Learning Lagrangian Interaction Dynamics with Sampling-Based Model Order Reduction
Authors: Hrishikesh Viswanath
Yue Chang
Aleksey Panas
Julius Berner
Peter Yichen Chen
Aniket Bera
Venue: Accepted by TMLR ()

Abstract: Simulating physical systems governed by Lagrangian dynamics often entails solving partial differential equations (PDEs) over high-resolution spatial domains, leading to significant computational expense. Reduced-order modeling (ROM) mitigates this cost by evolving low-dimensional latent representations of the underlying system. While neural ROMs enable querying solutions from latent states at arbitrary spatial points, their latent states typically represent the global domain and struggle to capture localized, highly dynamic behaviors such as fluids. We propose a sampling-based reduction framework that evolves Lagrangian systems directly in physical space, over the particles themselves, reducing the number of active degrees of freedom via data-driven neural PDE operators. To enable querying at arbitrary spatial locations, we introduce a learnable kernel parameterization that uses local spatial information from time-evolved sample particles to infer the underlying solution manifold. Empirically, our approach achieves a 6.6$\times$–32$\times$ reduction in input dimensionality while maintaining high-fidelity evaluations across diverse Lagrangian regimes, including fluid flows, granular media, and elastoplastic dynamics. We refer to this framework as GIOROM (\textbf{G}eometry-\textbf{I}nf\textbf{O}rmed \textbf{R}educed-\textbf{O}rder \textbf{M}odeling). All of our code and data is available at \url{https://github.com/HrishikeshVish/GIOROM}

Published in Transactions on Machine Learning Research (02/2026)
Learning Lagrangian Interaction Dynamics with Sampling- Based Model Order Reduction
Hrishikesh Viswanath
hviswan @xwrruueeppuuuuccccmmm edu
Purdue University
Yue Chang University of Toronto
chanegue.changgemmallauoomm wtoromooooo..oooooooooooo
Aleksey Panas University of Toronto
oleksey, sonkeyyy.oons@@@uuummittttooooo
Julius Berner Nuidia
jierner@nvidio.com
Peter Yichen Chen Massachusetts Institute Technology
pycocsail mit edu
Aniket Bera Purdue University
aniketbera edu
Reviewed on OpenReview: https: // /ttp:/ppppee opwnreviewwe.we. net/ forum? eXCQA EEaaG
Luganiia stepping
RES
Eaker Integration
Kemel-ROM
to Disc. 18 (Sel
Dise.
Interaction Networss (Loodee)
Trunforite
NO
Geofeery-Informedd 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 using neural autoregressive PDE operator de. The operator predicts accelerations A, from short history of particle velocities Vi-wı, which are integrated to update particle positions. A learnable online Kernel based ROM parametrizes the solution manifold, enabling efficient. querying at arbitrary spatial locations using local geometric information from the evolved particles.
Figure Overview GIOROM:

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Published in Transactions on Machine Learning Research (02/2026)
Abstract
Simulating physical systems governed by Lagrangian dynamics often entails solving partial differential equations (PDEs) over high-resolution spatial domains, leading to significant computational expense. Reduced-ordes modeling (ROM) mitigates this cost by evolving low-dimensional latent representations of the underlying system. While neural ROMs enable querying solutions from latent states at arbitrary spatial points, their latent states typically represent the global domain and struggle to capture localized, highly dynamic behaviors such as fluids. We propose a sampling- based reduction framework that evolves Lagrangian systems directly in physical space, over the particles themselves, reducing the number of active degrees of freedom via data-driven neural PDE operators. To enable querying at arbitrary spatial locations, we introduce learnable kernel paramcterization that uses local spatial information from time-evolved sample particles to infer the underlying solution manifold. Empirically, our approach achieves 6.6x-32x reduction in input dimensionality while maintaining high-fidelity evaluations across diverse Lagrangian regimes, including fluid flows, granular media, and elastoplastic dynamics. We refer to this framework as GIOROM Geeometry-InfOmmee Rkeduceed-Order Modeling). All of our code and data is available at http:p///g.gaaa.../.../aaaaaaaaa hotp://///nsishheeaaaaaa///A/O0
1 Introduction
Various physical simulations involve simulating the spatio-temporal evolution of continuous fields governed by partial differential equations (PDEs) of the form
J
(1)
f(xxt):(x77 T Rd
(2)
where f represents a multidimennsonal continuous vector field that depends on both space and time (e.g., position field, velocity field, etc.). The symbols V and (-) signify the spatial gradient and time derivative, respectively. Here, Ω Rd and T R denote the spatial and temporal domains, respectively. To solve such system, we define the solution operator J a + f, that maps function of known obscrvations a : L' x T + Rd to f, where we assuine and f to lie on Banach function spaces defined on bounded domains D' E Rdd and D E Rd respectively. We let M denote the manifold of solutions over all parameters and time and denotes the dimonsion (2 or 3).
To numerically solve the equation, the system is discretized spatially and temporally. Spatial discretization allows for the approximations f(X,t) fp(X,t), t), and, a(X, t) ap(X,t), where P represents the P-point spatial discretization in Rd, transforming X to (P- di-dinnnsio vector. Similarly, we introduce temporal samples [T=) that, for given sample evolution using a set of input state variables a; = ar((x f) to obtain the solution f++1 = fr(xxt 1) at the next state.
{x E Pdd, we can compute the spatiotemporal
In this work, we focus on particle interaction dynamics modeled using Lagrangian systems Zefran & Bullo (2005), which the temporal evolution of interacting particles (represented as point-clouds) is governed by equations of motion derived from a Lagrangian functional encoding kinetic and potential energy contributions.
It has been shown that the computational cost of solving J scales with the resolution P, making it computationally expensive to use full-order PDE solvers, which operate directly on P. d degrees of freedom
Chen et al. 2023 2021
Reduced order modeling approaches are used to overcome this computational bottleneck. Traditional ROMs achieve computational speedup by projecting high dimensional state variables onto low dimensional latent subspaces, typically via methods like Proper Orthogonal Decomposition (POD) or reduced basis techniques. This enables approximation of these dynamics on fewer degrees of freedom (e.g. Q < P Pdd, reducing the
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RR
f. E R
(a) Prior ROM Techniques
k(Ew)
FIEE Namerr
Wrd Discretization
Rdd Discretization En
2
Particle Trajectory
ti
trajectory
Latz Sic
x E R
Particle Trajectory
E RPd
[
Kernel octeork
(xt RRPd
(At0)=1
2
B PP Discretization
Pd Discretization 4
(b) GIOROM
x E RPd f. R
Figure 2: Overview of GIOROM. This figure illustrates the temporal evolution of Lagrangian interaction systems using reduced-order modeling. (a) Prior discrelization invariant methods evolve a global latent. state & using PDE time-steppers and reconstruct the full-field solution using neural decoders g. This means, they require explicit PDE forms to compute the temporal evolution (b) In contrast, our approach employs a data-driven neural stepper to compute sparse field estimates { R"d, which are then used by the kernel to parameterize the solution manifold. In this case, the model does not require explicit PDE form. f represents the deformation field depicted above. While the decoder processes the latent vector representing the entire spatial field, the kernel network evaluates over local spatial regions.
computational cost Lucia et al. (2004). These approaches operate under an offline-online paradigm: the expensive offline phase constructs a global basis from full-order simulations, and the inexpensive online phase evolves low-dimensional coefficients spatio-temporally using reduced forms of governing equations. To recover the solution field from the evolved latents, modern ROM approaches typically leverage continuous implicit neural representations such neural fields. While effective in many regimes, these global latent representations often fail to capture complex, fast-evolving local dynamics within dynamic systems such fluids, because of their lack of locality. Furthermore these approaches are not fully data-driven. They still rely on access to the underlying governing equations for spatiotemporal evolution.
To address this, we propose a new ROM framework that replaces projection-based reduction with a more direct, sampling-hased reduction that enables reduced evolution directly in physical space. Specifically, we evolve the peatio-tempore dynamics of small set of representative particles (samples) embedded in the physical domain, bypassing the need for global latent state. This evolution can be facilitated using data-driven neural Lagrangian solvers Gonvallz et al. 20200) reducing dependencies explicit. forms of governing equations.
Furthermore, this direct 'particle-space' evolution preserves locality while also reducing computational complexity by minimizing the number of active degrees of freedom (i.e. number of particles processed) during the cvolution. This allows for distinct advantage invariant kernel-based manifold parameterization can be defined over the evolved particle sample set to query arbitrary points, thus retaining the modular approach of traditional ROMs without sacrificing the locality of the physical space. Crucially, unlike traditional ROMs, which perform temporal evolution in reduced space, we follow a hybrid approach of evolving the Lagrangian particles in sampled space, while leveraging kernel based manifold parameterization as a reduced spatial representation to parameterize the high resolution spatial field. The temporal evolution is achieved with the discretization agnostic neural operator.
constructing the solution manifold: discretization
The framework remains fully agnostic the underlying neural operator architecture, enabling to exploit the strengths of established neural physics models that accurately model particle dynamics and generalize
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across PDE parameterizations. Through reduced sampling of particle sets, GIOROM further enhances these models, leveraging their robustness to discretivation variations to improve their inference speeds.
Thus, our proposed GIOROM framework introduces a fully data driven ROM framework that retains essential ROM characteristics: modularity, reduced-order evolution, and continuous implicit spatial parameterization of the solution field, while performing time-stepping directly in physical space. The formulation remains independent of the underlying discretization.
2 Related Works
Reduced-order models ROMs reduce computational cost by projecting tigh--dimensionl systems onto low-
dimensional manifolds Berkooz et al.
(1993);
Holmes et al. (2012): Lee & Carlberg (2020); Peherstorfor (022)).
often via sample selection An et al.
(2008). These low dimensional states are explicitly and independently
evolved using time-steppers. Neural field- based ROMs Pan et al. (002) Yin et al. (2023): Wen et al. (2023):
Chen et al. (2023); Chang et al. (2023) enable discretizatiion-anooiie learning, supporting generalization across
geometric discretizations, but are still intrusive, roquiring exact PDE formulation to time-step. Data-driven
based techniques to achieve order-reduction Guo & Hest (2019) ; Peeherstoferr & Willcox (2015); [Beaabe et al. (2025). but these are however, not discretization invariant and require retraining to adapt to different discretizations. Kernel-based ROM techniques Honcine 22011)
reduced-order modeling use
Kou& & Zhang (2177); Raveh (2001] : Munteanu et al. (2005); Salvador et al. (2021) leverage kernel mel hods
such as Kernel-PCA or Kernel-POD, but they are discretization-deeedeen Additional works are in section
Our work can be seen as discretization invariant kernel-integral ROM parameterization for reduced space
that is both discretization invariant and non-intrusive.
Neural physics solvers
Neural solvers have advanced simulations across fluid dynamics Sanchez Goneza--Gzzzza
et al.
(2 Kochkov et al. (202)); Vinuesa & Brunton (202T Mao et al. (2020); Shukla et. al. (2024) 3)
Hao
ct al.
2024], , solid mochanics Geist & Trimpe (2021): ): Capuano & Rimoli (2019); Jin et al. (2023) climate
modeling Pathak al. [2022]. and robotics Ni & Qureshi (2022); Kaczmarski et al. (2023). These models
range from data-driven to inysice-inormedd architectures Raissi et al. (2019) ; Sirignano & Spiliopoulos 2018):
Richter & Berner (2022) ; Nam et al. (2024). CNNs are effective on regular grids Lee & Carlberg (2020): Maulik
et al. (2021p; Stoffel
| et al. ) Bamer et al. 2202)) while GNNs generalize to irregular meshes, enabling
applications in mesh-based dynamics Pfaff et al. [2020);
; Cao et al. (002): Han et al. (2022); Fortunato el al.
(202)). Lagrangian systems Sanchez-Gonzaleer et al. (2020), parametric PDEs Pichi et al. (2024), and rigid body physics Kneifl et al. (2024). Lagrangian modeling, which depends on particle-interaction are typically modeled with GNN based message passing. However, GNNs such as GNS and Meshgraphnets scale poorly with graph size due to node-wise message passing defined over all degrees of freedom. While traditional ROMs parameterize latent-spaces, our proposed reduction strategy can leverage these models directly, while achieving order reduction.
Neural operators
Neural operators are class of discretizatioon-tnvaraat models for learning mappings
between infinite-dimensional function spaces. They have been applied to parametric PDFs Luu et al.
(02T);
et al.
2020c
2023):
Azizzadenesheli et al. [2024]; Rahman et al. (2024]
Liu-Schiaffni et al.
2024
Kovachki
et al.
(023))
Rahman et al. [2022a] Liu et al. (2022): Viswanath th et al.
2023); Shih et al.
2024) Goswami
et al.
2023).
fluid simulations Di Leoni et al. (2023)
: Wang et al. (2024)
Peyvan et al. (2024)
3D physics Xu
et al.
2024):
White et al.
(2023): Boney et al. (2023); Rahman et al.
2022b): He et al.
(2024). and even
cross-domain tasks in biology, robotics, and vision Liu et al. 2202
Dharuman et al. (022 ; Bhaskara
et al. (2023 Peng et al.
2023) Guibas et al. 2021 Rahman & Ych 2024]; Viswanath et al. 2022). While
operators such as FNOs, and Transolver Wu et al. (2024a), use projections within their frameworks, they Iearn direct end-to-end function mapping rather than evolving reduced state. Their architectures couple the projection and processing steps, meaning the online evaluation phase still depends on the full-order degrees of freedom of the input. However, DeepONet Lu et al. (2021) can be viewed as a form of Eulerian ROM solver. On the other hand, our work, bridges the gap between neural Reduced Order Models (ROMs), neural Lagrangian solvers and neural operators. We propose a principled framework for integrating operator learning within order reduction paradigm.
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Our proposed reduction strategy be used in Geometric Informed Neural Operator (GINO) Li et al. (2024) and the Unified Physics Transformer (UPT) Alkin et al. (2024) to reduce effective degrees of freedom, however, GINO is unsuitable for partiele-iterrationss The UPT family of architectures follow the encode- process-decode framework, with graph based layers facilitating Lagrangian inputs. But their architectures evaluate the solution field, rather than particle interactions and remain unsuitable for autoregressive particle dynamics.
Table 1: Contrasting neural PDE operators and neural ROM solvers: The upper half of the table represent features associated with neural physics solvers while the lower half represents features in neural ROM solvers
Category
GNS / MeshgraphNet
Transolver
GINO
UPT
CROM / LICROM
Ours
Discretization Invariance
Irregular Domain Support
Neural PDE Operator
X
Locality Preserving Reduced Space
X
X
X
Partiele Interaction Dynamics
V
X
X
/
Low-Dim State Representation
/2
X
V
J
Decoupled Latent Space Dynamics
X
EKKKK
AXXA
Autoregressve Reduced Evolution
X
X
Modularity
X
X
/
/
Effective Active DOFs (Online)
N
N
47 & N
by
V
T
r
Non- Intrusive
Partially
Notes:
FNO: While designed to discretiattion-nnvaraat, standard implemmentaions perform best on regular grids and may require adaptation for irregular meshes.
FNO: The "reduction" is in the learned operator, which is parameterized by a small number of Fourier modes, serving as the effective latent space, but not decoupled from the projection operator.
3 DINo: Can be considered 'partially intrusive' they designed around neural ODE formulation governing the latent space. DINo considers Initial Value Problems (IVP) where the trajectories follow the same dynamics but have different initial conditions.
GINO /UNO/UPT: Due to their ability to extrapolate inputs over arbitrary query points, their active dogrees of freedom can be effectively reduced to a spanse set of points using our proposed reduction strategy.
Transolver: The physics attention mechanism of Transolyer does not support Lagrangian imputs, but be combined with GNN based pre-processing layers to handle Lagrangian systems.
6 UPT: The UPT architecture models field characteristics Tagrangian systems instead tracking individual particles and is therefore unable to model autoregressive evolution, wherein the encoder requires particle positions as input.
3 Method: Kernel parameterization
We formulate the reduced-order model (ROM) not as a projection onto a fixed global basis, but as a continuous integral operator parameterized by an implicit kernel. We refer to this as kernel-integral ROM. Let H T,2 ) denote a Hilbert space state functions over bounded domain C Rdd and assume the solution f(.tt evolves on low-dimensional manifold embedded in H. f is assumedd lic on continuous, smooth solution manifold M. We do not make assumptions regarding access to its closed form and instead are provided with set of pointwise approximations {x], f fi where is the number of sparse samples.
To recover the continuous field structure from the sparse samples assumed to lic on the smooth solution manifold, we introduce a learnable kernel parameterization (dr,,) y) that allows the field value at a query point z to reconstructed as weighted aggregation of nearby Lagrangian samples. Since v is not assumed define a probability density, the reconstruction expressed as a normalized integral operator, enforcing a partition-of -nfttttoon---uuii--llle unity-like condition and ensuring consistency under uniform fields. This yields a nonlinear, location-dependent operator that can be viewed a continuous analogue of kernel regression, closely related
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to Nadaraya-Watson estimators Cai 2001 and graph-based kernel integral transforms. Unlike global projection-based ROMs, this formulation induces a spatially adaptive operator whose effective support and conditioning vary with the local sample geometry. Consequently, the particle features within the local support of a query point act as the degrees of freedom (DoF) for field evaluation, without the explicit orthonormality constraints required by global basis projections.
For query point Q, the reconstructed value is formally given by the ratio of expectations:
u(æ,t) Sn (xx, y) u(y, t) du(v) Sn (x v) du(y)
33)
where v a smoothing kernel and the denominator ensures density normalization, v R f denotes the known Lagrangian sample field, obtained via the PDE operator (i.e. time-stepper) de (discussed in section and du represents the weighted Lebesgue measure on Ω.
This formulation is similar to graph based kernel integral transform (as in GINO Li et al. (024 however, direct evaluation via radius-based graphs scales with the sample size and is computationally expensive for high-resolution Lagrangian simulations. To mitigate this, we adopt a projection stochastic sampling stralegy.
The Lagrangian samples are projected onto a fixed grid using trilinear basis functions on, producing a discrete density field D Dorid and a feature field Fgrid defined at the grid nodes g: (g) = E, 0n y/ (,)) (grid(g) E, dh (g. uj). The local integral operator is then approximated via stochastic interpolation of these grid fields around the query location x:
Eg [IF Foridd grid, x + E)
f(x11) ~ ū(x) = Ee [I(D (Drrii grid, x + {)] +
(4)
where -(:,-) denotes trilinear interpolation, & _NO0 is a stochastic perturbation, and is small regularization for numerical stability. The stochastic perturbation induces a localized averaging that acts as a Monte Carlo approximation of the convolution with a Gaussian smoothing kernel. This avoids the explicit construction of radius- based neighborhoods or dense graph connectivity. Similar to the radius graphs, this step discrotization convergent on the sample size r. We observe empirically that this formulation achieves computational improvements over radius based aggregation while achieving better performance. We discuss the exact implementation details of the parameterization in section H.2
Loss Objective The training objective for the learned field ú(z) is given by:
La = E, Σ [ū(z, t) - uer (2, 11
(5)
4 Method: Time-stepping on reduced sample set
We make a deviation this section to focus on obtaining the sample set {at, f which done using the operator de. With < P, we only require evaluating the full-order dynamics on sparse set of points, achieving computational speedups. Following the UPT family of transformer architectures Alkin et al. 2024 adapt the UPT backbone to enable autoregressive particle dynamics using the interaction network based encoder and docoder Battaglia et al. (016)). The interaction modules allow the operator framework to capture interaction dynamics and can be used in autoregressive generations akin to Seinche-GGozzll el al. (020))
However, the use of transformer processor enables speedups over GNS and MeshgraphNet style architectures. The modular nature of the architocture ensures the operator is agnostic to the choice of the transformer and
is compatible with any PDE operator transformers such as Hao al. (023);; Wu al. (2024a) Lee Oh
(202)): Alkin al. 2024).

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4.1 Step 1: Input representation
We consider reduced set of material points {a' with < P, for which we seek full-order evaluations without assuming specific sampling scheme. Importantly, we do not train de for fixed r. The model de is trained on coarse discretizations {a', a², f where [V < P. d, defined on a radius graph G (V, E). Here, V denotes particles and E contains undirected edges between nodes within radius Psy i.e. (I. y) E E d(a,y)= S Ps. The edges represent particle interactions.
For a subgraph H = (u,e), defined over v S V with edges induced by radius pr, we view G and H as discrete approximations of an underlying continuous manifold C C Rd Jin et al. (200)); : Burago et al. (2015). We observe that for reasonable sizes of H, we can tune the radius Pr and enforce that for x' E Vne. H (z', t) = $ t) ;= f", where őe.G represents the model operating on graph G. This is achieved by enforcing a discrete neural operator behavior within the GNN architecture. We require tuning of Pr as fewer
nodes alter local neighborhoods, which can disrupt message passing Garg et al. (2020) Gao & Isufi 2022
4.2 Step& 2: Heuristic for graph construction
For subgraph H defined on v S V, we require the connectivity radius pr 2 Ps to preserve paths between originally connected nodes, even after subsampling; r is chosen to optimize the kernel network performance/cos tradcoff, and consequently, Pr is tuned at inference to trade off accuracy and speed.
We use mean aggregation in message passing to ensure consistent node features across discretizations. This is shown to behave like Monte Carlo approximation of kernel integrals, enabling the GNN to function as a neural operator under suitable graph construction Li et al. (2020b).
Prior sampling-based ROM approaches rely on stochastic,
suidual-guided strategies to select sample sets Chen et al. (2023). In contrast, we adopt simple sampling methods such as random or farthest-point selection. To ensure invariance to the sampling ratio for r := |wa/ /VV, we construct I random subgraphs per input domain to evaluate during training. This can be seen as a Nyström-type approximation, reducing variance and improving generalization across discretizations.
Empirically, we find that overly small |e| degrades accuracy even with large Pr, implying lower bound to retain spatial structure. This threshold guides inference- tateence-timeeeee subgraph construction (section M Conversoly, excessive Pr introduces redundant edges and deteriorates performance duc to loss of locality.
4.3 Step 3: Spatio-temporal time-stepping
Equipped with our data structure H, we now discuss the architecture of de that enables the mapping af fit We define at and f to be point-wise function values defined on the node set u E Rrd
Extending the UPT family of architectures, our formulation can then be defined as de N NOT e H. where enc and dec denote the encoder and decoder, which operate the graph H. The NOT however operates only on the latent embeddings. The input for the model is the velocity sequence v defined over a time window w. Formally, of = v vi-wa, Vj E -Rr-d The output is then given as } = a where is the pointwise acceleration defined following Conchlz--onzalez et al. (2020) with data-driven losses as follows Le mine Σ, (de)(a)) fill Additional training details and hyperparameter information is provided in section and section
We obtain f4, by Euler integration. The model is trained
5
Dataset
Our dataset consists of four 3D physical systems- - Newtonian fluids (Water), Drucker-Prager elastoplasticity (Sand), von Mises yield (Plasticine) and purely Elastic deformations. Unless otherwise noted, we assumee the discretization to be point clouds.
We used the nclaw simulator (a al. 2023 to generate 100 trajectories for each of these systems with random initial velocity conditions with free-slip boundary condition, typically used in MPM simulations. The
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At between consecutive time frames is 5e ss We additionally used datasets provided by Banchez-Gonzaler et al. 2 WaterDrop, Water-3D, Sand-3D, Sand, Goop and MultiMaterial. Additional training details and model hyperparameters can be found in Appendix H.2
6 Experiments
This section is structured to evaluate the two core contributions of GIOROM: the locality of the kernel- integral ROM formulation (space) and the computational efficiency of the PDE agnostic dynamics (time).
Manifold parameterization and generalization (space) We first investigate the fidelity of the ROM basis functions by benchmarking kernel formulation against prior ROM basis representations. We calegorize these baselines based on their latent space design:
Rdd
Kernel ROM Evaluations in RPd
Time Axis
Autoencoder
PCA
LiCROM
20K
62K
Ground
Truth
CROM
GIOROM
GT
(a)
(b)
Figure 3: (a) Comparison of reduced-order modeling (ROM) techniques on fluid-in-container simulations: The visualization reveals that the baseline models PCA (rollout MSF: 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-integral ROM on high-resolution particle systems: Contrasting the behavior of the kernel-integral ROM on largo-scale simulation containing approximately 2 million particles, discretized voxelized grids. Using randomly sampled (r 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 к under resolution refinement.
Continnous Space INR ROMs. These baselines represent architectures that traditionally rely on explicit PDE forms to evolve the latent space dynamics. We evaluate the global bases utilized in CROM Chen et al. 2023)
CoLORA Berman & Peherstorfer (2024). and LiCROM, alongside the local graph kernel integral transform (GKI) from GINO Li et al. (2024). While Transolver, FNO and GINO include projection layers, they are trained end-to-end in operator setting, but the integral transform (GKI) serves as a discretization-convergg kernel formulation that operates directly in sample space, forming a key baseline for local parameterization methods. To evaluate the spatial expressivity of these manifolds, we employ a common time-stepper to evolve the sample space trajectory for all static ROMs. The full-order state subsequently reconstructed via the respective basis encoders. This allows us compare spatial parameterizations of the baseline projection ROMs and overcome their PDE dependencies for temporal rollouts.
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Continuous time latent dynamics ROMs. We further evaluate architectures such DINo Yin et al. (2023 and CORAL Serrano et al. (2023), which integrate non-linear autoencoders with intrinsic Neural ODEs. For these models, we train the intrinsic Neural ODE to evolve the latent states temporally. Reconstruction metrics are then computed on the decoded full-order states.
End-to-end computational efficiency (time) Following the manifold validation, we evaluate the end-to- end performance of the full GIOROM pipeline against GNS Saanche-Goozzller et al. (2020), the state-of-the-an full-order Lagrangian graph solver. In this context, GNS serves as high-fidelity oracle for rollout accuracy. These experiments demonstrate that our latent time-stepping on the reduced manifold achieves rollout. stability comparable to the full-order solver.
6.1 Results: Manifold parameterization
Generalization We showcase quantitative analyses in table 2 We observe that GIOROM achieves the lowest Relative L2 error and Chamfer distance in the four physical systems on previously unseen trajectories (i.e. initial conditions). The performance gap is most pronounced in the WATER-3D dataset, a regime characterized by topological breaks and highly dynamic transitions. The Icarnable latent dynamics baselines (DINo, CORAL) exhibit. high variance and error rales (> 30%) in these multi-trajectory settings, whereas the continuous space INR ROMs exhibit better performance. We additionally present the results WATER-3D in fig. 3a The baselines we compared include (1) PCA (Principal Component Analysis), also known as POD (Proper Orthogonal Decomposition), (2) Autoencoders Lee & Carlberg 2020) (3) LICROM
Chang el. al. 2023 and (4) CROM Chen et al. (023))
Computational footprint We further analyze the trade-off between performance and computational cost. While PCA is trivially the fastest (< 0.5 ms) with negligible memory, fails to capture non-linear deformations, resulting in high Chamfer distances. Among the non-linear deep learning methods, the Jax based GIORÓM offers the most favorable resource-to-accuraay ratio. It requires order of magnitude less peak memory (17-267 MB) compared to the graph-kernel baseline GKI (462-2970 MB) and the latent ODEs, while operating at inference speeds (0.79- 10..9955.4 5.47 ms) that approach the linear baseline.
Impact of reduced space To quantify the manifold capacity across ROM architectures, we analyze error convergence with respect to the reduced space size. For global ROMs (e.g., PCA, CROM), this corresponds strictly to the fixed latent dimension Q. In contrast, GIOROM does not project the global state into single latent vector; instead, it distributes information onto a sparse grid. To compare these architectures, we view each active grid cell as a local reference vector for the query points within its support. Accordingly, we define a heuristic for reduced size Sett as the ratio of total Lagrangian feature size to the active grid voxels: Sell x Np x C/Nactive: where N₂ is the Lagrangian sample size and C is the feature sizc. This represents heuristic for the expected number of features that contribute to each active cell. This framework represents relaxation of the strict projection- based ROM paradigm, where time-stepping is confined entirely to the reduced latent space. In our hybrid formulation, we decouple these complexities: the spatial parameterization is governed by the sparse grid in equation A while the time-stepping is defined over the Lagrangian samples.
Figure illustrates the convergence behavior under this metric. It can be seen that in the case of topologically stable regimes such as PLASTICINE-3D and ELASTICITY-3D, GIOROM performs similarly to global bases. This attributed to the preservation of relative particle positioning, which allows global linear bases to effectively capture smooth deformations. However, we observe that global bases outperform GIOROM in extremely sparse regimes. This occurs because in such settings, each active voxel is effectively influenced by a single particle. In contrast, global ROMs utilize learned basis functions (e.g., principal components) making them more expressive. We present visual comparisons for different regimes in fig. 17 fig. 18 fig. 19 and
fig. 20
Computational scalability To empirically verify the computational footprint in section 3 we benchmark GIOROM against the Graph based GKI bascline Li et al. (2024). We analyze scalability with respect to two factors: the interaction radius 8 and the sample space particle count.
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Sensitivity to interaction radius (memory scaling) We compared the memory footprint of the neighborhood
graph structure (GKT) versus our grid-projection hash (GIOROM) for fixed particle set (N = 20, 000) while varying the normalized smoothing radius ó. As shown in fig. a GKI exhibits explosive memory growth, scaling from 2.8 MB at 8 = 0.01 to 1.37 GB at 8 = 0.2. In contrast, GIOROM maintains a constant 7.0 MB memory footprint regardless of
Scaling with sporsity (N) We further decompose the wall-clock inference time into 'Structure Build' (Graph Search vs. Grid Projection) and 'Function Evaluation' for particle counts ranging from N 2, 000 to 35, 000. This is shown in fig. I and fig. c Our results reveal that for GKI, the graph construction is the dominant bottleneck, accounting for 97% of the total runtime at N = 35k (176.9 ms build vs. 5.1 ms compute). GIOROM effectively eliminates this bottleneck. Furthermore, while GKI memory scales linearly with N (reaching 83 MB at N = 35k), GIOROM's structure memory remains fixed at 7 MB, determined by the grid resolution. Lastly, as the grid is discretization convergent, it exhibits degradation with extreme reduction particle counts. We analyze this in section M.2 and provide additional ablations on kernel parameters.
Memury Scaling
Time
Keruury Sceling
18X
CX030 ICRIN
K0x GB1 Panved Pen
CECnOX Nemery
+Cx GKANN Scnoul Iax
18X
200 10000 TEL
01.00 02.44
0..0
5.000
1.23
2.500
3.77
2.20
1080
210.0
32.00
20023 20.03
19K4
30000
15:00
-0.02
20300
2004
2.54
Inteuction (oll
Figure 4: Computation overhcads in implicit kernel based ROMs These plots showcase the overheads in performing radius based graph integral transform, with plot (a) highlighting the scaling as the radius increases, plots (b) and (c) highlighting the time and memory overheads as the particle count scales. In contrast, grid projection maintains fairly stable computation overhead.
6.2 Results: End-to-end dynamics and scalability
Comparison with full-order solvers. We first evaluate the ability of the reduced model to preserve dynamic stability compared to the full-order bascline. table 3 reports the one-step and rollout MSE of GIOROM against. the GNS oracle across diverse physical systems. We show that operating on sparsified graphs, reduced by factors of up to 32x (e.g., WATER-3D, SAND-3D). The error margins remain within the same order of magnitude as the GNS basclinc, even for complex multi-matorial interactions. Additional results and discussions are provided in section K and section M
Computational improvements While Graph Neural Networks (GNNs) effectively capture spatial inter- actions in Lagrangian systems, the iterative message- nessage-passing operations introduce significant. computational bottlenccks. This overhead scales poorly with edge density. GIOROM mitigates this by adopting a UPT-stylc Encoder-Process- Decoder architecture, where global temporal mixing is handled by a Transformer processor rather than recursive message passing the graph edges. We evaluate this in two settings, first by increasing the number of edges by tuning the graph radius, and second by increasing the number of nodes by tuning the sampling rate.
Graph Radius table ! isolates the impact of the connectivity radius on inference time. As the radius increases from 0.040 to 0.100, the inference time of GNS degrades drastically (from 43.5s to 386.0s). In contrast, GIOROM time-stepper exhibits far more robust scaling behavior, maintaining a speedup factor that widens from 2x at low connectivity to ze 3.5x at high connectivity (109.3s vs 386.0s).
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Table 2: Per-Dataset Quantitative Evaluation. We report Rel. L2. Chamfer, and VRAM. GIOROM achieves the best geometric fidclity with constant-timg efficiency. Bold is best, underlined is second. All the results are computed on reduced space with 32 effective active degrees of freedom
DATASET
MODEL
REL L2 ERROR (%)
CHAMFER (La)
TIME (us)) J
AVG MEMORY
PEAK MEMORY
(x10 "
0.02 (MB) 4
(MB) J
PCA
16.15% 0.96%
19:25
0.42
4.42
DINo
31.8%
434.74
478.00
CORAL
21.85% 8.46%
141.81
2.71
407.87
408.00
CoLORA
9.08% 1 3.90%
21.05
1.25
407.87
408.00
PLASTICINE-3D
CROM
3.19% + 0.62%
4.58
1.49
407.87
408.00
LICROM
2.31% + 0.46%
3.78
4.23
474.80
476.00
GKI
3.69% +: 3.89%
8.64
3.95
461.87
462.00
GIOROM (Ouns)
1.98% 0.57%
17.28
13.37%
1.91%
26.32
0.51
1.69
DINo
15.36%
7.78%
319.17
3.43
533.51
331000
CORAL
25.80% 8.06%
06.11
3.57
407.62
408.00
CoLORA
12.31% + 4.31%
23.56
1.59
407.62
408.00
SAND-3D
CROM
4.21% + 0.56%
6.61
1.55
407.62
408.00
LICROM
3.41% + 0.65%
6.26
9.15
$13.52
522,00
8.46% + 5.24%
23.41
962.40
GIOROM (Ours)
2.68% 0.91%
2.55
3.25
2.07
132.24
PCA
15.84% 3.60%
57.52
1.69
DINo
58.97% + 13.47%
483.62
3.25
375.15
592.00
CORAL
53.71% + 15.16%
$23.06
4.15
428.61
430.00
CoLORA
51.5%% + 12 12.43% 43%
139.11 11
2.77
129.62
430,00
WATER-30
CROM
24.00% 5.75%
138.02
2.12
428.85
430,00
LICROM
22.01%
548.00
GKI
15.16%
6.95%
15.33
79.89
2970
GIOROM (Ours)
9.24% + 4.13%
8.67
3.51
9.33
266.72
PCA
39.41% 0.66%
63.04
0.58
0.01
1.05
DINo
41.18% + 10.01%
165.95
4.50
617.47
629.00
CORAL
46.24% + 12.17%
242.50
5.09
441.60
442.00
CoLORA
9.83% +: 3.90%
10.10
3.66
441.60
442.00
ELASTICITY-30
CROM
4.25%
0.91%
442.00
LICROM
3.43%
1.27%
5.77
7.98
547.74
549.75
GKI
7.66% 1: 0.80%
6.89
42.53
1788
1788
GIOROM (Ours)
2.04% + 0.49%
3.03
5.47
9.60
267.81
Graph Nodes table 5 highlights a critical distinction between number of nodes and runtime cost. GIOROM consistently achieves 2x-2.5x faster wall-clock inference across all graph sizes. For the largest system (7056 points), GIOROM completes the 200-step rollout. 47.68 compared to 111.6s for GNS.
Discretization Convergence We analyze the sensitivity of the learned operator spatial discretization using the ELASTICITY-3D system (78k particles). As illustrated in fig. 5 the rollout error remains remarkably stable (MSE ~ 10-4) across sparsily levels ranging from 0.125x down to 0.031 x, confirmiing mesh-independent physics learning up to a 32x compression ratio. Performance degrades only below this threshold due to over-sparsification; notably, in extreme regimes (0.007x), increasing the connectivity radius beyond 0.15 yields diminishing returns, indicating that long-range edges cannot compensate for lost local geometry. Computationally, the method exhibits linear scaling in wall-clock time and peak GPU usage (inclusive of dynamic graph construction overhead) as the system sparsified, ensuring that the cost of rebuilding the latent topology does not negate the speedups from dimensionality reduction.
7 Discussions and Conclusion
Adaptive Time Discretization. While our current implementation utilizes fixed-step latent solver, the continuous manifold formulation is naturally compatible with adaptive temporal integration. A future direction could involve exploring the continuous time dynamics as used DINo and CORAL, such as with Neural ODE (NODE) allowing, the solver to dynamically adjust computational effort during periods of complex dynamics.
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Table 3: Comparative performance of GIOROM (Ours) against the GNS baseline across diverse physical
systems. Our model operates on compressed latent graph of size r, where Scale (N/r) represents the reduction in graph complexity compared to GNS. All error metrics are computed on the full particle set IR
FULL SIZE
LATENT
ONE STEP-MSE (xc
ROLLOUT MSE (xc=a)
PHYSICAL SYSTEM
(N)
(7)
SCALE
Ours
GNS
Ours
GNS
WATER-3D
1.7k
32x
5.23
38.6
17.4
WATER-2D
0.12k
8.3x
0.524
1.31
6.70
7.55
SAND-3D
32k
1k
32x
4.87
3.38
0.25
0.31
SAND-2D
2k
0.3k
6.6x
8,50
6.50
1.34
1.21
GOOP-2D
1.9k
0.2k
9.5x
1.31
3.02
0.94
0.55
PLASTICINE
5k
1.1k
4.5x
0.974
1.02
0.50
0.71
ELASTICITY
78k
2.6k
30x
0.507
0.20
MULTI-MAT 2D
2k
0.25k
2.30
1.99
9.43
Sparsity vs. GPU usage (in GB)
Sparsity vs. Time (in s)
120
Time infer 120 sleps (in
R.0.00
CPU Usage (GI3)
8=0.50
80
0.0000
seconds)
-.0000
0.020
80
R=00.000
20
R 0.00060
R 0.080
0.050
R.0.060
0.5x
0.25x
0.125x
0.062x
0.081x
0.01Bx
0.007%
0.5k
0.25x
0.128x
0.062x
0.031x
0.015x
0.007%
Sampling Ratio
Sampling Ratio
Sparsity vs. Rollout MSE
Radius vs. Rollout MSE
0.005
R-0000
0.35
0.3
S
0.007X
Rollout MSC
Rollout MSE
0.016
0.01
0.0007x
0.0000
0.050
R.0.000
R=0.000
R-0.000
0.00000
-300000X
5.0.007
S 0.007
0.08TX
0.007X
0.02 0.04
0.06
0.16
0.45
Sampling Ratio
Radius
Figure 5: Effect of sparsity on rollout error (Elasticity dataset, 78K particles). Plots show performance of de across varying sparsity levels, expressed fraction of the full system. GPU usage and computation time include pre- 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 rollout as a function of sparsity at fixed radius. Bottom-left plot reports rollout MSE, showing that performance remains stable (~1e-4) for sparsity levels 0.125x, 0.062x, and 0.081x. Below this, performance degrades due oversaarsifiattinn Boutom-right plot shows that for high sparsification (0.007x), increasing the graph radius beyond 0.15 does not improve performance.
Failure modes of ROM Performance degradation in ultra-sparse regimes is observed in our approach: when the sampled particle count drops such that the active voxel count approaches 0, the estimator becomes ill-conditioned, forcing the network to rely entirely on the prior, which lacks local physical guidance. We provide more discussions and analysis of limitations in section M
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CIOROM (Ours)
DINO LICROM
PCA CROM
CORAL COLORA
GNO
PLASTICINE (MPM)
SAND (MPM)
10
104
Relative L2 Error (%)
32
16
32
WATER (SPH)
ELASTICITY (FEM)
10a
6 10
x 10
3x 10
2 10
10
Reduced
Figure 6: Impact of ROM space different physical regimes The above plots showcase the effect of reduced space sparsity on different regimes, ranging from highly dynamic fluids (Water-3D) to regimes with minimal deformation (Plasticine-3D) and elastic behaviors (Elasticity-3d) The sharp degradation in GIOROM performance observed for fewer than attributed to extreme sparsity the Lagrangian sampling, which results in empty support neighborhoods for the kernel intogration.
Table 4: Contrasting the change in computation time with the increase in connectivity radius for a graph with 7056 points. The times shown represent the overall time needed to infer all 200 time steps. We compare our time-stepper with other neural network based physics solvers.
CONNECTIVITY RADIUS
MODEL
TIME STEPS
NUMBER OF SPATIAL POINTS
0.040
0.050
0.060
0.070
0.080
0.090
0.100
OURS
200
7056 points 7056 points
20.1s
34.3s
47.6s
65.8s
89.7s
104.1s
109.3s
GNS
200
43.5s
73.58
111.6s
162s
226.2s
305.9s
386.0s
Limitations and Generalization We show in our work that we can scale to millions of points and model physical regimes. We additionally provide results for generalization cases in fig. 21 long horizon dynamic trajectories, multi-object elastic collisions and phase changes (melting).
GIOROM, like other learning-baseo ROMs, exhibits reduced performance under extreme distr--iisttiiuttion
conditions [Li et al.] 2020b Chen et. al. 2023) and extreme sparsifications (section M.2 Furthermore,
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Table 5: Contrasting the change in computation time with an increase in graph size at a fixed radius of 0.060. The times shown represent the overall time needed infer 200 time steps. We compare our time-stepper against other neural network based physies solvers
MODEL PARAMETERS
CONNECTIVITY
MATERIAL
TIME STEPS
1776 POINTS
4143 POINTS 5608
GRAPH
POINTS
7056 POINTS
OURS
4,3122447 1,592,987
Phaticine Platticne
3.9h
14.5=
GNS
0,000
200
7.8
38.3s
08.75
111.Gs
while currently designed for continuous systems, future work may extend GIOROM to explicitly address discontinuities (Belhe et al.) 2023 Goswami et al., 2022) and unbounded flows. While we currently only leverage Euler inlegration schemes, studying the impacis of higher-order methods, such as Runge-Kutta 4th order on accuracy or stability of ROM is an exciting future direction. We also restrict our problem setups to consistent time-discretizations. Understanding the behavior of neural time-steppers on varying time discretization is another consideration for future work.
Conclusion We show that our reduced-order modeling framework for learning Lagrangian dynamics on sparse imputs achieves computational improvements over existing neural solvers (lable while preserving high simulation fidelity across diverse physical systems, outperforming traditional ROM approaches, particularly, on dynamic fluid simulations. The end-to-end data-driven approach can open doors to incorporating model-order reduction to simulations generated by generative models, with no PDE priors.
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A Appendix
B Additional related works
Time scries dynamical systems Simulating temporal dynamics in an auto-regressive manner is a
particularly challenging (ask due error accumulations during long rollout Wikner et al. (2024); List el. al.
(024). There have been many works that learn temporal PDEs and CFD, including Majid & Tudisco
[2024] Liu ct al. (2024b); Sarkar ct al. (2024); Wu et al. (2024b); Jeon et al. (2024); Jiang et al. (2024): | Ma
el al. Jonnyyy et al. (2024). Some works have proposed neural net wetwkwbssse approaches to model 3D Lagrangian dynamics, such as Ummenhofer et al. (2020), who propose a convolutional neural network-based approach to model the behavior of Newtonian fluids in 3D systems. (Sncheez-onzzlez et al. (2020) propose a more general graph- based framework, but the network suffers from high computation time on very dense
graphs.
C Definitions
Full-order model A full-order model evolves the solution over all degrees of freedom present in the spatial discretization scheme (Hughes 2012) (c.g., P. d in RPd r d in RT These models do not perform any dimensionality reduction. They are therefore prohibitively slow when P is large. We denote the P-point discretization of the spatial field X by {-r) where x² represents particle in I& with negligible mass. Similarly, a sample set {= { forms an (r-d) vector.
Reduced-order model reduced-order model evolves the solution in reduced latent space R Q < P- d. Reduced-order techniques such as Chen et al. (203) leverage neural fields and projection-based ROM infer the continuous spatial function arbitrary spatial locations from the low-dimensional latents. In similar vein, for evaluation point æf and its local spatial neighborhood z N(xx), define parameterization (Lff,e) z f(xt,) () as k(xf, de. R (x) for of E IRd, F = N/(x*), o Rrdd We note here that x parameterizes M(f) and Φe.s = (f (e)())) We also note that f has a temporal dependency, i.e. ft represents the evaluation at t. However, we omit the time subscript for brevity.
Traditional ROM approaches define the latents to be a nonlinear low-dimensional manifold Chen et al. 2021 Such a low-dimension parameterization can be defined with a projection mapping, 9 Ro + 1R where Q & P. d. This defines a map to discrete field for every low dimensional latent vector *(t) E B such that gp gx()) + [i., f(P) Chen et al. (2023); ; Chang al. (2023) For point-wise evaluations, g( &;) low-dimensional latent representation of the continuous field f.
f(x', t) fi, with & serving
Discretization invariant ROM approaches Chen et al. (0023 Chang et al. (002) construct the latent state & from a known set of point-wise solution values Φ = (f:... f.....,,f fi") through a projection map π RPP + R often parameterized using PointNet-stylle models which enable continuous evaluation. However, temporal evolution, & + & is implemented via explicit numerical time-stopping schemes with time gradients provided by the exact PDE. As a result, these methods depend on explicit access both the PDE solution and the associated solver framework.
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Time integration To compute temporal dynamics these r-point spatial samples, we seck to evolve [1 + (f++)) In the discrete setting, we leverage an explicit Euler time integrator (Ascher & Petzold 1998) with step-size At,
f - St + + f - ft + +
(6)
(7)
If f represents the position field, then, the one and only unknown in the equation above is the acceleration
A = fi, which is necessary for computing the velocity V2+1 fi+1. We learn the acceleration field A = fi
using the parameterization Φe, which is then used in the explicit Euler update.
Background
D.1 Operator learning
Here, we summarize the important ingredients neural operators. For more details, please refer to ct al. (2020a). Operator learning is a machine lcarning paradigm where a neural network is trained map belween infinite-dimensioonal function spaces. Lel G V + A be a nonlinear map between the two function spaces V and A. A neural operator is operator parameterized by neural network given by Go: V- A, E IR, that approximates this function mapping in the finitediimensional space. The learning problem can be formulated as minøeR= E [l) - G(v)]B] where || llv is norm on V and D is a probability distribution on V. In practice, the above optimization is posed as an empirical risk-minimization problem, defined as mingges E (( 13.
Our data-driven discretization invariant reduced-orde modeling framework for Lagrangian systems is constructed by parameterizing the PDE solution operator J with data-driven neural parameterization Φe : f, trained from small set of point-wise samples {aj, fij. such that ft ~ ft and de Brd, where r.d < P d. We interpret Φe to be a surrogate for the solution operator, with {>1)== representing form of reduction of (P- dd)--diitt samples in X. The architecture for Φe leverages graph interaction network Battaglia et al. (2016) Sanchez-Goni Gonzalezz et al. (2020) to capture particle interaction dynamics. Figure 2 illustrates this distinction by comparing particle trajectories in elastic deformation systems modeled using
our approach with discretization invariant CROM Chen et al. (2023).
D.2 Kernel methods and manifold learning
Kernel methods have been applied in manifold learning, where they serve as tools for constructing operators that act on functions defined over sampled data. Many such methods including diffusion maps and Laplacian eigenmaps- can be interpreted as variants of kernel PCA Izenman (2012).
Belkin and Niyogi Belkin & Niyogi (2003) and Coifman et al. Coifman & Lafon (0066) studied radially symmetric kernels of the form k(I, y) h (L=-مll) where h R4 + IR is a decreasing function, typically
Gaussian. These kernels are local. As such, they are isotropic and spatially invariant Berry & Sauer (016
Neural operator literature defines the kernel based kernel integral transform:
(Kf)())=)) - 1/0k/k(a, dy.
which is central to both manifold learning and neural operator frameworks. It maps input functions f Ω + Rd to output functions via integration against k, and serves as the foundation for approximating function-to-function mappings.
In the context of manifold learning, local radially symmetric kernels induce a geometry on the embedded manifold. As the sample density increases, the kernel implicitly defines a metric structure through its interactions over neighborhoods on the manifold Berry & Sauer (2016). Consequently, such kernels not only enable dimensionality reduction but also act as geometric priors over data manifolds.
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Finally, this kernel framework extends naturally to operator learning, where local solution maps, such those arising from hyperbolie PDEs, can be effectively approximated using locally supported kernels Liu-Schiafini et al. 2024).
E
Consistency & stability
To analyze the theoretical properties of the proposed GIOROM, we study the error decomposition of the discretized integral operator. The continuous operator is defined as H[f Jn x(x, f)) dy. Its discrete approximation H introduces error from three distinct sources: Lagrangian sampling, grid discretization, and neural approximation.
Let u(x) denote the true drift field and ((x) its reconstruction produced by GIOROM.
Monte-Carlo Consistency The projection of Lagrangian particles onto the grid can be interpreted as a Monte- Carlo approxination of the integral operator Callisch] (1998). Let {y: be i.i.d. samples drawn from density p on Ω. Then, for any square-integrable function the Monte-Carlo estimator satisfies
1 / f(y) dply)
Varp(S) VN
E
Σ f(y.)
(8)
N
where the expectation taken over the sampling of {y:). Consequently, the sampling error decays at the standard Carlo rate Robert & Casella (2004).
Grid Convergence The use of trilinear splatting and interpolation corresponds to a tensor-product linear B-spline basis. Assuming f E (?)) and normalized partition-of-unity interpolation scheme, standard approximation theory yields the interpolation error bound
Il - Ilgria fll (a) S (Ax)) [|>)
where Ar denotes the grid spacing and C' is a constant independent of Ar Strang & Fix 1973) De Boor 1978).
Total Error Combining the sampling and grid discretization errors, the estimator is consistent. As the number particles ~ xo and the grid resolution Ar + O, the discrete operator converges to the contimuous limit, up to the approximation error of the neural parameterization. Specifically,
C1 CC(AAr) + VN
| ull
(10)
where 60 denotes the approximation error induced by the finite expressivity and optimization of the neural networks Hornik al. 1989
Sampling Condition To select subset of Lagrangian markers, we employ uniform random subsampling.
While less deterministic than covering- based methods, random sampling preserves the i.i.d. assumption required for the Monte Carlo consistency bound derived above. For random samples in bounded domain
Q C Rd, the fill distance (covering radius) hx,0 converges probabilistically as hx,n [ogg N 1/d ensures that the spatial coverage becomes dense as N +
which
However, random sampling may locally violate the coverage condition in sparse regimes due to stochastic clustering. In practice, stability requires that the local particle density remains sufficient relative to the grid stencil. To quantify this, we define a heuristic condition analogous to the Particle- Perlle--er-Gaae Per--err-eeee PerlCell (PPC) criteria found in PIC literature Birdsall & Langdon (1991) : Jiang al. (2015): Nowf N-Vateniil 2 This condition ensures that the average number of particles contributing an active grid cell remains sufficient to avoid aliasing artifacts, though it does not constitute a sufficient. theoretical guarantee.
Nastive
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E.1 Graph interaction network
The graph interaction nelwork proposed in Ballaglia el al. (0016) learns relation-centric function / that encodes spatial interactions between the interacting nodes within a system as function of their interaction attributes This can be represented as 6t+11 fR( T2,,, A node-centered function predicts the temporal dynamics the node as a function of the spatial interactions as follows 21,4-1 fo(e++1, T1.e). In a system of m nodes, the spatial interactions are represented as graph, where the neighborhood is defined by ball of radius This graph is represented relationships between them. The interaction between them defined I(G) Io(a(G, X, ['(((,,, Ti ri))))) Where is aggregation function that combines all the interactions, X is the set of external effects, not part of the system, such as gravitational acceleration, etc.
G(O,R) R), where O is the collection of objects and R is the
F Algorithms
The following section presents key algorithms that helpful for implementing the concepts presented in the paper
Algorithm GNN Time-Stepper Inference
Input:
G (V, E): input graph with nodes V and edges
x NVV]x, categorical node types
p R VVxx; recent positions and velocities
R x]xxe edge attributes (displacement, distance)
edge_ index E N2x[El; sender and receiver indices
recent_por e glvixd; current position of each node
Output:
RIvixr predicted acceleration (or equivalent output)
Step 1: Node and Edge Feature Initialization
2: h. + concat concat (EmbedTypeex)).2))
Emned categorical lype and concatenate with pos/ /ollvclyity
3: h, + IodeInputMLP(hh.)
4: he < EdgeInputMLP(c)
Step 2: Message Passing (Encoder)
h.mp_layers do he + GNNLayer, edge_index, he node_oddddaaa dist) end for
for i=
Step 3: Global Transformation via GNOT Layer
10: h + GNOTLayer(h,,, recent recent__esss..ee
11: Step 4: Message Passing (Decoder)
12: for i = 1 to n_mp_layers do
13: he hg +- CNNLayerOut,(h.. edge_indxx, he, wode_nddee_st
14: end for
15: Step 5: Node-wise Output Projection
16: à (o-oDDuuttuMMM)((.
17: return
G Architecture outline
The schematic in fig. illustrates the full sequence of operations performed during a single time-step of the proposed method. It delincates the interaction between the timo-stopping mechanism, the kernel-basod reduced-order model (kernel (kernel-integral integral ROM), the Euler integration scheme, and the spatial sampling strategy used to update the system state.
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Algorithm Learnable Kernel- Kernel-Integral ROM (GIOROM)
Require: Source stale (x,, u.). Query points Xy: Grid Res R, Noise o, Paths K
Ensure: Predicted drift field iy
1. Feature Encoding
V, + (u,
> Lift drift to latent features (MLP)
2. Grid Splatting (Monte-Carlo Integration)
4: Initialize grid G E |x with zeros
5: for each particle in source set do G Vp A(xp)
Accumulate features via trilinear weights
end for
3. Stochastic Sampling (Feynman-Kac)
for each query point j in X, do
10: Sample noise ,I,...,Ex ~ .N(00
11: h, < R Lk=1 Interp G, xj Ek)
Convolve & Sample
12: 4. Residual Decoding
13: x (Dy) + Interp(u,, x,)
b Decode & Add Residual
14: end for
return 0g
P i=1
Sampling
Time derivative (compute vel)
{apx1)) i=1
P i=1
{x
Kernel ROM
Neural Time Stepper
p
P
(f1)"
{xl
{ x
Kernel ROM input Initial state for kernel neighborhood
Euler Integration
{xt
' + i=1
Figure 7: Single-step flow diagram: Overview of the forward pass used to infer the system state at the next time-step via kernel-integral ROM, timo-stepping, Euler integration, and spatial sampling. As de takes as input a window of past-states, we require the same set of samples for a trajectory evolution
H Experimental setup details
H.1 PDE operator setup and hyperparameters
Data representation To train de, we create window w point cloud velocity sequences as the input defined on the nodes, with the pointwise acceleration the output. We define (x() E Redd be the pointwise positions of particles within d-dimensional system at time t. A sequence of T time steps denoted
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Xo.2 = (Xu..... ................................ xT). In particular, (xf,..., xi) are the individual partieles within the system at time t. We define velocity at time n as as X Similarly, acceleration at time n is defined as a, = V₁ Vt-1- In all these cases, At is set to one for simplicity. To train the model, we follow the same procedure as Goazalzz et al. 2020). We additionally encode the particle types (water, sand, plasticine, etc.) as
embeddings, which useful for multimaterial simulations. The graph is constructed as a radius_graphe defined within pytorch geometric library.
Boundary representation To enforce the boundaries of the system, the node feature includes the past w velocity fields as well as the distance of the most recent position field to the upper (bw) and lower (bt) boundaries of the computational domain, given by D = [[xr - bb)/p, (b. - x)/pp, Vr, where p is the radius of the graph.
H.2 Hyperparameters
The models were implemented using Pytorch library and trained on CUDA. The graphs were built using Pytorch Geometric module. All models were trained on NVIDIA RTX 3060 GPUs.
de time-stepper The graph constructed using radius radius_graph defined in Pytorch Geometric. The node features and the edge features, which include the distance from the boundary points, are encoded into latent vectors of size 128 using 2 MLPs. The encoder uses two layers of interaction network. The latents are then processed by two layers of Neural Operator Transformer. The decoder layers are symmetric to the encoder layers. The NOT block uses 4 attention heads, branch and trunk sizes of 32, and an output dim of 128.
Learnable Kernel-Integral ROM We implement the kernel integration using Jax with a differentiable, grid-based Monte- Carlo approximation. Unlike standard grid methods, we interpret the Lagrangian particles not just mass points, but as Monte-Carlo samples of underlying continuous latent field. The architecture approximates the integral transform (x) = y)(y)) through learnable threc-stage pipeline parameterized by 0 = (Wene, 3:
Learnable Feature Encoding: First, we lift the raw Lagrangian drift up x2 x, into a high- dimensional latent feature space. This mapping is parameterized by shallow MT.P encoder Eg comprising two hidden layers of width 64 with GELU activations:
vp = Eo(u)) GELU(W. eneuup up+ + benc),
(11)
where vp E Rdd represents the learned feature weight for particle p. We set the latent feature dimension dy = (3 learned features + density channel) to balance expressivity with memory efficiency.
Grid Splatting & Monte-Carlo Integration: We approximate the continuous field (x(x on a regular Euclidean grid G of resolution 64 64 x 64 covering the normalized domain Ω [0, 17a. This step acts as discretized Monte-Carlo integration of the particle features onto the grid manifold:
G = Σ Vp A(x, - xi),
(12)
PEN())
where G3 is the feature vector at grid vertex i, and A(.) is the trilinear basis function. To enforce smoothness and emulate a contimuus Gaussian kernel operator, we apply a stochastic convolution step during sampling. We define the smoothed field value al query point X, as the expectation over a noise dist distribution & ~~ N/ o²1):
v(x,)) = Eg [Interp(G, x,, + E)
(13)
In practice, equation 13 is approximated via Monte-Carlo sampling with K = 8 paths per query, effectively convolving the grid features with a Gaussian kernel of width a = 0.42 (in grid units).
Residual Decoding: Finally, the smoothed grid features are mapped back to the physical drift space via a decoder MLP Dd (width 64, GELU). To presorve high-frequency details lost during the smoothing step, we employ a residual connection:
ig DD(((xx))
Uinterps
(14)
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where Mitterp is the direct trilinear interpolation of the drift field. The entire pipeline is trained end-to-end with Adam optimization (T-R=10-9) by minimizing the 1.2 reconstruction loss L = Eg [1, un 112
Optimizers Optimization is done with Adamax optimizer, with an initial learning rate of le-4, weight decay of 1e-6 and a batch size of 4. The learning rate was decayed exponentially from 10-1 to 10-6 using a scheduler, with a gammaa of 0.1 0/56/
Additional dataset details
We model the following classes of materials elastic, plasticine, granular, Newtonian fluids, non-Newtonian fluids, and multi-material simulations. Notably, this framework is compatible with data generated using different solvers such as Finite Element Method (FEM) Hughes (2012), Material Point Method (MPM) Jiang
et al. 2016), or Smooth Particle Hydrodynamics (SPH) Monaghan (1922].
Plasticine (von Mises Yield) Using the NCLAW simulator, generated 100 trajectories of 400 time steps (dt = 5e 4) with random initial velocilies and 4 different geometries Stanford bunny, Stanford armadillo, blub (goldfish), and spot (cow). The trajectories are modeled using Saint by
elastic model, given
P U(2pe + Atr(e))U"
(15)
where À and a are Lamé constants, P the second Mirchof stress and the strain. is obtained by applying SVD to the deformation gradient F UEVT. The von Mises yield condition denoted by
87 = |el|
Ty
(16)
2/
where 6 is the normalized Henky strain, is the yield stress.
Granular material (Drucker Prager sand flows) We trained the model on 2 datasets to simulate granular media. We generated 100 trajectories at 300 time steps, using NCLAW simulator and on the 2D Sand dataset released by Pfaff al. (2020). The Drucker-Prager elastoplasticity is modeled by the same Saint Mmaant-Kirchhoff elastic model, given by Equation 15 Additionally, the Drucker-Prager yield condition is applied such that
(3) + 2p)tr(() >
tr(e) or 87 || + a
(17)
2p
where, a 2
2sin6 sind and 0 is the frictional angle of the granular media.
To simulate clasticity, we generated simulations using moshes from Thingi10k datasot Zhou & Jacobson (2016). We generated 24 trajectories, with 200 time steps, for goometries train the model. The elasticity is modeled using stable neo-Hookean model, proposed in Smith et al. (2088). The energy is denoted by
Elasticity
V= " 1Ic- 3) > (J - a)²
" log +
(18)
where Ic refers to the first right Cauchy- -rreen---eeeeee invariant and J is the relative volume change. a and À are Lamé constants. The corresponding Piola-Kirchoff stress is given by
P p= Ic + 1 F X(+ - a) ØJ
(19)
OF
where F the deformation gradient.
Newtonian fluids For Newtonian fluids, In the 2D setting, use WaterDrop dataset created by Pfaff ct al. 2020), which is generated using the material point method (MPM). For the 3D setting, we generated 100 trajectories with random initial velocity, each spanning 1000 time steps at a dt of 5e 3. 3. This dataset was prepared using the NCLAW framework. These are modeled as weakly compressible fluids, using fixed corotated elastic model with = 0. The Piola-Kirchoff Stress given by
P= =J( 1)F-T
(20)
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Non-Newtonian fluids
To train the model on non-Newtonian fluids, we used the Goop and Goop-3D
datasets.
Multimaterial 2020).
We simulated multi-material trajectories in 2D using the dataset published by Pfaff et al.
J Training setup
Time-stepper We follow the exact training procedure outlined in Sanchez- Gonzalez et al. (2020) for de. We use the 1-step loss function over a pair of consecutive time steps and + 1, imposing a strong inductive bias towards a Markovian system. Each system is trained for 5 million steps.
The model is validated by full rollouts on 10 held-out validation sets per material simulation, with performance measured by the MSE between predicted particle positions and ground-truth particle positions.
kernel-integral ROM To train the kernel-integral ROM, we define the neighborhood over samples defined = 0, i.c. {= and (- For all the subsequent time-steps, enables us to evolve {f) using the spatial information of the state at t 0. The imput function for the model is point-wise positions at a given time-step t for non-fluid systems and deformation 2t To for fluid systems. We generate these using de. Each system is trained for 30,000 steps and evaluated on unseen discretizations and trajectories.
leverage this neighborhood. This
Additional results
2D simulations Figure presents qualitative comparisons between ground truth and (P-d)-point predictions produced by GIOROM on various 2D simulations. The subfigures depict range of dynamic behaviors: (a) granular flow, (b) soft body motion under gravity, (c) external force acting highly elastic material, and (d) coupled interactions between granular media and Newtonian fluids.
predicted
(Q)
predicted
ground truth
ground truth
(b)
(d)
predicted
predicted
ground
Figure 8: 2D point-cloud simulations: (P d)-point inference results using GIOROM. (a) Granular flow;
(b) Soft body under gravity; (c) Elastic response to external force; (d) Coupled fluid-granular interactions.
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L
Ablations
Ablations on PDE operator Following Wu et al. (2024a); Alkin et al. (2024), we utilize interaction networks to model Lagrangian inputs. To justify this choice, we compare against. GNS and neural operator models built for Eulerian regimes. Table models on reduced sample sets. The performance is measured the average MSE accumulated over the entire duration. We compare against GINO (Eulerian) Li et al. (2024). General Neural Operator Transformer GNOT (Eulcrian) Hao et al. 22023 and Inducing Point Operator Transformer IPOT (Eulcrian) Loe &
@ represents the rollout performance of different Neural Operator
Oh 2024 Additionally, we compare against graph neural network based model GNS Sanchez-Gozzalez et al. 2020). We do not consider the Lagrangian field variant the UPT family as a baseline as it neither encodes particle interaction dynamics nor supports autoregression, making it incompatible with the datasets used in this work. All the operators struggle generalize to fluid simulations but interaction network basod models (GNS and GIOROM) exhibit the best performance.
Table 6: This table compares the rollout MSE of our PDE operator de against other neural PDE operators on reduced sampled sets I This highlights the importance of interaction network encoders and decoders for modeling particle dynamics.
MODEL
WATER-3D PLASTICINE ELASTIC
SAND-3D
GNS
0.0108
0.0038
0.0003
0.0008
GINO
0.09
0.07
GNOT
0.046
0.0052
0.0028
0.0085
IPOT
0.15
0.097
0.084
0.0075
de
0.0106
0.0008
0.0004
0.0009
Number of message-passineg layers We show that the key bottleneck in terms speed the message- passing operation within the Interaction Network encoder and decoder.
Table 7: This table shows that the number of messagc-passing layers results a negligible improvement in rollout Loss.
NUM. MESSAGE PASSING LAYERS
CONNECTIVITY
INPUT SIZE
INFERENCE TIME/STEP
LOSS
U.077
2247
3.6
0.0008
0.077
2247
3.8
0.0009
0.077
2247
4.2
0.0014
0.077
2247
4.3
0.0009
L.1
Sampling strategy and rollout loss
We compared different sampling strategies against. the rollout Loss (MSE). The results presented in Table
Table 8: Comparison of different sampling and graph construction strategies against rollout MSE on Water-2D
dataset
SAMPLING STRATEGY
GRAPH TYPE ROLLOUT MSE
RANDOM
RADIUS
0.0098
RANDOM
DELAUNAY
7.017
FPS
RADIUS
0.0097
FPS
DELAUNAY
8.04
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M Additional discussions
M.1 Understanding the correlation between sparsification and performance
We study the relationship between performance and sample size to understand how the structural fidelity of the interaction graph degrades the predictions under extreme sparsifications. Our analysis focuses on a small Water2D system (678 particles), where the graph structure can be explicitly visualized using networkx, as well larger Plasticine3D system. We consider different sampling levels to examine how graph sparsification impacts this relationship.
Figure 9 presents rollout MSE at two sampling ratios vatios-35% and 11%-for the Water2D system. At 35% sampling, we observe a trend: rollout MSE consistently drops with improving the connectivity. However, with further modifications to the radius, the MSE increases. This suggests that beyond a point, the addition of edges leads degradations. The visual rollout fig. T0 shows that improving connectivity improves the performance upto limit.
In contrast, 11% sampling, the rollout. MSE remain high regardless of radius, and clear trend emerges. As shown in fig. I1 the predicted rollouts diverge significantly from the ground truth. The third column in each frame represents the reduced-space ground truth at (r d) points, which itself is visually and physically distinct from the full-resolution system. This indicates that at extreme sparsily, the reduced graph fails to retain sufficient physical characteristics, rendering it unsuitable for accurate inference.
Figure 12 extends this analysis the Plasticinc3D system. At 22% sampling, we again observe that rollout MSE initially decreases, but the radius increases further, this trend reverses. However, al 3.3% sampling, there is no correlation.
Rollout MSE trends (Water-2D 85% sampling)
Rollout MSE trends (Water-2D 11% sampling)
-ooolouu
-i-Rellut
0.08
0.035
0.025
Rollout MSE
Rollout MSE
0.025
0.02
0.015
0.01
0.01
0.006
0.005
0.02
0.025
0.05
0.035
0.038
0.042
0.08
0.02
0.035
0.045
0.047
0.052
0.058
0.068
Graph Radius
Graph Radius
Figure 9: Water2D: Radius vs. rollout MSE. Left: 35% sampling. Right: At 11%, there is a weak correlation between them.
M.2 Limitations of kernel-integral ROM to extreme sparsifications
The kernel-integral ROM model employs an implicit neural representation to approximate discrete kernel integral transforms over sparse point-wise function evaluations. However, under extreme sparsification, the quality of this representation degrades significantly.
When the number of supervised function evaluations is severely limited, increasing the neighborhood does not lead to meaningful improvements. Instead, the model interpolates around the sparse active voxels, leading to point clustering near known estimates or interpolation artifacts that resemble piecewise-linear transitions. These effects arise not due to kernel parameter choices, but due to a fundamental lack of information in the reduced space: the interpolated function no longer reflects a smooth or physically meaningful structure.
This limitation is evident in fig. 13 where we interpolate a 2-million-point Dragon mesh from three different supervision levels: 700 points (0.035%), 20K points (1%), and 60K points (3%). At 700 points, the
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N: 678 E: 13296
N: 240 E: 597
N: 240 E: 2974
N: 240 E: 7572
R: 0.027 g:
R: 0.020
R: 0.085
R: 0.060
Ground Truth R
Ground Truth Brd
Figure 10: Water2D at 35% sampling. First two columns: full-space ground truth and prediction on (P-d) points. Third column: reduced-space ground truth at (r-d) d) points. Right columns: predictions at varying radii Rollouts remain visually and physically consistent. with the full system at R-0.035
interpolated output exhibits dense clustering and abrupt discontinuities. The gap between 0.035% and 1% remains substantial. The top-row result makes clear that in the regime of extreme sparsification, the model fails to maintain coherent spatial representations, regardless of neighborhood density or kernel formulation.
This example illustrates the lower bound on sparsity below which the Icarned representation fails to extrapolate. In such cases, the reduced-space support does not sulficiently reflect the full system's 's physical characteristics, making the reconstruction problem ill-posed.
Sensitivity Analysis & Convergence To evaluate the impact of kernel parameteeizationn, we perform an ablation study on the two primary discretization parameters: the background grid resolution R and the Lagrangian sparsity factor S. Figure 14 summarizes the results the Elastic (Owl) dataset.
In Figure 14 (4a), we observe a clear convergence behavior with respect to the grid resolution. At coarse resolutions (R = 20), the grid fails to resolve high-frequency deformation gradients, resulting in higher reconstruction error (~ 5.6%). As the resolution increases, the error decays rapidly, effectively plateauing around = 64. This confirms that once the grid frequency exceeds the Nyquist limit of the underlying deformation field, the error becomes dominated by the neural approximation rather than spatial discretization, justifying our choice of R 64
elficient operating point.
Figure 14 () demonstrates the method's sensitivity to Lagrangian sampling density. Notably, the reconstruc- tion remains robust even as the available tracking markers become extremely sparse. Inereasing the sparsity factor from 12x to 100x (reducing the particle count by an order of magnilude) results in only a marginal increase in error (from ~ 1.9% to ~ 3.0%). This stability supports our theoretical claim that the method relies on the effective latent support of the grid manifold rather than raw particle density. The shaded regions indicate the standard deviation across test trajectories, showing that while higher sparsity introduces slightly more variance, the method avoids catastrophic failure modes in data-starved regimes.
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N: 678 E: 13296
N: 80 E: 36
N: 80 E: 107
N: 80 E: 537
R: 0.027 0:
R: 0.020
R: 0.047
R: 0.068
Ground Truth R
Ground Truth R
Ay
Figure 11: Water2D at 11% sampling. All predictions diverge from ground truth. Even at increased radi,, rollout MSE remains high. Reduced-space ground truth itself is physically dissimilar to the original system, limiting learnability.
Rollout MSE trends (Plasticine 22% sampling)
Rollout MSE trends (Plasticine 3.3% sample)
-ooloot MSE
MS-RollouMMEE
0.06
0.25
Rollout MSE
Rollout MSE
0.06
0.2
0.15
0.08
0.02
0.05
0.115
Graph Radius
Graph Radius
Figure 12: Plasticinc3D: rollout MSE. Left: 22% sampling shows consistent correlation. Right: At 3.3% sampling, correlation degrades as the reduced graph fails to encode sufficient structure for accurate rollout. At 35%, rollout MSE reduces 9.2c-4, while plateaus at 0.01 3.3%
M.3 GNNs as neural operator approximations
We now argue that Graph Neural Networks (GNNss, under a suitable construction, can interpreted as discretizations of kernel intogral operators. In particular, GNNs can be viewed data-driven approximations class of neural operators. However, GNNs are senstive input graph topologies, and cannot generalize to arbitrary graph structures defined discretization. This can be seen in fig. 15 where the model
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Kernel ROM Evaluations in R
Rd
Time Axis
Figure 13: Degradation under extreme sparsification. All rows show Bernel-integral ROM interpolation on a 2-million-point Dragon mesh. Top: 700 function evaluations (0.035%) lead to clustered and discontimuous outputs. Middle: 20K points (1%) Bottom: 60K points (3%) produces smoother reconstruction.
Impact Resolution
b. Impact Sparsity Factor
0.3%
4,5%
7.0%
4.0%
3.5%
Rolative L2 Errer
3.0%
2.5%
3.3%
2.0%
2.0%
20
40
50 70 90
100
80
100
(R
Spansity Foctor (1/S)
Figure 14: Ablation study on spatial resolution and sampling density for Elastic dynamics, (a) Error convergence with respect to grid resolution (R²). The method exhibits rapid convergence, platcauing R 64, indicating that the 64 grid sufficiently captures the latent deformation manifold. (b) Robustness Lagrangian sparsity fixed grid size of 64. The model maintains low reconstruction error (< 3%) even when the imput particle density is reduced by factor of 100 (Sparsity Factor S = 100), showing that in regimes where the topology is stable, the model is more robust to sparsification. Shaded regions denote #1 standard ion across the Lest. set.
generalize to different sampling strategies but different graph structures defined over the same sample size.
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N: 678, E:4316, Ro0.015
R: 0.015
N: 204, E: 3584, R: 0.050
N: 204, E: 3628, R: 0.050
N: 204, E: 504, NA
X
Whatt
sol
Full Order Graph
(b) Sampled Radius Graph several components
Randomly Sampled
(c) Farthest Point Sampled
Randomly Sampled Delaunay Graph
Radius Graph with single component
Radius Graph single component
Figure 15: This figure illustrates the critical role of radius- based graphs in maintaining discretization invariance and preserving connectivity in graph structures. Both random sampling and farthest-poinn point sampling produce consistent embeddings due to stable neighborhood definitions, whereas altering the graph construction method breaks the model.
M.3.1 GNNs as Monte-Carlo estimators
We discuss the following propositions made in Li et al. (2020b). Let. D C TR be a compact domain. For each # D, let Uy be a fixed Borel measure on D. We define a kernel integral operator of the form:
(a) = (Wo(x) + / no((, v)v(y) des())
where a IR + is a fixed nonlincarity applied elementwise, W E Rax is a learned matrix, and Ki : (g(d-1) + Raxx is a neural network parameterized by Φ. The arguments to Kg can include geometric and positional information such as (I, yz,y. y. || yl)).
In practice, do not have access to the continuum D, and instead work with finite point cloud (aI,..., σK} D. We approximate the measure Va by an empirical distribution over neighborhood N(x) around each ær typically defined via radius graph. Then, the integral operator is approximated by sum.
nole(a, (()(),)))(())) Σ (K)t)(x) a (wole)+ [V(r] yEA((z)
where e(x, y) denotes edge features constructed from (x, y). This yields the following message- message-passing update rule:
(x) =o (Wo((s)+ W(z)
Σ yEN(a)
(a(e(z,)))o'
We interpret this a discretization of the continuum operator Kes, where the integral is approximated by local aggregation, and the kernel Kg is represented by a shared neural network acting on edge features.
The key modeling decision is that Ky defines a K x block kernel matrix, where each entry Ks (a 2y) is a matrix Rax=, The parameters are shared across all edge pairs. This sharing ensures that the operator is independent of the number and arrangement of discrete points, so long the neighborhoods are constructed consistently. Therefore, the learned operator exhibits discretization invariance.
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M.3.2 Discretization resolution?
Figure 16 presents four point clouds of identical size and geometry but with differing point configurations. Despite many non-overlapping locations, the predictions across all four discretizations remain consistent. This visual evidence supports the notion of learned model generalizing across different samplings of the same underlying domain. It is important to distinguish this from resolution, which concerns the density number of points used to represent the domain. Discrelization refers to the specific arrangement of fixed number of points; resolution, by contrast, changes the number of points altogether. This is subtle difference.
Disc. 1
Disc. 2
Disc. 3
Disc. 4
Time
Water 3D
Figure 16: Discretization invariance on the same geometry. We construct four different point clouds sampled from the same domain, with equal size but varying locations. On cach, we build radius graph with fixed connectivity radius and apply the same learned GNN layer. Despite the different discrete realizations, the resulting outputs are consistent.
Implicit handling of self-contact Training data were generated using MPM solvers that handle self- contact. implicitly via a background Eulerian grid, applicable to both solids and fluids. As result, the model implicitly learns self-contact behavior from data. Future work could explore improved sampling strategies in regions prone to self-collision for enhanced resolution.
N Results
In this section, present the visual comparisons against baseline ROM architectures.
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GROUND
CROM
DINO
GKI
LICROM
GIOROM (OURS)
Figure 17: Comparisons on Water Dataset. Qualitative evaluation of the reconstructed flow dynamics for the WATER-3D dataset. GIOROM effectively preserves fine-scale splashing features compared to global ROM basclines.
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GROUND
CROM
DINO
GKI
LICROM
GIOROM (OURS)
Figure Comparisons on Plasticine Dataset. Qualitative results for the PLASTICINE-3D dataset, demonstrating the model's ability handle viscoplastic deformations. GIOROM maintains structural coherence.
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GROUND TRUTH
COLORA
DINO
GKI
LICROM
GIOROM (OURS)
Figure 19: Comparisons on Elasticity Dataset. Visual comparison for the ELASTICITY-3D (Owl) dataset. The results highlight the method's performance continuous elastic deformation.
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GROUND TRUTH
COLORA
CORAL
DINO
GKI
LICROM
GIOROM (OURS)
Figure 20: Comparisons on Granular Sand Dataset. Evaluation on the SAND-3D dataset, characterized scattering and discrete particle behavior. GIOROM successfully reconstructs the granular flow patterns.
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Figure 21: 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 10-2. (Middle) Elastic collisions (Jelly) over 500 time steps with 84K particles; rollout MSE: 6.4 x 10-5, (Bottom) Phase transition in plasticine- like material (Chocolate) over 2000 time steps with 24K particles; rollout MSE: 3.2 x 4
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Time Stepper Prediction
Kernel-ROM Inference
Time Stepper Prediction
Kernel-ROM Inference
Figure 22: Subspace vs full-space inference: Rows and show predictions from the time-stepper on the sampled space; Rows 2 and 4 depict Kernel-ROM output projected back to the full discretization.
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Prediction
Ground Truth
Prediction
Ground Truth
Figure 23: Ground-trtth comparisons: Rendered outputs for water and sand simulations contrasted against reference solutions.
42


Reviews:
[{"review_id": "Z3yDyXroAv", "reviewer": "Reviewer_nUUM", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a neural approximation framework for simulating physical systems governed by Lagrangian interaction dynamics, aiming to improve efficiency at high spatial resolutions. While many existing neural solvers adopt a reduced-order modeling (ROM) paradigm that evolves a low-dimensional latent representation, such global projection-based approaches can struggle to capture localized, rapidly evolving dynamics. The authors instead evolve a sparsely subsampled set of particles directly in physical space using a learned neural time-stepper, and reconstruct the full field via a learnable, locally defined kernel-based estimator. By decoupling reduced-order temporal evolution from spatial reconstruction, the method preserves locality while substantially reducing the number of active degrees of freedom during simulation. Experiments across several Lagrangian systems, including fluids and elastoplastic materials, demonstrate favorable accuracy–efficiency trade-offs and stable long-horizon rollouts.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The experiments show improved accuracy over ROM baselines as well as clear computational efficiency gains, providing convincing support for the paper’s claims.", "audience_yn": "Yes", "audience_explanation": "Neural operators for approximating PDE solvers and data-driven PDE modeling constitute a rapidly growing area of research, with broad potential applications across many domains, particularly in AI for Science. Addressing scalability to high-resolution fields remains one of the most important practical bottlenecks in this area, and the motivation of the paper directly targets this challenge.", "requested_changes": "- The paper motivates the proposed approach by arguing that reduced-order modeling (ROM) methods are inherently prone to losing important local information due to projection onto low-dimensional spaces. It would be helpful for the authors to elaborate on this claim. In particular, is this criticism intended to apply only to the specific ROM baselines considered in the paper (e.g., PCA- or autoencoder-based global latent models), or does it extend more broadly to latent space learning approaches in general? In many other areas of machine learning, learning continuous latent representations has proven highly effective, with sufficient flexibility to reconstruct fine-grained local details. Clarifying whether and why such representations are especially challenging to employ for physical systems governed by Lagrangian dynamics would strengthen the motivation and positioning of the proposed method.\n\n- In Figure 3 (upper-left and upper-right), both GPU usage and runtime exhibit noticeable non-monotonic behavior, with apparent outliers around the 0.25× sampling ratio. Could you clarify the source of these bumps? \n\n- Is the Hilbert space assumption valid for practical scenarios?", "broader_impact_concerns": "I don't find any broader impact concern for this paper.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "OXUXhiy3f3", "reviewer": "Reviewer_ZH43", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes a method called GIOROM (Geometry-informed reduced order modelling), which is a framework for simulating Lagrangian particle dynamics in physical systems. The key idea is to replace the traditional projection-based reduced-order modelling with a sampling-based approach that evolves dynamics in physical space with a learnable kernel parameterization. \n\nStrengths:\n\n1. The method introduced seems novel in that it combines reduced-order modeling with direct physical space evolution, avoiding global latent representations that struggle with localized dynamics.\n\n2. Empirical results reported are convincing. \n\n3. Experiments are reported on multiple material types with detailed ablations studying sparsity effects, sampling strategies, and architectural choices.\n\n4. The approach is agnostic to underlying neural operator architecture, making it broadly applicable.\n\nWeaknesses:\n\n1. Paper lacks theoretical justification about the following: Convergence guarantees for the kernel approximation, Error bounds as a function of sampling density and/or justification for when the method should work. \n\n2. Extreme sparsity failure is poorly characterized. \n\n3. The reported baselines are rather limited ROM baselines are under-tested, and no comparison with recent learning-based ROM methods (beyond 2023). \n    \n4. Computational cost analysis is incomplete in that the memory scaling with radius could be prohibitive and there is no wall-clock time comparisons for complete simulations.\n\n5.  Experiments on only 100 trajectories per system seem modest for deep learning\n  \n6. There is no discussion of generalization to different boundary conditions or domain sizes.\n\n7. Writing can be made a little lighter.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The claims made are failry supported with empiral evidence (some gaps exists, see questions).", "audience_yn": "Yes", "audience_explanation": "This paper deals with an important sub-area of deep learning research, namely, the neural PDE operators, and therefore is of interest to the community.", "requested_changes": "1. Provide empirical lower bounds on sampling density as a function of system properties (e.g., velocity gradients, material stiffness, geometry complexity).\n\n2. Please consider testing all ROM methods across all material types, not just water, by including recent neural ROM methods.\n\n3. Consider adding a table showing when GIOROM vs. other methods is preferable.\n\n4. Even approximate error bounds or convergence conditions would strengthen claims significantly.\n\n5. Presentation can be reorganized -  critical figures (8-9) to can be moved to the main paper\n\n6. Code and datasets should be released for reproducibility along with hyperparameter selection strategies.\n \n7. Consider discussions/ demonstration of adaptive time-stepping.\n\n8. Fix notation inconsistencies. \n\n9. Discuss computational requirements for graph construction overhead.\n\n10. Provide ablation on the window size for velocity sequences.", "broader_impact_concerns": "NA", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Q9EGtzNp1a", "reviewer": "Reviewer_QeMW", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces GIOROM, a novel sampling-based, discretization-invariant reduced-order modeling framework for Lagrangian physical systems that overcomes key limitations of traditional ROMs and neural PDE solvers. Instead of evolving a global latent state, GIOROM directly evolves a small set of representative particles using a data-driven autoregressive neural time-stepper built from interaction networks and neural operator transformers, drastically reducing active degrees of freedom while preserving local dynamics. To recover full-resolution fields, the method introduces a learnable, locality-aware kernel parameterization that reconstructs the solution manifold at arbitrary spatial points, outperforming projection-based ROMs in highly dynamic regimes such as fluids.", "claims_and_evidence_yn": "No", "claims_explanation": "Table 2 presents the performance of GIOROM on several physical systems. However, the results reported in this table are shown only for GIOROM, without comparison to any baseline methods. As a result, it is unclear whether the reported MSE values are sufficiently good to support the author’s claim that the proposed method is valid and effective for these physical systems. Although the paper discusses a wide range of related work, the experimental comparison is limited. In Table 3, the authors compare their method only against other neural PDE operators, and the evaluation metric is restricted to rollout MSE. This comparison would be more convincing if Table 4 also included computational cost metrics (e.g., runtime or memory usage) alongside MSE. In addition, more qualitative comparisons should be provided through visualizations of predicted trajectories or fields for different methods, rather than limiting such visualization only in Figure 2 (a).", "audience_yn": "Yes", "audience_explanation": "Many individuals in TMLR’s audience would likely be interested in the proposed method and its application to the physical systems.", "requested_changes": "- In Figure 1, it is difficult to clearly understand the differences between prior ROM techniques and GIOROM. In particular, the distinction between the “PDE stepper” and the “Neural stepper” blocks is unclear, as is the role and input of the decoder g and the kernel network k. The purpose of the dotted-line box on the right-hand side of the figure should also be more explicitly explained. \n\n- Additional ROM baselines, such as CROM, should be included in Table 2 for a more comprehensive comparison. Moreover, the effect of sparsity on rollout error, as shown in Figure 3, should also be compared against these alternative ROM methods.", "broader_impact_concerns": "There is no ethical concern.", "additional_comments": "- In Figure 2, the fluid predictions produced by other baseline methods appear to extend outside the simulation domain. Is this physically plausible, or could it be an artifact of the visualization or model behavior?\n\n- In Figure 3, there appears to be an inverse trend when the sampling ratio decreases from 0.5 to 0.25 in both the “Sparsity vs. GPU” and “Sparsity vs. Time” plots. Could the authors explain the reason for this unexpected behavior?", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)