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GIOROM

Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reduction

Transactions on Machine Learning Research (TMLR) 2026 · Expert Certification

Hrishikesh Viswanath¹ Yue Chang² Aleksey Panas² Julius Berner³ Peter Yichen Chen⁴ Aniket Bera¹

¹Purdue University ²University of Toronto ³NVIDIA ⁴Massachusetts Institute of Technology

Paper arXiv Code BibTeX

GIOROM overview — **Geometry-informed reduced-order modeling in physical space.** Rather than evolving a global latent representation, GIOROM advances a reduced set of Lagrangian particles directly in physical space with a neural autoregressive PDE operator $\phi_\Theta$, which predicts accelerations $\mathbf{A}_t$ from a short history of velocities $\mathbf{V}_{t-w:t}$. A learnable kernel-based ROM parametrizes the solution manifold, so the field can be queried at arbitrary locations from local geometric information around the evolved particles.

TL;DR

GIOROM simulates Lagrangian systems by evolving only a sparse sample of the particles with a data-driven neural PDE operator, instead of a global latent state. A learnable kernel then reconstructs the full field at any location from the evolved particles, cutting active degrees of freedom by 6.6×–32× while staying accurate on fluids, granular media and elastoplastic materials.

6.6–32×fewer active degrees of freedom than the full particle system

2–3.5×faster 200-step rollouts than GNS on the same graph

7 MBconstant structure memory, vs. up to 1.37 GB for graph kernels

≈2Mparticles reconstructed from a 1–3% sample

Sampling-Based Reduction

Classical ROMs project the state onto a fixed global basis and evolve its coefficients with reduced forms of the governing equations; neural-field ROMs then decode the latent state back to the full field. These global latents struggle with complex, fast-evolving local dynamics such as fluids, and still need the PDE to time-step. GIOROM instead evolves a small set of representative particles embedded in the physical domain with a data-driven Lagrangian solver, which preserves locality and needs no explicit governing equations, then reconstructs the full field from those particles with a discretization-invariant kernel.

Global latent ROM vs. GIOROM — (a) Prior discretization-invariant methods evolve a global latent state $\hat{\mathbf{x}}$ with PDE time-steppers and decode the full field with a neural decoder $g$, which requires the explicit PDE. (b) GIOROM uses a data-driven neural stepper to compute sparse field estimates $\{\hat{u}^i\}_{i=1}^r$, from which a kernel $\kappa$ parameterizes the solution manifold over local spatial regions.

Method

Kernel-integral ROM

Instead of a projection onto a global basis, the reduced-order model is a continuous integral operator with a learnable kernel $\psi$. Given particle samples $v \approx u$ produced by the time-stepper, the field at a query point $x$ is a normalized, locally weighted aggregation of nearby samples, a continuous analogue of Nadaraya–Watson kernel regression:

$$u(x,t) = \frac{\int_\Omega \psi(x, y)\, v(y,t)\, d\mu(y)}{\int_\Omega \psi(x, y)\, d\mu(y)} .$$

Evaluating this over radius graphs scales poorly, so the samples are projected onto a fixed grid with trilinear basis functions, giving a feature field $F_{grid}$ and a density field $D_{grid}$, and the operator is approximated by stochastic interpolation around $x$, where $\xi \sim \mathcal{N}(0, \sigma^2 I)$ acts as a Monte Carlo Gaussian smoothing kernel:

$$\bar{u}(x) = \frac{\mathbb{E}_{\xi}\left[\mathcal{I}(F_{grid}, x + \xi)\right]}{\mathbb{E}_{\xi}\left[\mathcal{I}(D_{grid}, x + \xi)\right] + \epsilon} .$$

Time-stepping on the reduced sample set

The sparse particles are advanced by a neural PDE operator that adapts the UPT encode–process–decode architecture to autoregressive particle dynamics, with interaction-network encoder and decoder on a radius graph over the sample and a transformer processor in latent space: $\phi_\Theta := IN^{dec} \circ NOT^{process} \circ IN^{enc}$. From a window of velocities $\mathbf{v}_{t-w:t}$ it predicts per-particle accelerations, and positions follow by Euler integration. Mean aggregation in message passing makes the graph layers behave like a Monte Carlo approximation of kernel integrals, so the operator stays consistent as the sample changes; training on several random subgraphs per domain makes it invariant to the sampling ratio. The framework is agnostic to the choice of time-stepper.

Single-step flow diagram — One step of inference: kernel-integral ROM, time-stepping on the sample, and reconstruction of the next state.

Results

Reconstruction against neural ROMs

ROM comparison on fluid in a container — Fluid in a container. PCA (rollout MSE 0.083), autoencoder (0.091), LiCROM (0.033) and CROM (0.079) drift from the true fluid boundary, overshooting the container; GIOROM stays faithful at 0.0091.

Discretization convergence on 2 million particles — Discretization convergence on ≈2 million particles: from random samples of 62K and 20K particles (3% and 1%), rollout MSE is 0.0097 and 0.0088.

Rel. $L_2$ error (%) ↓	Plasticine-3D	Sand-3D	Water-3D	Elasticity-3D
PCA	16.15	13.37	15.84	39.41
DINo	31.83	45.36	58.97	41.18
CORAL	21.85	25.80	53.71	46.24
CoLoRA	9.08	12.31	54.54	9.83
CROM	3.19	4.21	24.00	4.25
LiCROM	2.31	3.44	22.01	3.43
GKI	3.69	8.46	15.16	7.66
GIOROM	1.98	2.68	9.24	2.04

Unseen trajectories, 32 effective active degrees of freedom. GIOROM's peak memory is 17–268 MB, against 462–2970 MB for the graph-kernel baseline GKI and ≈400–630 MB for the latent-ODE ROMs.

Effect of the reduced space

Error against reduced-space size per regime — Error against reduced-space size, from highly dynamic fluids (Water-3D) to minimal deformation (Plasticine-3D) and elasticity. GIOROM matches global bases on topologically stable regimes; below about 8 effective degrees of freedom the Lagrangian sample is so sparse that kernel neighbourhoods go empty and accuracy drops.

Against a full-order solver

System	$N$	$r$	Scale	Rollout MSE (Ours)	Rollout MSE (GNS)
Water-3D	55k	1.7k	32×	38.6	17.4
Water-2D	1k	0.12k	8.3×	6.70	7.55
Sand-3D	32k	1k	32×	0.25	0.31
Sand-2D	2k	0.3k	6.6×	1.34	1.21
Goop-2D	1.9k	0.2k	9.5×	0.94	0.55
Plasticine	5k	1.1k	4.5×	0.50	0.71
Elasticity	78k	2.6k	30×	0.20	0.30
Multi-material 2D	2k	0.25k	8×	9.43	12.1

Rollout MSE (×10⁻³) computed on the full particle set, with GIOROM evolving a graph up to 32× smaller than GNS. Errors stay within the same order of magnitude as the full-order oracle.

200-step rollout, 7056 points	radius 0.040	0.060	0.080	0.100
GIOROM	20.1 s	47.6 s	89.7 s	109.3 s
GNS	43.5 s	111.6 s	226.2 s	386.0 s

A transformer processor replaces recursive message passing, so the speedup widens from about 2× to 3.5× as connectivity grows.

Scalability

Memory and time overheads of graph kernels vs. grid projection — (a) A radius-graph integral transform grows from 2.8 MB to 1.37 GB as the radius increases, while grid projection stays at 7 MB. (b, c) As the particle count grows, graph construction dominates GKI's runtime (97% at 35K particles) and its memory grows linearly; grid projection stays nearly flat.

Effect of sparsity on rollout error — Elasticity, 78K particles. Rollout error stays near $10^{-4}$ from 0.125× down to 0.031× of the full system, i.e. up to 32× compression, and degrades only below that; at 0.007×, raising the graph radius past 0.15 no longer helps. Time and peak GPU memory include rebuilding the graph at every step.

Rollouts

Operator on the sparse sample

→

Kernel ROM upsampling

→

Rendered simulation

full-order

vs. ground truth yellow: ground truth

reduced-order

Long-horizon rollouts — Rollouts until equilibrium. Top: water, 1000 steps over 55K particles (MSE $1.1\times10^{-2}$). Middle: elastic collisions (jelly), 500 steps over 84K particles ($6.4\times10^{-5}$). Bottom: phase transition in a plasticine-like material (chocolate), 2000 steps over 24K particles ($3.2\times10^{-4}$).

Limitations

In ultra-sparse regimes, when the sampled particle count leaves almost no active grid voxels, the estimator becomes ill-conditioned and the network falls back on its prior without local physical guidance. Like other learned ROMs, GIOROM degrades under extreme out-of-distribution conditions. It currently targets continuous systems with consistent time discretizations and Euler integration; discontinuities, unbounded flows, higher-order integrators and adaptive time-stepping are left to future work.

Citation

@article{viswanath2026learning,
  title   = {Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reduction},
  author  = {Hrishikesh Viswanath and Yue Chang and Aleksey Panas and Julius Berner and Peter Yichen Chen and Aniket Bera},
  journal = {Transactions on Machine Learning Research},
  issn    = {2835-8856},
  year    = {2026},
  url     = {https://openreview.net/forum?id=vXCQA1EzaG},
  note    = {Expert Certification}
}