full-field solution using neural decoders $g$. (b) In contrast, our approach employs a data-driven neural stepper to compute sparse field estimates $\{\hat{{\boldsymbol{f}}}^i\}_{i=1}^r \in \mathbb{R}^{r \cdot d}$, which are then used by the kernel-ROM $\kappa$ to parameterize the local solution manifold. ${\boldsymbol{f}}$ represents the deformation field depicted above.
• Figure 2: (a) Comparison of reduced-order modeling (ROM) techniques on fluid-in-container simulations: The visualization reveals that the baseline models PCA (rollout MSE: 0.083), Autoencoder (0.091), LiCROM (0.033), and CROM (0.079), exhibit noticeable deviation from the true fluid boundaries, including overshooting beyond the container. GIOROM maintains physical fidelity with a lower rollout MSE of 0.0091. (b) Demonstration of discretization convergence of kernel-ROM on high-resolution particle systems: Contrasting the behavior of the kernel-ROM on a large-scale simulation containing approximately 2 million particles, discretized as voxelized grids. Using randomly sampled $(r\cdot d)$-points of 62K and 20K particles (sampling percentages of 3% and 1%), the rollout MSEs are 0.0097 and 0.0088, respectively. These results indicate discretization convergence of $\kappa$ under resolution refinement.
• Figure 3: The overall architecture of GIOROM. The neural time-stepper $\phi_\Theta$ (Encoder-Processor-Decoder) predicts the acceleration of a Lagrangian system $\mathbf{A}_{t}$ at time $t$ from the past $w$ velocity instances $\mathbf{V}_{t-w:t}$. From the predicted accelerations, we leverage standard Euler integration to obtain the predicted positions. The rightmost kernel-ROM is used to efficiently evaluate the deformation field at arbitrary locations. This is summarized in the flow diagram \ref{['fig:full-flow']}
• Figure 4: Effect of sparsity on rollout error (Elasticity dataset, 78K particles). Plots show performance of $\phi_\Theta$ across varying sparsity levels, expressed as a fraction of the full system. GPU usage and computation time include pre-processing overhead for graph construction at each rollout step. Top-left plot reports peak GPU usage across graph sizes (R=radius, S=sample ratio); top-right shows net computation time for a rollout as a function of sparsity at fixed radius. Bottom-left plot reports rollout MSE, showing that performance remains stable ($\sim$1e-4) for sparsity levels 0.125×, 0.062×, and 0.031×. Below this, performance degrades due to oversparsification. Bottom-right plot shows that for high sparsification (0.007×), increasing the graph radius beyond 0.15 does not improve performance.
• Figure 5: Rollout trajectories for three physical systems evaluated until equilibrium. (Top) Water simulation with high dynamism, rolled out for 1000 time steps over 55K particles; rollout MSE: $1.1 \times 10^{-2}$. (Middle) Elastic collisions (Jelly) over 500 time steps with 84K particles; rollout MSE: $6.4 \times 10^{-5}$. (Bottom) Phase transition in plasticine-like material (Chocolate) over 2000 time steps with 24K particles; rollout MSE: $3.2 \times 10^{-4}$.