LaM-SLidE: Latent Space Modeling of Spatial Dynamical Systems via Linked Entities
Authors: Florian Sestak
Artur P. Toshev
Andreas Fürst
Günter Klambauer
Andreas Mayr
Johannes Brandstetter
Venue: NeurIPS 2025 poster ()
Keywords: diffusion models
flow-matching
latent space models
latent diffusion models
dynamical systems
simulation
generative modeling
molecular dynamics
unified representations
Primary area: deep_learning
TL;DR: We propose a method for modeling dynamical systems, that bridges efficient latent space modeling with entity tracking by introducing identifier representations that maintain entity traceability within a latent system representation.

Abstract: Generative models are spearheading recent progress in deep learning, showcasing strong promise for trajectory sampling in dynamical systems as well. However, whereas latent space modeling paradigms have transformed image and video generation, similar approaches are more difficult for most dynamical systems. Such systems -- from chemical molecule structures to collective human behavior -- are described by interactions of entities, making them inherently linked to connectivity patterns, entity conservation, and the traceability of entities over time. Our approach, LaM-SLidE (Latent Space Modeling of Spatial Dynamical Systems via Linked Entities), bridges the gap between: (1) keeping the traceability of individual entities in a latent system representation, and (2) leveraging the efficiency and scalability of recent advances in image and video generation, where pre-trained encoder and decoder enable generative modeling directly in latent space. The core idea of LaM-SLidE is the introduction of identifier representations (IDs) that enable the retrieval of entity properties and entity composition from latent system representations, thus fostering traceability. Experimentally, across different domains, we show that LaM-SLidE performs favorably in terms of speed, accuracy, and generalizability. Code is available at https://github.com/ml-jku/LaM-SLidE.

LaM-SLidE: Latent Space Modeling of Spatial Dynamical Systems via Linked Entities
Florian Sestak!
Artur Toshev?
Andreas Fürst'
Günter Klambauer*
Andreas Mayr". Equal contribution
Johannes Brandstetter
ELLIS Unit, LIT AT Lab, Institute for Machine Learning, JKU Linz, Austria 2 Department of Engineering Physics and Computation, TUM, Germany
3 Emmi Al GmbH, Linz, Austria, Cllinical Research Institute for Medical AI, JKU Linz, Austria (klambauer mayr, brandstetter)nn j.k.utt
Abstract
Generative models are spearheading recent progress in deep learning, showcasing strong promise for trajectory sampling in dynamical systems well. However, whereas latent space modeling paradigms have transformed image and video gen- eration, similar approaches are more difficult for most dynamical systems. Such systems - from chemical molecule structures to collective human behavior are described by interactions of entities, making them inherently linked connectivity patterns, entity conservation, and the traceability of entities over time. Our ap- proach, LAM-SIIDE (Latent Space Modeling of Spatial Dynamical Systems via Linked Entities), bridges the gap between: (1) keeping the traceability of individual entities in a latent system representation, and (2) leveraging the efficiency and seala- bility of recent advances in image and video generation, where pre-trained encoder and decoder enable generative modeling directly latent space. The core idea LAM-SLIDE is the introduction of identifier representations (IDs) that enable the retrieval of entity properties and entity composition from latent system represen- tations, thus fostering traceability. Experimentally, across different domains, show that LAM-SLIDE performs favorably in terms speed, accuracy, and gener- alizability. Code is available wtthhtp://// :// thub. com/m.ommm//momomo ml- Jau/Laa-
Introduction
Understanding dynamical systems represents a fundamental challenge across numerous scientific and engineering domains [47
In this work, we address spatial dynamical systems, characterized by scenes of distinguishable entities at defined spatial coordinates. Modeling temporal trajectories of such entities quickly becomes challenging, especially when stochasticity is involved. prime example is molecular dynamics [47), where trajectories of individual atoms are modeled via Langevin dynamics, which accounts for omitted degrees of freedom by using stochastic differential equations. Consequently, the trajectories of the atoms themselves become non-deterministic.
46 73
conventional approach to predict spatial trajectories entities is to represent scenes as neighbor- hood graphs and to subsequently process these graphs with graph neural networks (GNNs). When using GNNs nearby are connected by an edge in the neighborhood graph. Neighborhood graphs have extensively been used for trajectory prediction tasks [51] especially for problems with large number of indis- tinguishable entities. [e.g., 8I 633. Recently, GNNs have been integrated into generative modeling frameworks effectively capture the behavior of stochastic systems 9g
32 [T [6), each entity is usually represented by node, and the spatial entities
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

Despite their widespread use in modeling spatial trajectories, GNNs hardly follow recent trends in latent space modeling, where unified representations 44 together with universality and scalability of transformer blocks [91] offer simple application across datasets and tasks, a behavior commonly Notably, recent breakthroughs in image and video generation can be attributed to latent space generative modeling [38] 15]. In such paradigms, pre-trained encoders and decoders are employed to map data into a latent space, where subsequent modeling is performed, leveraging the efficiency and expressiveness of this representation. This poses the question:
observed in computer vision and language processing [24 25
Can we leverage recent techniques from generative latent space modeling to boost the modeling of stochastic trajectories of dynamical systems with varying number of entities?
Recently, it has been shown [5] that it is possible to model the bulk behavior of large particle systems purely in the latent space, at the cost of sacrificing the traceability individual particles, which is acceptable even favorable for systems where particles are indistinguishable, but challenging for, e.g., molecular dynamics, where understanding the dynamics of individual atoms essential.
Graph Neural Networks
LaM SLidE
Latent Diffusion (conditional generation)
based tracking
pntatir Inaget
3
2D
D
1+
f + At
Figure Overview of our approach. Left: Conventional graph neural networks (GNNs) model time-evolving systems (e.g., molecular dynamics) by representing entities as nodes and iteratively updating node embeddings and positions to capture system dynamics across timesteps. Right: Latent diffusion models employ encoder-decoder architecture to compress input data into lower-dimensionss latent space where generative modeling is performed. Latent diffusion models, frequently enhanced with conditional information such text, excel generative tasks; however, due to their fixed input/output structure, they are not directly adaptable to physical systems with a varying number of entities. Middle: Our proposed approach LAM-SLIDE bridges these paradigms by: (1) introducing identifiers that allow traceability of individual entities, and (2) leveraging a latent system representation.
To leverage latent system's 's representation, we need to be able to trace individual entities within the system. The core idea of LAM-SLIDE is the introduction of identifiers (IDs) that allow for the retrieval of entity properties, e.g., entity coordinates, from the latent system representation. Consequently, we can train generative models, like flow-matching [54 [7] 2) purely in latent space, where pre-trained decoder blocks map the generated representations back to the physics domain. An overview given Section [ Qualitatively, LAM-SLIDE demonstrates flexibility and performs favorably across a variety of different modeling tasks.
Summarizing our contributions:
Latent system representation: We propose LAM-SLIDE for generative modeling of stochastic trajectories, leveraging latent system representation.
Entity structure preservation: We introduce entity structure preservation recover the encoded system states from the latent space representation via assignable identifiers.
Cross-domain generalization: We perform experiments in different domains with varying degrees of difficulty, focusing on molecular dynamics, human motion behavior, and particle systems. LAM-SLIDE performs favorably with respect all other architectures and showcases scalability with model size.

---

2 Background & Related Work
Dynamical systems. Formally, we consider random dynamical system to be defined by state space S, representing all possible configurations the system, and an evolution rule & R x S + S that determines how state S evolves over time, and which exhibits the following properties for the time differences 0, f1, and, lz:
((0,s) =8 (f + f)
(f2, (f1,s))
We note that Φ does not necessarily need to be defined on the whole space R x S, but we assume this for notational simplicity. The exact formal definition of random dynamical systems more involved and consists of base flow (noise) and a cocycle dynamical system defined on a physical phase space [ We skip the details, but assume that deal with random dynamical systems for the remainder of the paper. The non-deterministic behavior of such dynamical systems suggests the use of generative modeling approaches.
Generative modeling. Recent developments in generative modeling have captured widespread interest. The breakthroughs of the last years were mainly driven by diffusion models 87 88
paradigm that transforms a simple distribution into a target data distribution via iterative refinement steps. Flow Matching [54] [5]] has emerged powerful alternative diffusion models, enabling
simulation-free training between arbitrary start and target distributions [55]. It has also been extended data manifolds [99. This approach comes with straighter paths, offering faster integration, and has been successfully applied across different domains like images [27] audio [92], videos 771,, protein design [1). and roboties (13].
Latent space modeling. Latent space modeling has achieved remarkable success at image and video generation 27), where pre-trained encoders and decoders map data into a latent space, and back into the physics space. The latent space aims to preserve the essential structure and features of the original data, often following compositional structure D A & 85 5 where the encoder & maps the input signal into the latent space, the approximator A models a process, and the decoder maps back to the original space. Examples of approximators include conditional generative modeling techniques, such as generating an image given a text prompt (condition) (77). This framework was recently used for 3D shape generation, where the final shape in the spatial domain is then constructed by querying the latent representations over a fixed spatial grid [TOT] 100]
3 LaM - SLidE
Encoder
Decoder
Input Token
मلनन
NN
input tokens
K
Features
Da
Z
Cross Attention
LayerNorm
LayerNorm
K
Cross Attention
Pos
D,
L latents
D.
D2
Figure 2: Architecture of our secoder-decodee structure (First Stage): Left: The encoder maps N
input tokens to a latent system representation by cross-attending to learned latent query tokens. The decoder reconstructs the input data from the latent representation using the assigned IDs. Right: Structure of the input token, consisting of an ID, spatial information, and features (see also Figure
We introduce an identifier pool and an identifier assignment function which allows us to effectively map and retrieve entities and from latent system representation. The ID components preserve the relationships between entities, making them traceable across time-steps. LAM-SLIDE follows an encoder E approximator A decoder D paradigm.

---

3.1 Problem Formulation
State space. We consider spatial dynamics. Our states E S describe the configuration of entities within the scene together with their individual features. We assume that a scene consists of N entities e with E ,.....N. N. properties m, E Pi (e.g., atom type, etc.). We denote the set of entities as F = {ei..... ,......... Fa consider states s at discretized timepoints t. Analogously, we use x/. m to describe coordinates and properties at time f. respectively. We refer to the coordinate concatenation [x], x of the N entities in s' as X E R N= Analogously, we use M E IR" N to denote [mi... m'yl When properties are conserved over time, i.e. M' M', we just skip the time index and the time-wise repetition of states and use M R Nx P.,,, We concatenate sequences of coordinate states X' with I E I. T to tensor X E xxTxxxDD which describes a whole sampled coordinate trajectory of system with 7 time points and N entities. An example of such trajectories from dynamical systems are molecular dynamics trajectories (e.g. Figure Notation is summarized in 6
An entity e, described by its spatial location x, E R and some further
We
Prediction task.
We predict a trajectory of entity coordinates X[T.+I: T] X E ( given short (observed) initial trajectory X Tol
X X
X E RTxxNxD together with general (time- (ime-invariant) entity properties M. Here, To denotes the length of the observed trajectory and T T To denotes the prediction horizon.
X ,..,., , ,'...,,""
3.2 Entity Structure Preservation
We aim to preserve the integrity of individual entities in a latent system representation. More specifically, we aim preserve both the number of entities as well as their structure. For example, in the case of molecules, we want to preserve both the number of atoms and the atom composition. We therefore assign identifiers from an identifier pool to each entity, allowing us to trace the entities by their assigned identifiers. The two key components are: (i) creating fixed, finite pool of identifiers (IDs) and (ii) defining a unique mapping between entities and identifiers.
Definition 3.1. For a fixed u E N, let I = {0, 1..., u - 1} be the identifier pool. An identifier i is an element of the set
Definition 3.2. Let E be a finite set of entities and I an identifier pool. The identifier assignment pool is the of all injective functions from F. to I:
I {ida(-) E + I Ve,, ej E :e E, = ida(es) # ida(e))). An identifier assignment function ida(-) is an element of the set I.
Proposition 3.3. Given an identifier pool I and finite of entities F,, identifier assignment pool I as defined by Definition 3.2 is non-empty if and only if E < [Z).
Proof, see App. F.1
Proposition 3.4. Given an identifier pool I and a finite set of entities E such that |E| < [I], the identifier assignment pool I as defined by Definition 3.2 contains finitely many functions.
Proof, see App.
Notably, since ida(-) may be selected randomly - the specific choice of the mapping ida(-) can be arbitrary the only requirement is that injective mapping between entities and identifiers is established, i.e., each entity uniquely assigned to an identifier, but not all identifiers need to be assigned to an entity. Further, Proposition 3.3 suggests using an identifier pool that is large enough, such that model learned on this identifier pool can generalize across systems with varying numbers of entities.
Example aspirin. Aspirin CgHgO, consists of 21 atoms. thus the identifier pool I needs to have least 21 unique identifiers. We select an assignment function ida(-). arbitrary, and use it to assign each atom unique identifier. Notably, e.g., for molecules, we do not explicitly model molecular bond in- formation, as the spatial relationship between atoms (interatomic distances) implicitly captures this information. Figure shows an arbitrary fixed identifier assignment for aspirin; we illustrate different IDs by colored fingerprint symbols.
Figure Example as- pirin: IDs are assigned to the atoms of the molecule.

---

3.3 Model Architecture
Since predicting continuations of system trajectories is a conceptually similar task to generating videos from an initial sequence of images, we took inspiration from Blattmann et al. [15] in using latent diffusion architecture. We also took inspiration from Jaegle al. [43] decompose our model architecture as follows: To map the state of the system composed of N entities to a latent space containing learned latent tokens (E RD:),, ) we use a cross- cross-attention mechanism. In the resulting latent space, we aim to train an approximator to predict future latent states based on the embedded initial states. Inversely to the encoder, we again use a cross-attention mechanism to retrieve the physical information of the individual entities of the system from the latent system representation. To wrap up, LAM-SLIDE is built up by an encoder & approximator A decoder architecture, D o A &. A detailed composition of & and D is shown in Figure
The encoder aims to encode a state of the system such that the properties of each entity en can be decoded (retrieved) later. At the same time, the structure of the latent state representation Z' E TR. I.xD, is constant and should not depend on the different number N of entities. This contrasts with GNNs, where the number of latent vectors depends on the number of nodes.
Encoder.
To allow for traceability of the entities, first embed each identifier i in the space DD. by learned embedding IDEmb I+ RD,,, We map all (n ,,....N) system entities e, to u, E IR as follows:
ida())
(arbitrary identifier assignment) Vn E 1, N
Un IDEmbbida(e.))
(5)
The inputs to the encoder comprise the time dependent location x E RD= properties m, E R=, and identity representation u, R of each entity cn as visualized in the right part of Figure concatenate the different types of features across all entities of the system: X' = [xf,,.,"j x [m1,-., .., my] and U [u,,-., u.).
2 We
M
The encoder & maps the input to latent system representation via
& X MX UJ E R (VxxD.+tt.D..) Z' BLXDD
realized by cross-attention 91 75
between the input tensor E IR DDx which serves as keys and values, and fixed number of L learned latent vectors R queries. The encoding process depicted on the left side of Figure 2
43 which serve as the
The aim of the decoder D is to retrieve the system state information X' and M from the latent state representation Z' using the encoded entity identifier embeddings U. The decoder maps the latent system representation back into the coordinates and properties of each entity via
Decoder.
D z E IR LXD, x U E UUERLX, LLD, + X MJ E N (D..
As shown in the middle part of Figure 2 representation Z' serves as the keys and values in the cross-attention mechanism, while the embedded identifier un acts as the query. Applied to to all (n = 1,..., N) system entities e,, this results in the retrieved system state information X and M. Using the learned identifier embeddings as queries can be interpreted as form of content- based retrieval and associative memory
D is realized by cross-attention layers. The latent space
6 40 [5].
Approximator. Finally, the approximator models the system's 's time evolution in latent space, i.e., it predicts a series of future latent system states Z +1: 77 of initial latent system states Z [1 Iol
TT+ Z' Zi...., Iz,, given series
zz 2...., Z
A: Z T.) E TTxLLLD To+ TI TIXLXD.
Given the analogy of predicting the time evolution of a dynamical system to the task of synthesizing videos, we realized A by a flow-based model. Specifically, we constructed based on the stochastic
interpolants framework (2 60)
We are interested in time-dependent processes, which interpolate between data 0 ~ Pt from target data distribution and noise ~ Po = (0,
o, 0, to,,
(6)
5

---

where т E [0. 1] is the time parameter of the flow (to be distinguished from system times t). and o, are differentiable functions in т, which have to fulfill a + o² 0 VT E [0, 1}, and, further (r
0, and, 01) = do = 1. The goal is to learn parametric model (o 7), s.t., So (l.)) 01 o
[ dr is minimized. Within the stochastic interpolants framework, we identify or with a whole trajectory
Z T] [ Tol ZTT+1: ri E RTxLxDs
Since the generated trajectories should be condi- tioned on the latent system representations of ini- tial time frames ZlI: To), we extend Vy with con- ditioning argument C RT*xIx,; making ef- fectively a conditional vector field uy: Txx D. [0, x Tx r> Tx The tensor struc- ture of C is the same as the one for Z. For the first time steps, both tensors have equal values, i.e., Cl Tul Z(: Tol, The remaining tensor entries ClT++1:T] I] filled up with mask tokens, see Fig- u The latent model is structured as series trans- former blocks [90, 677, alternating between the spa- tial and temporal dimensions (see App. [.4). We parametrized our model using data prediction ob- jective 55 see App. C.1 For architectural details
DD-baedd decading
Figure 4: Left: The latent model receives conditioning via known tokens (observed timesteps) and mask tokens (for prediction). This example shows conditioning on one time- frame to predict three future ones. Right: ID--based decoding, where the predicted atom positions are decoded by the assigned IDs.
see App.
3.4 Training Procedure
The training process follows a two-stage approach similar to latent diffusion models [77]. First Stage: We train the encoder and decoder D. to reconstruct entities from latent space using assigned IDs, see Figure 2). Second Stage: We train the approximator A on the latent system representations produced by the frozen encoder & (details in App. E.3 and Figure
4 Experiments
In order to evaluate LAM-SLIDE we focus on three key aspects: (i) Robust generalization in diverse domains. We examine LAM-SLIDE is generalization in different data domains in relation to other methods, for which we utilize tracking data from human motion behavior, trajectories of particle systems, and data from molecular dynamics (MD) simulations. (ii) Temporal adaptability. We evaluate temporal adaptability through various conditioning/ppediitionn horizons, considering single/mulli-free frame conditioning and short/long serilonng-emmmmmmo predictions; (iii) Computational efficiency and scalability. Finally, we assess the inference time of LA M-SLIDE, and evaluate performance with respect model size. The subsequent sections show key findings. For implementation details and additional results see App. and for information on the datasets see App. E.1
Metries. We utilized the Average Discrepancy Error (ADE) and the Final Discrepancy Error (FDE), defined ADE(X, X) '-ー1. Ei ΣNV X -X FDE(X, X) \ Σ I
X 2. capturing model performance across predicted future time steps and the model performance specifically for the last predicted frame, respectively. These metrics represent well-established
evaluation criteria trajectory forecasting 94 95]
For the MD experiments on peptides (Tetrapeptides). we use the Jensen- Shannon divergence (JSD) over the distribution of torsion angles, considering both backbone (BB) and side chain (SC) angles. To capture long temporal behavior, we use Time- limeelagged Independent Component Analysis (TICA) 770) focusing the slowest components TIC 0 and TIC 1. To investigate metastable state transitions, make use of Markov State Models (MSMs) 74 6651
For inference time and scalability, we assess the number of function evaluations (NFE) and report the performance of our method for different model sizes. A comprehensive overview of the conditioning and prediction horizons for each experiment is shown in Table 7

---

Table Results on generation on N-body dataset, in terms of ADF/FDE averaged over runs.
Table 2: Results on the ETH-UCY dataset for
pedestrian forecasting. Results are in terms of mi- nADE/minFDE out of 20 runs.
Particlo
Spring
Gravity
ETH
Hotel
Univ
Zaral
Zara2
Average
Linear'
1,07/2.28
0.31/8661
0.52/1.16
0.42/0.55
0.32/072
0.53/1.14
SB -T: 30
RF 53
EGNN 82
0.186
0.395
0.479
0.936
0.426
1,050
0,0865
0.1001
0,0145
0.2364
0,0389
0,2043
0.0231
0.338
0.310
0.791 1,630
0,830
0.709
SGAN 30
PECNet 61
SoPhie 18
Trj+ X0
0.5640.98
0.20V1.43
0.54/0.87
0.54/0.94
0.64/1.09
0.46/0.9
0,16/0.28
0.18/0.24
0.76/167
0.25/0.47
0.35/0.60
0.28/0.55
0.56/1.18
2.54/1.24
0.23/0.55
0.21/0.42
0.300.53
0.22/0.39
0.33/0.67
0.160.33
0.38/078
0.31/0.64
0.160.32
0.1749.00
0.54/1.15
0.22/0.50
0.46/0.91
0.29/0.48
0.22/0.50
EgMotion 94
0.141
0.310
0.0134
0,0358
0.302
0.671
MID> 31
0.5000.76
0.16/0.24
0.28/049
0.25/0.41
0.19/0.35
0.27//445
SVAE 95
0.378
0.732
0.0200
0,0209
0.582
1.101
SVAE 99
0,4700766
0.14/0.2
0.25/0.77
0.20/0.37
0.149028
0.2410.42
GooTDM
0.110
0.258 0.0030
0.0079
0.256
0.613
Goo 3ST
0.4600664
0.13/0.21
0.24/0.45
0:21/0.39
0.16/0.0
0.24/0.40
LAM-SLID
0.104
0.238 0.0070
0.0135
0.157
0.406
LAM SLIDE
0.45KL.5
0.13/0119
0.26/047
0.210.355
0.17 /
0.24/0.41 0.41
Results from Han el. [55]
Results from Han ct
35
4.1 Pedestrian Trajectory Forecasting (ETH-UCY)
Experimental setup. For human motion behavior, we first consider the ETH-UCY dataset [68] 53 which provides pedestrian movement behavior, over five different scenes: ETH, Hotel, Univ, Zaral, and Zara2. We use the same setup as Han et al. [35], Xu et al. (94 (55. in which the methods obtain the first 8 frames conditioning information and have predict the next 12 frames. We report the min minADD//innDDE, computed across 20 sampled trajectories.
Compared methods. We compare LAM-SLIDE to eight state-of-the-aan art generative methods covering different model categories, including: GANs: SGAN [34], SoPhie [78]; VAEs: PECNet [61] Traj+ + 80). BiTrap 197]. SVAE [95]; diffusion models: MID 33 and GeoTDM [35] and linear
baseline. The baseline methods predominantly target pedestrian trajectory prediction, with GeoTDM and Linear being the exceptions.
Results. As shown in Table 2 our model performs competitively across all five scenes, achieving lower minFDE for Zaral and Hotel scene, and the lowest minADE on the ETH scene. Notably, in contrast the compared baselines, we did not create additional features like velocity and acceleration or imply any kind of connectivity between entities.
4.2 Basketball Player Trajectory Forecasting (NBA)
Experimental setup. Our second experiment examines human motion behavior in the context of basketball game- play. We utilize the SportVU NBA movement dataset [99], which contains player movement data from the 2015-2016 NBA season. Each recorded frame includes ten player positions (5 for each team) and the ball position. We con- sider two different scenarios: a) Rebounding: containing scenes of missed shots; b) Scoring: containing scenes of team scoring basket. The interaction patterns both in frequency and in their adversarial versus cooperative dynamics- exhibit different challenges as those in Sec- tion 4.1 The evaluation procedure by Xu et al. (95), which we use, provides 8 frames as input conditioning and the consecutive 12 frames for prediction. The performance reported by the minADD/minF'D metrics, which are computed across 20 sampled trajectories. For the reported metrics, only the trajectories of the players are considered.
Table 3: Results on the NBA dataset: Compared methods have to predict player positions for 12 frames and are given the initial 8 frames as input. Re- sults in terms of minADD/minFDE for the Rebounding and Scoring scenarios.
Rebounding
Scoring
Linear Traj++
2.15/5.09
2.07/4.81
80
0.98/1.93
0.73/1.46
BiTraP 97
0.83/1.72
0.74/1.49 1.99
SGNet-ED 93
0.78/1.55
0.68/1.30
SVAE 95
0.72/1.37
0.64/1.17
LAM -LMILLIDE
0.79/1.42
0.64/1.09
Results from Xu al. 95
Compared methods. The method comparison for this human motion forecasting task includes methods based VAES, Traj++ [80]. BiTrap (97), SGNet-ED 93). SVAE 95). and linear baseline.
Results. As illustrated in Table 3 our model shows robust performance across both scenarios, Rebounding and Scoring. For the Scoring scenario, LAM-SLIDE achieves parity with SocialVAE (34]
7

---

in terms of minADE and surpasses the performance in terms of minFDE. In the Rebounding scenario, we observe comparable but slightly lower performance of LAM-SLIDE compared SoC Trajectories sampled by our model are shown in Figure 13
4.3 N-Body System Dynamics (Particle Systems)
Experimental setup. We evaluate our method across three distinct N-Body simulation scenarios: a) Charged Particles: comprising particles with randomly assigned charges +1/ interacting via Coulomb forces 52 b) Spring Dynamics: consisting of N = particles with randomized masses connected by springs with a probability 0.5 between particle pairs [51]; and c) Gravitational Systems: containing N 10 particles with randomized masses and initial velocities governed by gravitational interactions 17.. For all three scenarios, we consider 10 conditioning frames and 20 prediction frames. In line with Han et al. 35 we use 3000 trajectories for training and 2000 trajectories for validation and testing, and we report ADE/RDE averaged over K runs.
Compared methods. We compare LA LAM-ELIE to seven different methods, including six equiv- ariant GNN based methods: Tensor Field Network [89], Radial Field [52].
(30]
EGNN 82), EqMotion (94), GeoTDM
35 and non-equivariant method: SVAE (95].
Results. LAM-SLIDE achieves the best performance in terms of ADE/FDE for the Charged Particles and Gravity scenarios and competitive second-rank performance the Spring Dynamics scenario, see Table I Unlike compared methods, LAM-SLIDE achieves these results without computing intermediate physical quantities such as velocities or accelerations. We present sampled trajectories in Figure 14
4.4 Molecular Dynamics Small Molecules (MD17)
Experimental setup. this experiment, we evaluate LAM-SLIDE on the well-established
MD17 [21] dataset, containing simulated molecular dynamics trajectories of 8 small molecules. The
size those molecules ranges from 9 atoms (Ethanol and Malonaldehyde) to 21 atoms (Aspirin). We use 10 frames conditioning, 20 frames for prediction, and report ADE/FDE averaged over K runs.
Compared methods. Similar to Section we compared LAM-SLIDE to: Tensor Field Net- work 89], Radial Field [52] Transfoereee and SVAE (95).
330 EGNN g2 EqMotion (4), GeoTDM 35
Results. The results in Table show the performance on the MD17 benchmark. LAM-SLIDE achieves the lowest ADE/DDE of all methods and for all molecules. These results are particularly remarkable considering that: (1) our model operates without an explicit definition molecular bond information, and (2) it surpasses the performance of all equivariant baselines, an inductive bias we intentionally omitted in LAM-SLIDE.
Notably, we train a single model on all molecules a feat that is structurally encouraged by the design of the performance on each one of them ("all->each" in the Table (3). Interestingly, we also observe consistent improvements in the GeoTDM performance. However, GeoTDM's performance does reach that of LAM-SLIDE. We also note that our latent model is trained for 2000 epochs, while GeoTDM was trained for 5000 epochs. Trajectories are shown in Figure
For ablation, also train GeoTDM (35] on all molecules and evaluate
4.5 Molecular Dynamics Tetrapeptides (4AA)
Experimental setup. To investigate LAM-SLIDE on long prediction horizons, we utilize the Tetrapeptide dataset from Jing al. [4] containing explicit-solvent molecular dynamics trajectories simulated using OpenMM [26]. The dataset comprises 3, 109 training, 100 validation, and 100 test peptides. We use a single conditioning frame to predict 10,000 consecutive frames. The predictions are structured as a sequence of ten cascading 1,000-step rollouts, where each subsequent rollout conditioned on the final frame of the previous. Note that, in contrast to the MD17 dataset, the methods predict trajectories of unseen molecules.
8

---

Table 4: Results on the MD17 dataset. Compared methods have to predict the atom positions of 20 frames, conditioned on 10 input frames. Results in terms of ADE/DDE, averaged over runs.
Aspirin
Benzene
Ethanol
Malonaldehyde
Naphthalene
Salicylic
Toluene
Uracil
ADE
FDE
ADE
FDE
ADE
FDB
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
RF 522
0,303
0.442 0.120 0.194
0.374
0,515
0.297
0.454
0.168
0.185
0.261
0.343
0.199
0.249
0.239
0.272
TEN
0.133
0.268 0.024
0.201
0.414
0.184
0.386
0.072
0.223
0.090
0.150
3-
30
0.294
0.556 0.027 0.056
0.188
0.359
0.214
0.456
0.069
0.103 0.189
0.312
0.108
0.184
0.107
0.196
EGNN
82
0,267
0.564 0.024 0.042
0.268
0.401
0.393
0.958
0.095
0.133
0.159
0.348
0,207
0.294
0.154
0.282
MoMoooon
94
0.185
0.246 0.029
0.043
0.152
0.247
0.155
0.249
0.073
0.092 0.110
0.151
0.097
0.129
0.088
0.116
SVAE
05
0,301
0.428
0.114
0.133
0.387
0,505
0.287
0.430
0.124
0.135
0.122
0.142
0.145
0.171
0.145
0.156
GeoTDM
0.107
0.193 0.023 0.039
0.115
0.209
0.107
0.176
0.064
0.087
0.083
0.121
LLIM
0.059
0.098
0.032
0.087
0.167
0.073
0.124
0.037
0.058
0.047
0.074
0.045
0.075
0.074
Results from Han al.
35
Compared methods. We compare LAM-SLIDE to the recently proposed method MDGen 44 which is geared towards protein MD simulations, and to a replicate of the ground truth MD simulation as baseline, which is lower bound for the performance.
Results. Table shows performance metrics of the methods (for details on those metrics see App. E.6 Figure 16 shows the distribution of backbone torsion angles, and the free energy surfaces of the first two TICA components, for ground truth vs simulated trajectories. LAM-SLIDE performs competitively with the current ttate-of-the-art method MDGen with respect torsion angles, which is a notable achievement given that MDGen operates in torsion space only. With respect to the TICA and MSM metrics, LAM-SL.DE even outperforms MDGen. Sampled trajectories shown in Figure 17
4.6
Computational Efficiency and Scaling Behavior
To assess computational efficiency, we compare the NFEs of our model to the second-best method, GeoTDM. Our results show that LAM-SI.IEE requires up 10x-100x fewer function evaluations depending on the domain. For a detailed discussion, see App. G.2
Table 5: Results the Tetrapeptide
dataset: Columns denote the JSD be- tween distributions of torsion angles (backbone (BB), side-chain (SC), and all angles), the TICA, and the MSM met- ric.
We further assess the scalability of LAM-SLIDE across different model sizes. Our results indicate that perfor- mance consistently improves with increasing parameter count, suggesting that method benefits from larger model capacity and could potentially achieve even better results with additional computational resources. Compre- hensive details of this analysis can be found in App. G
Torsions
TICA
MSM
Time
BR
0. 0.1 0.1
100ss
.103 .055 .076 201 .268 .208
Sh
MTD [4T
.130 .093 .109 230 316 .335
60s
LAM-SLIEE
.128
.224
Results from Jing et
54
4.7 Ablations
Identifier Pool Size. To understand how the size of the identifier pool affects the performance of the first stage model, we conducted additional experiments on the MD17 dataset. Maintaining fixed number of update steps across different configurations results in a modest increase in the reconstruction error for larger pool sizes. We attribute this to the reduced number of updates per individual identifier embedding. A comprehensive analysis is provided in App. G.4
Identifier Assignment. We assessed the sensitivity of our model to the specific entity-identifier assignment by evaluating five random assignments on the MD17 dataset. Results do not indicate negative impact on the performance of our model. This demonstrates the robustness our approach with respect to the specific assignment between entities and identifiers, see App. G.5 for more details.
Identifier Latent Utilization. We analyzed decoder attention patterns to understand how identifiers access latent information across different compression ratios. Our findings show that each identifier maintains a consistent addressing scheme across different molecule conformations, and the model adaptively utilizes attention heads based on the capacity of the latent space. In overparameterized

---

setting, the model favors single-head retrieval, while in the compressed setting utilizes both heads to avoid potential identifier collisions. See Appendix G.6 for detailed analysis.
Discussion
Limitations. Our experiments indicate that our architecture applies to diverse set of problems; however, a few limitations provide opportunities for future improvement. While our current approach successfully allows for compressing entities with beneficial reconstruction performance, our experi- ments indicate a tradcoff between the number of latent space vectors to encode system states and the performance of our latent model; for more details, see App. G.3
Conclusion. We have introduced LAM-SLIDE, novel approach for modeling spatial dynamical systems that consist of a variable number of entities within fixed-size latent system representation, where assignable identifiers enable the traceability of individual entities. Across diverse domains, LAM-SLIDE matches or exceeds specialized methods and offers promising scalability properties. Its minimal reliance on prior knowledge makes suitable for many tasks, suggesting its potential as a foundational architecture for dynamical systems.
Acknowledgments
We thank Niklas Schmidinger, Anna Zimmel, Benedikt Alkin, and Arturs Berzins for helpful discussions and feedback. The ELLIS Unit Linz, the LIT AT Lab, and the Institute for Machine Learning are supported by the Federal State Upper Austria. We thank the projects FWF AIRI FG 9-N (10.55776/FG9). ALAGreenHeetingGrrie (FFG- 899943), 4tters (HORIZON-CL6--D)) CLIMATE-01-01, FWF Bilateral Artificial Intelligence (10.5577//COE12). We thank NXAI GmbH, Audi AG, Silicon Austria Labs (SAL), Merck Healthcare KGaA, GLS (Univ. Waterloo), TÜV Holding GmbH, Software Competence Center Hagenberg GmbH, dSPACE GmbH, TRUMPF SE +
Co. KG.
References
[1] Abramson, J., Adler, J., Dunger, J., Evans, R., Green, T., Pritzel, A., Ronneberger, O., Willmore, L., Ballard, J., Bambrick, J., et al. Accurate structure prediction of biomolecular interactions with alphafold Nature, pp. 1-3, 2024.
[2] Albergo, M., Boffi, N., and Vanden- E. Stochastic interpolants: A unifying framework for flows and diffusions, 2023. ArXiv preprint ArXiv230308797, 2023.
[3] Albergo, M. S. and Vanden-Eijnden E. Building normalizing flows with stochastic interpolants. arXiv preprint arXiv:2209. 15571, 2022.
[4] Alkin, B., Fürst, A., Schmid, S. L., Gruber, L.., Holzleitner, M., and Brandstetter, J. Universal physics transformers: A framework for efficiently scaling neural operators. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
[5] Alkin, B., Kronlachner. T., Papa, S., Pirker, S., Lichtenegger, T., and Brandstet- ter, J. Neuraldem-reallffe simulation of industrial particulate flows. arXiv preprint arXiv:241 1.09678, 2024.
[6] Amari, S.-l. Learning patterns and pattern sequences by self-organizing nets of threshold elements. IEEE Transactions on computers, 100(1 100(11):1197-100: 1972.
[7] Ansel, J., Yang. E., He, H., Gimelshein, N., Jain, A., Voznesensky, M., Bao, B., Bell, P., Berard, D., Burovski, E., Chauhan, G., Chourdia, A., Constable, W., Desmaison, A., Vito,o, Z., Ellison, E., Feng. W., Gong, J., Gschwind, M., Hirsh, B., Huang, S., Kalambarkar, K., Kirsch, L.. Lazos, M., Lezcano, M., Liang, Y.. Liang. J.. Lu, Y., Luk, C., Maher, B., Pan, Y., Puhrsch, C., Reso, M., Saroufim, M., Siraichi, M. Y., Suk, H., Suo, M., Tillet, P., Wang, E., Wang, X., Wen, W., Zhang, S., Zhao, X., Zhou, K., Zou, R., Mathews, A., Chanan, G., Wu, P., and Chintala, S. PyTorch Faster Machine Learning Through Dynamic Python Bytecode Transformation and Graph Compilation. In 29th ACM International Conference
10

---

on Architectural Support for Programming Languages and Operating Systems. Volume (ASPLOS '24). ACM. April 2024. doi: 10.145//322065536446666 URL pttttp:://ddoc pytorch. ortorch.org/coo pytorcch.or//sssssss/tttotoooo aasstss/pytorchh//// pdf
[8] Arnold, L. Random Dynamical Systems. Monographs in Mathematics, Springer, 1998. ISBN 9783540637585.
[9] Arriola, M.. Gokaslan, A., Chiu, J. T., Yang. Z., Qi. Z.., Han, J., Sahoo, S. S., and Kuleshov, V. Block diffusion: Interpolating between autoregressive and diffusion language models. arXiv preprint arXiv:2503.09577 2025.
[10] Ba, J. L. Layer normalization, arXiv preprint ur0 2016.
[11] Battaglia, P. W., Hamrick, J. B., Bapst, V., Ganzalz- A., Zambaldi, V., Malinowski, M., Tacchetti, A., Raposo, D., Santoro, A., Faulkner, R., et Relational inductive biases, deep learning, and graph networks. arXiv preprint arXiv: /806.01 010600126, 2018.
[12] Biewald, L.. Experiment tracking with weights and biases, 2020. URL https: wandb com/ Software available from wandh.com.
[13] Black, K., Brown, N., Driess, D., Esmail, A., Equi, M., Finn, C., Fusai, N., Groom, L., Hausman, K., Ichter, B., et al. π 0_0: A caroo--lnguagg-actinnn flow model for general robot 2024.
control. arXiv preprint arXiv:2410.24160
[14] Black Forest Labs. Flux. https: github.....b...... com black howp/black-frrest-abb...
2023.
[15] Blattmann, A., Rombach, R., Ling, H., Dockhorn, T., Kim. S. W., Fidler, S., and Kreis, K. Align your latents: High-resolution video synthesis with latent diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 22575,, 2023.
[16] Bommasani, R., Hudson, D. A., Adeli, E., Altman, R., Arora, S., von Arx, S., Bernstein, M. S., Bohg, J., Bosselut, A., Brunskill, E., et al. On the opportunities and risks of foundation models. arXiv preprint arXiv:2108.07258. 2021.
[17] Brandstetter, J., Hesselink, R., van der Pol, E., Bekkers, E. J., and Welling, M. Geo- metric and physical quantities improve (3) equivariant message passing. arXiv preprins arXiv:2110.02905, 2021.
[18] Case, D., Aktulga, H., Belfon, K., Ben-Shalom, I., Berryman, J., Brozell, S., Cerutti, D., Cheatham, T., III, Cisneros, G., Cruzeiro, V., Darden, T., Forouzesh, N., Ghazimirsaeed, M., Giambaşu, G., Giese, T., Gilson, M., Gohlke, H., Goetz, A., Harris, J., Huang, Z., Izadi, S., Izmailov, S., Kasavajhala, K., Kaymak, M., Kovalenko, A., Kurtzman, T., Lee, T., Li, P., Li. Z., Lin, C., Liu, J., Luchko, T., Luo, R., Machado, M., Manathunga, M., Merz, K., Miao, Y.. Mikhailovskii, O., Monard, G., Nguyen, H., O' Hearn, K., Onufriev, A., Pan, F., Pantano, S., Rahnamoun, A., Roe, D., Roitberg, A., Sagui, C., Schott- Verdugo. S., Shajan, A., Shen, J., Simmerling, C., Skrynnikov, N., Smith, J., Swails, J., Walker, R., Wang, J., Wang. J., Wu, X., Wu, Y., Xiong. Y., Xue, Y., York, D., Zhao, C.,, Zhu, Q., and Kollman, P. Amber 2024, 2024.
[19] Chen, R. T. and Lipman, Y. Riemannian flow matching on general geometries, arXiv e-prints, pp. arXiv-230, 2023.
[20] Chen, R. Q. torchdiffeq. 2018. URL /ttp///tw////ggtt..o/oo/////aaaa
[21] Chmiela, S., Tkatchenko, A., Sauceda, H. E., Poltavsky, I., Schütt, K. T., and Müller, K.-R. Machine learning of accurate energy-conserving molecular force fields. Science advances, (5)) 2017.
[22] Costa, A. d. S., Mitnikov, I., Pellegrini, F., Daigavane, A., Geiger, M., Cao, Z.. Kreis, K., Smidt, T., Kucukbenli, E., and Jacobson, J. Equijump: Protein dynamics simulation via so (3)-equivariant stochastic interpolants. arXiv preprint arXiv:2410.09667. 2024.
[23] Dehghani, M., Djolonga, J., Mustafa, B., Padlewski, P., Heek, J., Gilmer, J., Steiner, A. P., Caron, M., Geirhos, R., Alabdulmohsin, I., et al. Scaling vision transformers to 22 billion parameters. In International Conference on Machine Learning, pp. 7480-7512. PMLR, 2023.
11

---

[24] Devlin, J. Bert: Pre-training of deep bidirectional transformers for language understanding. arXiv preprint arXiv:1810.04805,, 2018.
[25] Dosovitskiy, A. An image is worth 16x16 words: Transformers for image recognition at scale.
arXiv preprint arXiv:2010. /929, 2020.
[26] Eastman, P., Swails, J., Chodera, J. D., McGibbon, R. T., Zhao, Y., Beauchamp, K. A.. Wang, L.-P., Simmonett, A. C., Harrigan, M. P., Stern, C. D., al. Openmm 7: Rapid development of high performance algorithms for molecular dynamics. PLoS computational biology, 13(7): e1005659, 2017.
[27] Esser, P., Kulal, S., Blattmann, A., Entezari, R., Müller, J., Saini, H., Levi, Y., Lorenz, D., Sauer, A., Boesel, F., et al. Scaling rectified flow transformers for high-resolution image synthesis, 2024. URL https://arxiv. org/abs/2403.022222 032006 2024.
[28] Falcon, W. and The PyTorch Lightning team. PyTorch Lightning, March 2019. URL https
//github. www.ttt........pgeeeeeee woomm/ighniigg-.... Ligghtnninngg///gghtiinn
[29] Ford, G., Kac, M., and Mazur, P. Statistical mechanics of assemblies of coupled oscillators. Journal of Mathematical Physics, 6(4):504-515, 1965.
[30] Fuchs, F. B., Worrall, D. E., Fischer, V., and Welling, M. Se tra-trrnsformmr:: 3d roto- translation equivariant attention networks. arXiv preprint arXiv:2006. /0503, 2020.
[31] Gardner Jr, E. S. Exponential smoothing: The state of the art Journal of forecasting, 4(1): 1-28, 1985.
[32] Gilmer, J., Schoenholz, S., Riley, P. F., Vinyals, O., and Dahl, G. Neural message passing for quantum chemistry. In International conference on machine learning, pp. 1272-1272.
PMLR, 2017.
[33] Gu, T., Chen, Li, Lin, C., Rao, Y., Zhou, J., and Lu, Stochastic trajectory prediction via motion indeterminacy diffusion. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 17113-17122, 2022.
[34] Gupta, A., Johnson, J., Fei-Fei, L.., Savarese, S., and Alahi, A. Social gan: Socially acceptable trajectories with generative adversarial networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 2255-2264, 2018.
[35] Han, J., Xu, M., Lou, A., Ye, H., and Ermon, S. Geometric trajectory diffusion models. arXiv preprint arXiv:2410. 13027, 2024.
[36] Heusel, M., Ramsauer, H., Unterthiner, T., Nessler, B., and Hochreiter, S. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances neural information processing systems, 30, 2017.
[37] Ho, J., Jain, A., and Abbeel, P. Denoising diffusion probabilistic models. Advances in neural information processing systems, 33:6840-6851, 2020.
[38] Ho, J., Chan, W., Saharia, C., Whang, J., Gao, R., Gritsenko, A., Kingma, D. P., Poole, B., Norouzi, M., Fleet, D. J., et al. Imagen video: High definition video generation with diffusion models. arXiv preprint arXiv:2210.02. 023003,,, 2022.
[39] Hoel, H. and Szepessy, A. Classical langevin dynamics derived from quantum mechanics. arXiv preprint arXiv: (wXiv::906.00000 09858, 2019.
[40] Hopfield, J. Neural networks and physical systems with emergent collective computational abilities. Proceedings of the national academy of sciences, 7955):255--2558 1982.
[41] Huguet, G., Vuckovic, J., Fatras, K., Thibodeau-LLaueer E., Lemos, P., Islam, R., Liu, C.-H., T., Bronstein, M., et al. Sequence-augmented se (3)-flow
Rector-Brooks, J., Akhound-Sade matching for conditional protein backbone generation. arXiv preprint arXiv:2405.20::
2024. [42] Husic, B. and Pande, V. Markov state models: From an art to a science. Journal of the American Chemical Society, 140(7):2386-2399 2018.
12

---

[43] Jaegle, A., Borgeaud, S., Alayrac, J.-B., Doersch, C., Ionescu, C., Ding, D., Koppula, S., Zoran, D., Brock, A., Shelhamer, E., et al. Perceiver io: A general architecture for structured inputs & outputs. arXiv preprini 10Xiv:2107.1473 2021.
[44] Jing, B., Stärk, H., Jaakkola, T., and Berger, B. Generative modeling of molecular dynamics trajectories. arXiv preprint 20t.:::400977880.0 2024.
[45] Joshi, C. K., Fu, X., Liao, Y.L.,, Gharakhanyan, V., Miller, B. K., Sriram, A., and Ulissi, Z. W. All-atom diffusion transformers: Unified generative modelling of molecules and materials. arXiv preprint arXiv: 250::036333, 2025.
[46] Jumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., Ronneberger, O., Tunyasuvunakool, K., Bates, R., Zidek, A., Potapenko, A., et al. Highly accurate protein structure prediction with alphafold. nature, 596(7873):583-588 2021.
[47] Karplus, M. and Petsko, G. A. Molecular dynamics simulations in biology. Nature, 347(6294); 631-639, 1990.
[48] Kingma, D. and Gao, R. Understanding diffusion objectives the elbo with simple data augmentation. Advances in Neural Information Processing Systems, 36, 2024.
[49] Kingma, D. P. Auto-encoding variational bayes. arXiv preprint arXiv: 2013.
[50] Kingma, D. P. Adam: A method for stochastic optimization. arXiv preprint arXiv: 14Xiv91412.6980. 2014.
[51] Kipf, T., Fetaya, E., Wang, K.-C., Welling, M., and Zemel, R. Neural relational inference for interacting systems. In International conference machine learning, pp. 2688 2697. PMLR, 2018.
[52] Köhler, J., Klein, L., and Noé, F. Equivariant flows: sampling configurations for multi-body systems with symmetric energies. arXiv preprint arXiv:1910.00753, 2019.
[53] Lerner, A., Chrysanthou, Y., and Lischinski, D. Crowds by example. In Computer graphics forum, volume 26, pp. 655-664. Wiley Online Library, 2007.
[54] Lipman, Y., Chen, R. T., Ben-Hamu, H., Nickel, M., and Le, M. Flow matching for generative modeling. arXiv preprint 1rXiv:2210.02777, 2022.
[55] Lipman, Y., Havasi, M., Holderrieth, P., Shaul, N., Le, M., Karrer, B., Chen, R. T., Lopez- Paz. D., Ben-Hamu, H., and Gat, I. Flow matching guide and code. arXiv preprint arXiv:2412.066: 06264, 2024.
[56] Liu, J., Yang, C., Lu, Z., Chen, J., Li, Y., Zhang, M., Bai, T., Fang, Y., Sun, L., Yu, P. S., et al. Towards graph foundation models: A survey and beyond. arXiv preprint arXiv:2310.11822 2023.
[57] Liu, X., Gong, C., and Liu, Q. Flow straight and fast: Learning to generate and transfer data with rectified flow. arXiv preprint arXiv:2209.088 20000303, 2022.
[58] Loshchilov, I., Hutter, F., et al. Fixing weight decay regularization in adam. arXiv preprint (.Ni::1711.0500001 5, 2017.
[59] Lou, A., Meng, C., and Ermon, S. Discrete diffusion modeling by estimating the ratios of the data distribution. URL http::/arxiv. org/abs/23100 10..i/231100000 16834, 2024.
[60] Ma, N., Goldstein, M., Albergo, M. S., Boffi, N. M., Vanden-Fijnden E., and Xie, S. Sit: Exploring flow and diffusion- generative models with scalable interpolant transformers. arXiv preprint arXiv:2401.0874 2024.
[61] Mangalam, K., Girase, H., Agarwal, S., Lee, K.-H., Adeli, E., Malik, J., and Gaidon, A. It is not the journey but the destination: Endpoint conditioned trajectory prediction. In Computer VisVon-F.CC 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part II 16. pp. 759-776. Springer, 2020.
13

---

[62] Mao, H., Chen, Z., Tang, W., Zhao, J., Ma, Y., Zhao, T., Shah, N., Galkin, M., and Tang, J. Position: Graph foundation models are already here. Forty-first International Conference Machine Learning, 2024.
[63] Mayr, A., Lehner, S., Mayrhofer, A., Kloss, C., Hochreiter, S., and Brandstetter, J. Boundary graph neural networks for 3d simulations. Proceedings of the AAAI Conference on Artificial Intelligence, 37(8):9099-1177, Jun. 2023, doi: 10. 10.1609aaaai.v333 v3718 260922.
[64] Micheli, A. Neural network for graphs: A contextual constructive approach. IEEE Transactions on Neural Networks, 20(3):498-01 511, 2009. doi: 10.1
[65] Noé, F., Wu, H., Prinz, J.-H., and Plattner, N. Projected and hidden markov models for calculating kinetics and metastable states of complex molecules. The Journal of chemical physics, 139, 2013.
[66] Paszke, A., Gross, S.. Massa, F., Lerer, A., Bradbury. J. Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., al. Pytorch: An imperative style, high-performance deep learning library. Advances in neural information processing systems, 32, 2019.
[67] Peebles, W. and Xie, S. Scalable diffusion models with transformers. In Proceedings of the IEEE/CVF International Conference on Computer Vision, pp. 4195-4205, 2023.
[68] Pellegrini, S., Ess, A., Schindler, K., and Van Gool, L. You'll never walk alone: Modeling social behavior for multi-target tracking. In 2009 IEEE 12th international conference computer vision, pp. 261-268. IEEE, 2009.
[69] Perez, E., Strub, F., De Vries, H., Dumoulin, V., and Courville, Film: Visual reasoning with general conditioning layer. In Proceedings of the AAAI conference artificial intelligence, volume 32, 2018.
[70] Pérez-Hernández, G., Paul, F., Giorgino, T., De Fabritiis, G., and Noé, F. Identification slow molecular order parameters for markov model construction. The Journal of chemical physics, 139(1), 2013.
[71] Polyak, A., Zohar, A., Brown, A., Tjandra, A., Sinha, A., Lee, A.. Vyas, A., Shi, B., Ma. C.-Y., Chuang, C.-Y,, et al. Movie gen: A cast of media foundation models, arXiv preprint arXiv:2410.13720, 2024.
[72] Ponder, J. W. and Case. D. A. Force fields for protein simulations. Advances in protein chemistry, 66:27-85, 2003. ISSN 0065-3233. doi: 10. 1016/50065-3233300 (333506655-223333366002XXX
[73] Price, I., Ganzhez- A., Alet, F., Andersson, T. R., El-Kadi, A., Masters, D., Ewalds, T., Stott, J., Mohamed, S., Battaglia, P. et al. Probabilistic weather forecasting with machine learning. Nature, 637(8044):84-90) 2025.
[74] Prinz, J.-H., Wu, H., Sarich, M., Keller, B., Senne, M., Held, M., Chodera, J. D., Schütte, C., and Noé, F. Markov models of molecular kinetics: Generation and validation. The Journal of chemical physics, 134(17), 2011.
[75] Ramsauer, H., Schäfl, B., Lehner, J., Seidl, P., Widrich, M., Gruber, L., Holzleitner, M., Adler, T., Kreil, D., Kopp, M. K., G, K., and Hochreiter, S. Hopfield networks is all you need. International Conference on Learning Representations, 2021.
[76] Rogozhnikov, A. Einops: Clear and reliable tensor manipulations with einstcin-like notation. In International Conference on Learning Representations, 2021.
[77] Rombach, R., Blattmann, A., Lorenz, D., Esser, P., and Ommer, B. High-resolution image synthesis with latent diffusion models. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 10684-10695, 2022.
[78] Sadeghian, A., Kosaraju, V., Sadeghian, A., Hirose, N., Rezatofighi, H., and Savarese, Sophie: An attentive gan for predicting paths compliant to social and physical constraints. In Proceedings of the IEFF/CVE conference on computer vision and pattern recognition, pp- 1349-1358, 2019.
14

---

[79] Sahoo, S.. Arriola, M., Schiff, Y., Gokaslan, A., Marroquin, E., Chiu, J., Rush, A., and Kuleshov, V. Simple and effective masked diffusion language models. Advances in Neural Information Processing Systems, 37:130136-1323 100366--31888 130184, 84, 2024.
[80] Salzmann, T., Ivanovic, B., Chakravarty, P., and Pavone, M. Trajectron++: Dynamically- feasible trajectory forecasting with heterogeneous data, In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23-28, 2020, Proceedings, Part XVIII 16, pp. 683-700. Springer, 2020.
[81] Sanchez-Gonzalez. A., Godwin, J., Pfaff, T., Ying, R., Leskovec, J., and Battaglia, P. Learning to simulate complex physics with graph networks. In International conference machine learning, pp. 8459-8468. PMLR, 2020.
[82] Satorras, G., Hoogeboom, E., and Welling, M. B(n) equivariant graph neural networks. In Meila, M. and Zhang, T. (eds.), Proceedings the 38th International Conference on Machine Learning, volume 139 of Proceedings of Machine Learning Research, pp. 9323-9332. PMLR, 18 24 Jul 2021.
[83] Scarselli, F., Gori, M., Tsoi, A. C., Hagenbuchner, M., and Monfardini, G. The graph neural network model. JEEE Transactions on Neural Networks, 20(1):11-80, 2009. doi: 10.1 200.9TNNN.000220055500
[84] Scherer, M. K., Trendelkamp- Schroer, B., Paul, F., Pérez-Hemández., G., Hoffmann, M., Plattner, N., Wehmeyer, C., Prinz, J.-H., and Noé, F. Pyemma 2: software package for estimation, validation, and analysis of markov models. Journal of chemical theory and computation, 11(11):55255544 2015.
[85] Seidman, J., Kissas, G., Perdikaris, P., and Pappas, G. J. Nomad: Nonlinear manifold decoders for operator learning. Advances in Neural Information Processing Systems, 35:5601-5613, 2022.
[86] Sestak, F,, Schneckenreiter, L., Brandstetter, J., Hochreiter, S., Mayr, A., and Klambauer, G. Vn-egnn: E (3)-equivariant graph neural networks with virtual nodes enhance protein binding site identification. arXiv preprint arXiv:2404.07/1 240400994, 2024.
[87] Sohl-Dickstein, J., Weiss, E., Maheswaranathan, N., and Ganguli, S. Deep unsupervised learning using nonequilibrium thermodynamics. In International conference on machine learming, pp. 2256-2265. PMLR, 2015.
[88] Song, Y., Sohl-Dickstee J., Kingma, D. P., Kumar, A., Ermon, S., and Poole, B. Score-based generative modeling through stochastic differential equations. In International Conference on Learning Representations, 2021.
[89] Thomas, N., Smidt, T., Kearnes, S., Yang, L., Li, L., Kohlhoff, K., and Riley, P. Tensor field networks: Rotation-an rransaaton-equiivarian neural networks for 3d point clouds. arXiv preprint arXiv: /802.0880 2018.
[90] Van Den Oord, A.. Vinyals, O., et al. Neural discrete representation learning, Advances in neural information processing systems, 30, 2017.
[91] Vaswani, A. Attention is all you need. Advances Neural Information Processing Systems, 2017.
[92] Vyas, A., Shi, B., Le, M., Tjandra, A., Wu, Y.-C., Guo, B., Zhang, J., Zhang, X., Adkins, R., Ngan, W., et al. Audiobox: Unified audio generation with natural language prompts. arXiv preprint arXiv:2312. 15821, 2023.
[93] Wang, C., Wang, Y., Xu, M., and Crandall, D. J. Stepwise goal-driven networks for trajectory prediction. IEEE Robotics and Automation Letters, 7(2):2716-277 2022.
[94] Xu, C., Tan, R. T., Tan, Y., Chen, S., Wang, Y. G., Wang. X., and Wang, Eqmotion: Equivariant multi-agent motion prediction with invariant interaction reasoning. In Proceedings of the IEEF/CVF Conference on Computer Vision and Pattern Recognition, pp. 1410-1420, 2023.
15

---

[95] Xu, P., Hayet, J.- and Karamouzas, I. Socialvae: Human trajectory prediction using timewise latents. In European Conference on Computer Vision. pp. 511-528. Springer, 2022.
[96] Yadan, O. Hydra - framework for elegantly configuring complex applications. Github, 2019.
[97] Yao, Y., Atkins, E., JobssoonnRoooooon,, M., Vasudevan, R., and Du, X. Bitrap: Bi-directional pedestrian trajectory prediction with multi-modal goal estimation. IEEE Robotics and Automa- tion Letters, 6(2):1633-1477 1470, 2021.
[98] Yu, Z., Huang, W. and Liu, Force-guided bridge matching for full-atom time-coarsened dynamics of peptides. arXiv preprint arXiv:2408. 15126, 2024.
[99] Yue, Y., Lucey, P., Carr, P., Bialkowski, A., and Matthews, I. Learning fine seee-graind spatial models for dynamic sports play prediction. IEEE International Conference on Data Mining, pp. 670-679, 2014.
[100] Zhang, B. and Wonka, P. Lagem: A large geometry model for 3d representation learning and diffusion. arXiv preprint arXiv:2410.01295, 2024.
[101] Zhang, B., Tang, J., Niessner, M., and Wonka, P. 3dshape2vecset: A shape representation for neural fields and generative diffusion models. ACM Transactions on Graphics (TOG), 42 (4):1-16, 2023.
[102] Zwanzig, R. Nonlinear generalized langevin equations. Journal of Statistical Physics, 9()): 115-220, 1973.
16

---

Appendix
Table of Contents
Notation
18
Architecture Details
19
Architecture Overview in Detail
19
B.
Identilier Assignment
B.3
Encoder and Decoder
19
B
Latent Flow Model
20
B.5
Python Pscudocode
21
Additional Information on Stochastic Interpolants
23
Parametrization
23
D
Additional related work
24
D.I
Molecular Dynamics (MD)
................................
24
D.2
Relationship Graph Founddaion Models
................................ ................................
25
D.3
Relationship to Video and Language Diffusion Models
25
E
Experimental Details
26
Datasets
26
Condition Prediction Horizon
................................
26
Implementation Details
26
F.4
Loss Functions)
27
E.5
Hyperparameters
27
E.6
Evaluation Details
28
E.7
Computational Resources
28
E.8
Software
28
Proofs
34
F.
Proof Proposition 3.3
34
F2
Proof of Proposition 3.4
34
G
Additional Experiments
35
Number of Parameter
35
G.2
Number of Function Evaluations (NFEs
................................
35
Number Learned Latent Vectors
36
Identifier Pool Size
36
G.5
Identilier Assignment
36
G.6 6
Identifier Latent Utilization
37
Visualizations
42
17

---

A
Notation
Table 6: Overview of used symbols and notations.
Definition
Symbol/Notation
Type
continuous time
overall number (sampled) time steps
T
number of observed time steps (when predicting later ones)
To
N
mumber future steps (prediction horizon)
T7
time index for sequences of time steps
N
system state space
S
applicationdepeedeen set, to be further defined
system state
entity
symbolic
number of entities
N
entiy index
I..
set of entities
E
spatial entity dimensionality
Dx
R
entity feuture dimensionality
D.n
entity location (coordinate)
R.
entity properties (entity features)
R
identifier repe representation dimensionality
number latent vectors
latent vector dimensionality
N
trajectory of system (locations of entities time)
X
RTxXNDD,
entity locations f
X
RNx2,
entity of trajectory at trjeectory in latent space latent system
Z'
R
time invariant features of entities matrix of identifier embeddings projection matrices
M
RNKD.
U
RND.
Q.K.V
not specified: depends on number of heads etc.
identiferr assignment function encoder
ida(-)
+ I
E()
R (N(00_ + BIx0. glxu, RR NKNO RTxEXD, 14 gTXLXD,
decoder
D(
approximater (time dynamics model)
A
loss function
var.
time parumeter flow-based model
noise distribution
On
R RTXEDD, RTxLED,
de-noised de-masked trajectory
flow basd model 'velocity prediction' (ucural net)
0y(O,.
R RTxEND, x 2
flow based model 'data prediction" (neural net)
Og ())
RTXLND, x xR R+ 2TxLDD,
neural network purameters
undef.
18

---

B Architecture Details
B.1 Architecture Overview in Detail
Figure shows expanded view of our architecture, and how the different components of LAM- SLIDE interact.
First Stage Autoencoder
Encoder
Decoder
tokens
Ansertion
Lapehoen
Lwsitton
K
ittents
Second Stage Latent Flow-bas based model
Input
1)
02
Example: Decoding
Ve 0 7)
0,
Figure 5: Expanded architectural overview. First Stage: The model is trained to reconstruct the encoded system by querying the latent system representation by IDs. Second Stage: Latent flow- based model is trained to predict multiple masked future timesteps. The predicted system states are decoded by the frozen decoder.
B.2 Identifier Assignment
We provide pseudocode for the identifier creation in Algorithm | of already assigned IDs, maintaining unique IDs across all entities. From practical perspective, we sample IDs randomly so that all entity embeddings receive gradient updates.
This algorithm prevents the reuse
Algorithm 1: Identifier Construction
Input :number of entities N; identifier pool size [I] where N < Z[: embedding dimension D, Output: U e x
U < empty matrix of size N D.
2 + (}
// track assigned identifiers
for i + to N do
r RandomSample(Z/ \ S)
S < SU {r)
e, <- Embedding(r)
// learnable embeddings
U[i] f-
return
B.3 Encoder and Decoder
We provide pseudocode of the forward passes for encoding (E) to and decoding (D) from the latent system space LAM-SLIDE in Algorithm and Algorithm 3 respectively. general, encoder and decoder blocks follow the standard Transformer architecture [91 with feedforward and normalization layers. To simplify the explanation, we omitted additional implementation details here and refer readers to provided source code.
19

---

Algorithm 2: Encoder Function & (Cross-Attention
Input
imput data XMU = XX M, U] E (=*x(D+
Output
latent system state Z RLXD;
Internal parameters : learned latent queries Zmi E RLLD
K <- Linear (MMU)
V <- Linear XMU)
Q Linear mii
return LayerNorm Attentionn( K. v)) // without learnable affine parameters
Algorithm 3: Decoder Function D (Cross-Attention))
Input :lttent system representation E LR entity representation u E R., drawn from
U E Nx
Output: [x. m
R Pm
Z LayerNorm(2
// without learnable affine parameters
2 K { Lincar Z
V Linear Z))
4 + Linear(u)
return Attention[(4]) ([g), K, V)
For the decoding functionality presented in Algorithm 3 we made use of multiple specific decoder blocks depending on the actual task (e.g.. for the molecules dataset, we decoder block for atom positions and one decoder block for atom types).
B.4 Latent Flow Model
We provide pseudocode of the data prediction network Og forward pass in Algorithm The latent layer functionality is given by Algorithm [ The architecture of the latent layers (i.e., our flow model) based on Dehghani al. [23]. with the additional usage adaptive layer norm (adaLN) 699 as also used for Diffusion Transformers [67]. The implementation is based on ParalleIMI.P block code from Black Forest Labs [14) which was adapted to use along the latent dimension well as along the temporal dimension (see Figure (). The velocity model is obtained via reparameterization as outlined in App. C.I
Algorithm Latent Flow Model Oy (data prediction network)
Input
noise-interpplated data Ointer x diffusion time used for interpolation: conditioning C E IR" x conditioning mask B {0, 1x
Output:prediction original data (not interpolated with noise) o E TxXxD, è Embed(T)
2 + Linear( (ineer(Onttrr)) Onter) + Linear (C) + (m)ed(B)
for i +- 1 to nwn_naaapp mum_layers do
LatentLayer(o,.7 T) 0,,9,7 Linear(SiI.U(7))
return o o + O MLP(a O LayerNorm(o) B)
Algorithm 5: LatentLayer
Input :o E Lx'' diffusion time embedding -
Output:updated E R LxC
1 o +- PtweellMLLAttttttnnnnnne WithRoPE(o,, t, dim = 0)
ParalleIMI.PAtttntioonnn PAtteentoon T, dim 1)
return O
20

---

Latent Layer
Parallel-MLP Block (temporal)
Att-out
MLP-Out
Attention
Layerkorm
LayerVorm
Bass
Parallel-MLP Block
MLP-In
(latent)
Scalie Shift
Layerhoorm
MLP
Positional Encoding (RoPE)
Input Tokens
Conditioning
Figure 6: Left: LatentLayer of our method, consisting of latent and temporal ParalleIMLP block. Right: Zoomed in view of the ParalelIMLP block
Using einops notation, the latent layer in Figure 6 can be expressed as:
o' - rearrange (B L) T D2 + (B T) L Dz)
of - (o',T)
of < rearrange(o'; (B T) L D. (B L) T D.)
o' ('0,))
with parameters sets & and 6, where for the latent block, the time dimension gets absorbed into the batch dimension, and for the temporal block, the latent dimension gets absorbed into the batch dimension.
B.5 Python Pseudocode
This section shows Python pseudocode for training and inference for the latent approximator model.
def latent_layer (z, t, z_cond):
## Latent layer of the approximator (high level)
2 2 cobed_cood cxbee_))ond(_.cond) t t_ebbed embed_timee
x Dz, attention calculated over ref [4,5] attention_spatial (z, t_embed)
rearrange "T L Dz L Dz")
# x T Dz, attention calculated over ref [4, 5] z ireetionnttemprral (z, t_embed)
rearrange (z, "L T Dz Dz")
return z
21

---

### Training ####
T x N Dn (N-number of entities, atom e.g. Aspirin 21x4
ids
T binary mask [1, 0, 0, ...,, e.g., one conditioning frame mask
Encoding and conditioning setup
T x L. Dz Encoder state into latent vector 21 encode (x, ids)
x Contains only the conditioning frames, the other frames are masked out and set to 0. 2_cond mask_frames (21, mask)
Sample 20 from normal distridtion 20 torch lanen_like(z)
Sample t uniformly torch uniform 1)
Create interpolation 2t = t 21 (1 t) 20
Loss
zi_hat approxinator (zt, t, z_cond)
loss torch.meat (zt_hat z1) #*2)
loss backvard
**
### Inference ####
x Dn, but con-conditioning frames are zero xond
Transform model trained data prediction into velocity prediction model.
velocity_model data_ totaatoo.vllocity.node (approximator)
T L Dz, contains only the conditioning
frames/ missing frames set
2 cond (x ids)
z0 corch..radd__11t.
Solve ODE torchdiffeq (https://ghh :/ggthhbb com//ttiicte//torcdddffq))
z1_pred solver (velooityovii 20, z_cond)
x hat decoder (z1_pred)
22

---

C Additional Information on Stochastic Interpolants
General interpolants. The stochastic interpolants framework
60 defined without reference
to forward SDE, which allows a lot of flexibility; any choice of a₁ and satisfying the following conditions, possible:
1. a? + o > 0:
2. Qy and are differentiable for all E [0, 1]
3. og = 01 0 and 01 = 00
Interpolants. Two common choices for or and or the Linear and Generalized Variance- Preserving (GVP) path:
Linear:
0ty =
0 = 1 -
GVP: ay = sin
0, = COs
(8)
2
C.1 Parametrization
Our latent flow-based model implemented via data prediction objective 55 48 with the aim have small differences:
[0) - o1
(9)
The velocity model ég is obtained by reparameterization according to Lipman et al. [55]
d ar
ord, + a
0o r) 8o(o;
(10)
ar
where
80(o;)) -0, -0, 19,2(0 2( (r-ōn(:T))
(11)
For integration, employed the torchdiffeg package [20] which provides solvers for differential equations.
23

---

D Additional related work
D.1 Molecular Dynamics (MD)
The most fundamental concepts used to describe the dynamics of molecules nowadays are those given by the laws of quantum mechanics. The SchrÄúdinger equation is partial differential equation that gives the evolution of the complex-values wave function & over time t: ih Or: Ot H (t)p. Here i is the imaginary unit with 22 -1, h is the reduced Planck constant, and (T is the Hamiltonian operator at time 6, which is applied to a function vó and maps to another function. It determines how a quantum system evolves with time, and its eigenvalues correspond to measurable energy values of the quantum system. The solution to SchrAüiingee's equation in the many-body case t): X
(particles 1, :,...::. is the wave function ......... xt
1-1 IR R - C which we abbreviate
as ((x) t). It's the square modulus [xx(x)] t)|² 26 (fx) t))((x), .t) is usually interpreted as probability density to measure the positions X...... XN at time f. whereby the normalization
condition J (xx) dx) ...ix...dee dxx holds for the wave function tp.
Analytic solutions of vi for specific operators H() are hardly known and are only available for simple systems like free particles hydrogen atoms. In contrast to that are proteins, which consist of many thousands of atoms. However, already for much smaller quantum systems, approximations are needed. A famous example is the SSppenhimeeee approximation, where the wave function of the multi-body system is decomposed into parts for heavier atom nuclei and the light-weight electrons, which usually move much faster. In this case, one obtains a SchrAüdinger equation for the electron's movement and another SchrÄüdinger equation for the nucleus's movement. A much faster option than solving a second SchrAúdinger equation for the motion of the nuclei is to use the laws from classical Newtonian dynamics. The solution of the first SchrAúdinger equation defines an energy potential, which can be utilized to obtain forces F, on the nuclei and to update nuclei positions according to Newton's equation of motion: F, m, q (t) (with m, being the mass of particle and (t) describing the motion trajectory of particle over time t).
Additional complexity in studying molecule dynamics is introduced by the environmental conditions surrounding molecules. Maybe the most important property is temperature. For biomolecules, it is often of interest to assume that they are dissolved in water. To model temperature, a usual strategy is to assume a system of coupled harmonic oscillators to model heat bath, from which Langevin dynamics can be derived [29 102). The investigation of the relationship between quantum- mentttnnnec modeling of heat baths and Langevin dynamics remains a current research topic, with various aspects, including the coupling of oscillators and the introduction of Markovian properties with stochastic forces. For instance, Hoel & Szepessy (39] studies how canonical quantum observables are approximated by molecular dynamics. This includes the definition of density operators, which behave according to the quantum Liouville-von Neumann equation.
The forces in molecules are usually given the negative derivative of the (potential) energy: F. E. In the context of molecules, E is usually assumed to be defined by force field, which is a parameterized sum of intra- and intermolecular interaction terms. An example is the Amber force field 72 18]:
E
Σ ko 22 + bands
Σ angles
ko(O - 00)
(12)
N
Aij R
Bij
9igs cRij
VI + cos(no) 7))
+
R
dihedrals
j=
Here kb, To, ko, Op, Vn 7, Aig: Bij. e, 4. serve force field parameters, which are found either empirically or which might be inspired by theory.
Newton's 's equations of motion for all particles under consideration form a system of ordinary differential equations (ODEs), to which various numerical integration schemes, such as Euler, Leapfrog, or Verlet, can be applied to obtain particle position trajectories for given initial positions and velocities. If temperature is included, the resulting Langevin equations form a system of stochastic differential equations (SDEs), and Langevin integrators can be employed. It should be mentioned
24

---

that it is often necessary to use very small integration timesteps to avoid large approximation errors. This, however, increases the time needed to find new stable molecular configurations.
D.2 Relationship to Graph Foundation Models
From our perspective, LAM-SLIDE bears a relationship to graph foundation models [GFMs; 56 62] Bommasani et al. [66] consider foundation models be trained on broad data scale and to be adaptable to wide range of downstream tasks. Mao et al. [62] argue that graphs are more diverse than natural language or images, and therefore, there are unique challenges for GFMs. Especially, they mention that none of the current GFM have the capability to transfer across all graph tasks and datasets from all domains. It is for sure true that LAM-SLIDE not a GFM this sense. However, it might be debatable whether LAM-SLIDE might serve domain- or task-specific GFM. While we primarily focused on trajectory prediction task and are, from that point of view, task-specific, we observed that our trained models can generalize across different molecules or scenes, which may seem quite remarkable given that is common practice to train specific trajectory prediction models for individual molecules or scenes. Nevertheless, it was not our aim in this research to provide GFM, as we believe that this would require further investigation into additional domains and could also necessitate, for instance, examining whether emergent abilities might arise with larger models and more training data [56].
D.3 Relationship to Video and Language Diffusion Models
We want to elaborate our perspective on the relationship between LAM-SLIDE and recent advances video [15] and language diffusion models 779 [9]. At their core, these approaches share
fundamental similarity: they can be conceptualized
form of unmasking.
In video diffusion models, the model unmasks future frames; in language diffusion models, the model unmasks unknown tokens. Both paradigms learn to recover information that is initially obscured in the sequence, and importantly, both methods do so in parallel over the entire input sequence (9]. compared to autoregressive models, which predict single frame or token at a time.
Similarly, LAM-SLIDE represents each timestep concatenated). This perspective allows us to seamlessly incorporate recent advances from both video and language diffusion research into our modeling paradigm.
of latent tokens (or single token when
25

---

E
Experimental Details
E.1
Datasets
Pedestrian Movement. The pedestrian movement dataset, along with its data processing, is available
Basketball Player Movement. The dataset, along with its predefined splits, is available at Data processing is provided in our source code.
N-Body. The dataset creation scripts, along with their predefined splits, are available at https gggithub.ccmmhanjgg1111.
Small Molecules (MD17). The MD17 dataset is available at/http://ww www.ww...mlll..mm.mm org/ #datasets Preprocessing and dataset splits follow Han et al. (35] and can be accessed through their GitHub repository at fithtts::. //t:://gthbbbb com/hanjq17/Geol
The dataset comprises 5,000 training, 1000
validation, and 1000 test trajectories for each molecule.
Tetrapeptides. The dataset, including the full simulation parameters for ground truth simulations, is sourced from Jing et al. [44] and is publicly available in their GitHub repository at https //github. woom/bjing001////// test peptides.
The dataset comprises 3,109 training, 100 validation, and 100
E.2 Condition and Prediction Horizon
Table 7 shows the conditioning and prediction horizon for the individual experiments. For the Tetrapeptides experiments, we predicted 1000 steps parallel and reconditioned the model ten times on the last frame for each predicted block, this concept is similar to Arriola et al. [9]
Table 7: Number of conditioning and predicted frames for the different experiments.
Experiment
Conditioning Frames
Predicted Frames
Total Frames
Pedestrian trajectory forecasting (ETH-UCY) Baskethall (NBA)
12
20
12
20
N-Body
10
20
30
Molecular Dynamics (MD17)
20
Molecular Dynamics Tetrapeptides (4AA)
999
10 000
E.3
Implementation Details
Training procedure. (i) First Stage. In the first stage, we train the encoding and decoding functions & and 'D in auto-encoding fashion, i.e., optimize for precise reconstruction of the original system state representation from its latent representation. For discrete features (e.g.. atom type, residue type), we tend use cross-entropy loss, whereas for continuous features use regression loss (e.g., position, distance). The loss functions for each task are summarized in Appendix E.S Notably, the entity identifier assignment is also random. (ii) Second Stage. In the second stage, we freeze the encoder and train the approximator to model the temporal dynamics via the encoded latent system representations. To learn consistent behavior over time, we pass U from the encoder to the decoder D. To avoid high variance latent spaces, we used nayrr-normaiization (10) (see Appendix E.3)
Data Augmentation. ance/invariance with respect to spatial transformations, apply random rotations and translations to the input coordinates.
To compensate for the absence of built-in inductive biases such as equivari-
Identifiers. For the embedding of the identifiers we use torch mn. Embedding 666] layer, where we assign a random subset of the possible embeddings to the entities in each training step. See
also Algorithm
26

---

Latent space regularization. To avoid high variance latent spaces, Rombach et al. [77] relies on KL-rre.. imposing a small KL-penalty towards a standard normal on the latent space, as used in VAE 499 Recent work [100] has shown that layer normalization [1O] can achieve similar regulatory
effects without requiring an additional loss term and simplifying the training procedure. We adapt
this approach in our method (see the left part of Figure
Latent Model. For the latent Flow Model additionally apply auxiliary losses for the individual tasks, as shown in App. E.5 Where we decode the predicted latent system representations and pack-propagate through the frozen decoder to the latent model.
MD17. We train a single model on all molecules a feat that is structurally encouraged by the design of LLLDE For ablation, also train Geo 35 on all molecules and evaluate the performance on each one of them ("all->each" the Table (3). Interestingly. we also observe consistent improvements in the GeoTDM performance. However, GeoTDM's performance does not reach that of LAM-SLIDE.
employ the Atom 14 representation as used AlphaFold [I] In this representation, each entity corresponds to one amino acid of the tetrapeptide, where multiple atomic positions are encoded into single vector of dimension Dæ = 3 x 14. Masked atomic positions are excluded from gradient computation during model updates. This representation is computationally more efficient.
Tetrapeptides. For the experiments tetrapeptides in Section 4.5
E.4 Loss Functions
This section defines the losses, which we use throughout training:
Position Loss.
N
Spos C'os(X',X''
Σ
x
-
-
(13)
N
Inter-distance Loss.
(X,,
Σ
Σ
(Di((*)) - Di)()
(14)
N2
i=1 j=1
with
Di, (X') = || X
(15)
Cross-Entropy Loss. Depending the experiment, we have different CE losses depending on the problem, see App. E.5
1 N
--- yu log(pu)
Σ
LcE
(16)
Frame and Torsion Loss. For the Tetrapeptide experiments, we employ two additional auxiliary loss functions tailored to better capture unique geometric constraints of proteins, complementing our primary optimization objectives: a frame loss Lframe, which is based on representing all atoms withing a local reference frame II Algorithm 29], and torsion loss torsion loss Liors inspired by Jumper et al. [46].
E.5 Hyperparameters
Tabs. to 12
show the hyperparameters for the individual tasks, loss functions are as defined
App. For trained models, we use AdamW [50] (88) optimizer and use EMA 3I each
update step with decay parameter of 3 0.999.
27

---

E.6 Evaluation Details
Tetrapeptides. Our analysis of the Tetrapeptide trajectories utillzed РyEMMA 84 and followed the procedure as Jing et al. [4], incorporating both Time-lagged Independent Component Analysis (TICA) 770 and Markov State Models (MSM) 42). For the evaluation of these metrics, we relied on the implementations provided by [4]
E.7 Computational Resources
Our experiments were conducted using a system with 128 CPU cores and 2048GB of system memory. Model training was performed on 4 NVIDIA H200 GPUs, each equipped with 140GB of VRAM. In total, roughly 5000 GPU hours were used in this work.
E.8
Software
We used PyTorch 2 7 for the implementation of our models. Our training pipeline was struc- tured with PyTorch Lightning (28). We used Hydra [96] to run experiments with different
hyperparameter settings. Our experiments were tracked with Weights Biases [12].
28

---

Table 8: Hyperparameter configuration for the pedestrian movement experiments (Section 4.1
First Stage
Network
Encoder
Number of latents
2
Number of entity embeddings
8
Number of attention heads
2
Number of cross attention layers
Dimension latents D=
32
Dimension entity embedding
128
Dimension attention head
16
Decoder
Number of attention heads
2
Number of cross attention layers Dimension attention head
1
16
Loss
Weight
Lpos X, X) Lint (x,
Training
Learning rate
le-4
Learning rate scheduler
CosineAmmealingiiinnnll-ll
Batch size
256
Epochs
3K
Precision
32-Full
Batch size
1024
Second Stage
Setup
Condition
8 Frames
Prediction
12 Frames
Network
Hidden dimension Number of Layers
128
6
Auxiliary Loss
Weight
Cpos X x) Line (X, X)
0.25
0.25
Training
Learning rate Batch size Epochs Precision
1e-3
64
IK
BF16-Mixed
Inference
Integrator ODE steps
Euler
10
29

---

Table 9: Hyperparameter configuration for the basketball player movement experiments (Section 4.2
First Stage
Network
Encoder
Number of latents L
32
Number of entity embeddings
11
Number of attention heads
2
Number of cross attention layers
1
Dimension latents D=
32
Dimension entity embedding
128
Dimension attention head
16
Decoder
Latent dimension
32
Number of attention heads Number of cross attention layers
8
1
Loss
Weight
Lpos X, X)
Line (X, X)
1
LCE Co Team
Group
0.01
0.01
Training
Learning rate
1e-4
Learning rate scheduler
Cosine:Annalling(minni--ss
Optimizer
AdamW
Batch size
16
Second Stage
Setup
Condition Prediction
8 Frames
12 Frames
Network
Hidden dimension H Number of Layers
128
6
Auxiliary Loss
Weight
Epos X X)
0.25
Line (X, X)
0.25
Training
Learning rate Learning rate scheduler Batch size Epochs Precision
1e-3
CosineAmmealingmminntet// Ir=le-)
64
500
BF16-Mixed
Inference
Integrator
Euler
ODE steps
10
30

---

Table 10: Hyperparameter configuration for the N-Body experiments (Section 4.3
First Stage
Network
Encoder
Number of latents
16
Number of entity embeddings
10
Number of attention heads
2
Number of cross attention layers
1
Dimension latents D=
32
Dimension entity embedding
128
Dimension attention head
16
Decoder
Number of cross attention layers
1
Number of attention heads
Number of cross attention layers
16
Loss
Weight
pos X. X) Line X)
Training
Learning rate Batch size Epochs Precision
1e-3
128
2K
32-Full
Second Stage
Setup
Condition Prediction
10 Frames
20 Frames
Network
Hidden dimension Number of Layers
256
6
Auxiliary Loss
Weight
Cpos (X X) Line Line(X (X
0.0
0.0
Training
Learning rate Learning rate scheduler Batch size
lc-3
CosineAmmealinggmnn_ltetl (rapllnggmmi__li-17))))
64
Epochs Precision
1K
BF16-Mixed
Inference
Integrator
Euler
ODE steps
10
31

---

Table 11: Hyperparameter configuration for the small molecule (MD17) experiments (Section 4.4
First Stage
Network
Encoder
Number of latents L
32
Number of entity embeddings
8
Number of attention heads
2
Number of cross attention layers
Dimension latents D=
32
Dimension entity embedding
128
Dimension attention head
16
Decoder
Number of cross attention layers
Number of attention heads 2
Number of cross attention layers
16
Loss
Weight
Cpos (X, X) Line (X, X)
CCE(.)) Atom type
1
Training
Learning rate Batch size Epochs Precision
le-4
256
3K
32-Full
Second Stage
Setup
Condition Prediction
10 Frames
20 Frames
Network
Hidden dimension Number of Layers
128
6
Auxiliary Loss
Weight
Cpos (X X) Line XineX,X0 X)
0.25
0.25
Training
Learning rate
le-3
Learning rate scheduler
Cosine/nmeelinn(mmnnnIll.e
Batch size Epochs Precision
64
2K
BF16-Mixed
Inference
Integrator
Euler
ODE steps
10
32

---

Table 12: Hyperparameter configuration for the Tetrapeptides experiments (Section 4.5
First Stage
Network
Encoder
Number of latents T,
5
Number entity embeddings
Number of attention heads
Number cross attention layers
Dimension latents D=
96
Dimension entity embedding
128
Dimension attention head
16
Decoder
Number attention heads
Number of cross attention layers
Dimension attention head
16
LOss
Weight
Lpos (X, X)
Lint (X, X)
Lirame X, X)
Ciore X, X
0.1
( Residue type
0.001
Training
Learning rate
le-4
Batch size Epochs Precision
200K
32-Full
Second Stage
Setup
Condition
Frame
Prediction
10,000 Frames (10x rollouts)
Network
Hidden dimension Number of Layers
384
Auxiliary Loss
Weight
Cpos (X, X)
0.25
Lin (X, X)
0.25
Lirame (X,X) X
0.25
Training
Learning rate Optimizer Batch size Epochs Precision
le-3
AdamW
64
1.5K
BF16-Mixed
Inference
Integrator ODE steps
Dopri5 20
adaptive
33

---

Proofs
F.1 Proof of Proposition 3.3
The proof is rather simple, but we include it for completeness.
Proposition 3.3. Given an identifier pool I and a finite set of entities E, an identifier assignment pool T as defined by Definition 3.2 non-empty if and only if [E| [
Proof. Assume [ > [ By pigeonhole principle, any function f E -> I must map at least two distinct elements of E to the same element in J. Therefore, cannot be injective.
Conversely, if | < [Z]. we can construct an injective function from E to by assigning each element in E' unique element in I, which is possible because Z has at least as many elements E.
Therefore, an injective identifier assignment function ida(-) E only exists if E < [I]. Hence, the set is non-empty in this case and empty otherwise.
F.2 Proof of Proposition 3.4
Proposition 3.4. Given identifier pool and a finite set of entities E such that [E [Z] the identifier assignment pool as defined by Definition 3.2 contains finitely many injective functions.
Let Eb and m [Z]. then the set of infective functions is bounded and finite:
mn!
(m 1). (m +
(m nz)! (m)n & inf
(17)
Where (m)n is commonly referred as falling factorials, the number of injective functions from set of size n to a set of size m.
34

---

G Additional Experiments
To investigate the sensitivity of our model with respect to different hyperparameter settings, conducted additional experiments.
G.1 Number of Parameter
We conducted scaling experiments on both the MD17 and the Tetrapeptides (4AA) datasets to evaluate how LAM-SLIDE performance scales with model size. On MD17, we evaluate LAM-SLIDE using model variants with 1.7M, 2.1M, and 2.5M parameters. Our results show that, for nearly all molecules, performance improves with parameter count in terms of ADE/FDE, see Table I3] Similarly, on the Tetrapeptides dataset, we evaluate using model variants with 4M, 7M, 1 11M, and 28M parameters. All performance metrics show consistent improvement with increased model capacity, see Table These findings indicate the favorable scaling behavior of our method.
Table 13: Method comparison for forecasting MD trajectories of small molecules. Compared methods predict atom positions for 20 frames, conditioned on 10 input frames. Results are reported in terms of ADE/FDE, averaged over 5 sampled trajectories.
Aspirin
Benzene
Ettsnol
Malonaldchyyde
Naphtiallone
Salicylic
Tolaene
Uracil
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
RF
0.303
0.194
0.374
0.454
0.343
0.239
0.272
TPN N9
0.133 0.268
0.024
0.049
0.201
0.414
0.184
0.386
0.072
0.098
0.115
0.223
0.090
0.150
0.090
0.159
SBG
50
0.294
0.556 0.027
0.056
0.188
0.359
0.214
0.456
0.069
0.103
0.189
0.312
0.108
0.184
0.107
0.196
EGNN
0.267 0.564 0.024
0.042
0.268
0.401
0.393
0.958
0.005
0.133
0:159
0.348
0.207
0.294
0.154
0.282
EgMotion
[54]
0,185 0.246 0.029
0.043
0.152
0.247
0.155
0.249
0.073
0.092
0.110
0.151
0.097
0.129
0.088
0.116
0,430
0.145
GooTDM 1.M
0,107
0.193
0.023
0.039
0.115
0.209
0.107
0.176
0.064
0.087
0.083
0.120
0.083
0.121
0.074
0.099
GeoTDM 1.9M (all->euch)
0.091
0.024
0,140
0.104
0.164
0.074
0.114
0.000
0.102
0.059
0.098
0.021
0.032
0.087
0.167
0.073
0.124
0.037
0.058
0JM4
0.074
0.045
0.075
0.050
0.074
LAM-SLIDE
0.064
0.033
0.053
0.081
0.054
LAM-SLIDE 1.7M
0.074
0.117
0.025
0.037
0.110
0.195
0.097
0.159
0.053
0.074
0.063
0.091
0.064
0.094
0.064
0.089
Results from Han et al.
35
Table 14: Method comparison for predicting MD trajectories of tetrapeptides. The columns denote the ISD between distributions of torsion angles (backbone (BB), side-chain (SC), and all angles), the TICA, the MSM metric, and the number of parameters.
Torsions
TICA
MSM
Params
Time
BB
All
0.1 joint
(M)
100 ns"
.103
.055
.076
.211
.268
.208
3h
MDGen
.130
.093
.109
.230
.316
.235
34
60s
LAM-SLIDE
.128
0.122
0.125
.227
.315
.224
53s
LAM-SLIDE
.152
.151
.152
.331
.226
LAM-SLIDE
.183
.191
.187
.356
.235
ALIBE
.284
.331
.311
.339
.461
.237
Results from Jing et al.
G.2 Number of Function Evaluations (NFEs)
We conduct a comparative analysis on computational efficiency by measuring the number of function evaluations (NFEs) required to achieve the performance results reported in the main section of our publication. As shown in Table 15, our approach demonstrates remarkable efficiency compared to the previous ofth-offthe-art method, GeoTDM 35 across N-Body simulations, MD17 molecular dynamics, and ETH pedestrian motion forecasting experiments. Our model consistently requires significantly fewer NFEs than GeoTDM to reach comparable superior performance levels.
35

---

It is worth noting that flow-based models generally require fewer NFEs compared to diffusion- approaches, such as GeoTDM. However, this efficiency advantage does not come at the expense of performance quality [27 Indeed, the relationship between NFEs and performance is not strictly monotonic, as demonstrated in other domains. For instance, Esser et [27] achieved optimal image generation results in terms of FID [36] with 25 NFEs, demonstrating that computational efficiency and high performance can be achieved simultaneously with properly designed architectures.
Furthermore, in the case of the Tetrapeptides (4AA) experiments shown in Section 4.5 we employ an adaptive step size solver achieve the reported performance, which yields better results than an Euler solver. We use Dopri5 as implemented in the torchdiffeq package [20].
Table 15: Comparison of the number of function evaluations (NFEs) for LAM-SLIDE and GeoTDM.
N-Body
MD17
ETH
GeoTDM [35]
1000
1000
100
LAM-SLIDE
10
10
10
Results from Han et al.
(35]
G.3 Number of Learned Latent Vectors
We conducted experiments to quantify the relationship between model performance and the number of latent vectors L using the MD17 dataset. As shown in Table I6 performance increases with the number of latent vectors I.. Of particular significance is the performance at I. 21, which corresponds to the maximum number of entities, allowing to investigate whether this constitutes an upper bound on model capacity. Notably, performance continues to improve that model capacity scales favorably even beyond the number of entities. At L = 16, representing a compressed latent representation, our model remains competitive with the second-best method, GeoTDM [35].
= 32, indicating
We further analyze whether the improvement resulting from increasing L due solely to the encoder- decoder's 's reconstruction performance. Figure [ shows the reconstruction error for varying number of latent vectors L. Even with substantially fewer latent vectors than entities, the model achieves good reconstruction performance. This gap suggests that the performance gains from increasing I. are not due to improved reconstruction, but rather to the model's 's ability leverage the enlarged latent space representation more effectively.
G.4 Identifier Pool Size
We evaluate the impact of identifier pool size using the MD17 dataset. This dataset contains at most 21 atoms, in general, an identifier pool of [ 21 would be sufficient. However, for the results shown the main paper, we used [Z] == 32. conduct additional experiments by training multiple first-stage models with varying identifier pool sizes and report the reconstruction error measured by Euclidean distance. F shows that the reconstruction error increases with the size of the identifier pool, since larger pool results in fewer updates to each entity embedding during training.
investigate the impact a larger identifier pool I, we
G.5 Identifier Assignment
To assess the impact of different ID assignments on reconstruction performance, we run our model five times with different random ID assignments and measure the standard deviation across these assignments in terms of reconstruction error in Á. Results are shown in Figures 7 and g The low standard deviation across different ID assignments demonstrates the robustness of our model with respect to random ID assignment.
36

---

Table 16: Model performance in terms of ADF/FDE with respect to different number of latent vectors
Aspirin
Benzene
Ethanol
Malonaldehyde
Naphthulene
Salicylic
Toluene
Uracil
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
ADE
FDE
LAM-SLIDE (T. 0.354
0.483
0.213
0.264
0.420
0.541
0.366
0.537
0.210
0.223
0.253
0.286
0.316
0.418
0.243
0.275
SLIDE (L- 16) 0.099
0.146
0.039
0.049
0.108
0.187
0.095
0.149
0.070
0.989
0.075
0.101
0,073
0.103
0.071
0.093
LAM-SLIDE
0.082
0.054
0.059
0.057
0.083
LAM-SLIDE = 32) 0.059 0.098
0.021
0.032
0.087
0.167
0.073
0.124
0.037
0.088
0.047
0.074
0.045
0.075
0.050
0.074
G.6 Identifier Latent Utilization
We investigate how identifiers utilized by model, addressing three key questions: Does each identifier exhibit unique query pattern? How these patterns vary with latent space dimensionality? Do different attention heads learn distinct patterns?
We examined decoder attention weights in overparameterized (N and compressed (N 2 L) settings using models with L = 16 and L = 32 latent vectors trained on MD17. Our analysis focuses on Aspirin molecules (N 21 atoms) across different conformations.
The results Figs. to 12 reveal two main findings. First, identical atom identifiers produce consistent attention patterns across different conformations, with each identifier learning a unique addressing scheme distributed over latent vectors. This suggests that the identifiers function as "retrieval mechanisms" independent of stored content.
Second, attention head utilization adapts to the compression ratio. The overparameterized model (L 32) primarily uses single head, acting like hash table-like retrieval. The compressed model (L = 16) utilizes both heads with distinct patterns, likely to mitigate identifier collisions in the limited latent space.
0.04
Reconstruction Error in A
5,0325 020520)
0,0202 20,00320))
0.02
3.0085 000153)
000033)
(9.000004)
0.00
64
128
512
Size of Identifier Pool
Figure 7: Reconstruction error in À for the MD17 dataset. We report the reconstruction error for the encoder-decode model for different identifier pool sizes. The error bars show the standard deviation across five runs with different random ID assignments.
37

---

5,05E3 0Co417)
0.05
Reconstruction Error in A
0.03
0.0188 10.002076)
0.01
9.0010 000003)
0.0037 000032)
0.00
21
16
Figure 8: Reconstruction еπтог in À for the MD17 dataset. We report the reconstruction error for the encoder-decoder model for different number latent vectors L. The error bars show the standard deviation across five runs with different random ID assignments.
Sample Heat
Sample D- Head
Semple3
Sumsle1 Haad
Sample
Samile 3 Phaas
Sumple 3- haad
Laken Wetter
Latve
Figure 9: Attention patterns on the MD17 dataset for four random samples of the validation dataset, using model trained with 16 latent vectors. Each row corresponds to single sample and displays the attention patterns across individual attention heads. Each sample employs a distinct identifier assignment.
38

---

Zample o Head o
Samble D-1 hedd
Sample 2 0
lend
Sample 3 Mead
Samsle Head
Litee Wenter
Figure 10: Attention patterns on the MD17 dataset for four random samples of the validation dataset, using a model trained with 16 latent vectors. Each row corresponds to single sample and displays the attention patterns across individual attention heads. Each sample employs the same identifier assignment.
39

---

Zample lead
Samble D- head
31-
30
Sample
Sumslc Haad
10-
32-
2 60
12-
Sample -eaad
Summle 2 iied
KG
65
Figure 11: Attention patterns on the MD17 dataset for four random samples of the validation dataset, using a model trained with 32 latent vectors. Each row corresponds to single sample and displays the attention patterns across individual attention heads. Each sample employs a distinct identifier assignment.
40

---

Zample lead 0
Samble D- head
Sample 2 -eac
2-
Sample 3
Mead
Sample Hiadd
Letee
Attention patterns on the MD17 dataset for four random samples of the validation dataset, using a model trained with 16 latent vectors. Each row corresponds to single sample and displays the attention patterns across individual attention heads. Each sample employs the same identifier assignment.
Figure 12:
41

---

H Visualizations
Figure 13: Basketball player trajectories from the NBA dataset. We visualize each player's 's movement path up to the final frame and include the ball's trajectory in the visualization. Top row: Trajectories for scoring scenarios. Bottom row: Trajectories for rebound scenarios. (Court image source: htttss://gg thub. Comm/inouuk3//NBA-PPPP http://///...apppaoooooooooee//o
p
42

---

LAx
Figure 14: Trajectories from the N-Body dataset, predicted vs ground truth trajectories. Left: Charged particles. Middle: Spring dynamics. Right: Gravitational system.
43

---

Aspirin
Benzene
Ethanol
Malonaldehyde
Naphthalene
Salicylic
Toluene
Uracil
Figure 15: Molecular dynamics trajectories from the MD17 dataset, showing time-evolved structural predictions for each molecule. For every compound, we display four distinct trajectory predictions, with each prediction comprising 20 superimposed time frames to illustrate the range of conformational changes.
44

---

Sample
PNHP
PIDV
500
GG-NN
SFCH
HELI
Figure 16: Torsion angle distributions of the six backbone torsion angles, comparing molccular dynamics (MD) trajectories (orange) and sampled trajectories (blue); and Free energy surfaces pro- jected onto the top two lime-lageed independent component analysis (TICA) components, computed from both backbone and sidechain torsion angles.
45

---

seps
sePs Foems
sees Frwme
cers Foees
sePs Foee
sees Foee
6605 Freme
coPs Foeme
SSPE.
CMEE Frem
CMEr From
CrE From 3040
CME Froms
CME Frem 5080
crer Froms
crer Froms
OKL. Froms
OPE Frems M00
(EE -Trem 2000
Figure 17: Molecular Dynamics trajectories from the Tetrapeptides (4AA) dataset. showing
time-evolved structural predictions for ten frames at an interval of 1000 frames. Top: SDFS (Serine Aspartic Acid Phenylalanine Serine) peptide. Bottom: CPEE peptide (Cysteine Proline Glutamic Acid Glutamic Acid).
46

---

NeurIPS Paper Checklist
1. Claims
Question:
Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: Our main contributions are declared in the abstract, and we summarize the contributions in the introduction.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made in the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
It is fine to include aspirational goals as motivation as long as is clear that these goals attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification: We list the limitations in the discussion section and include additional section in the appendix investigating model performance with respect parameter size and inference performance.
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification. asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
3. Theory assumptions and prool's
Question: For each theoretical result, does the paper provide the full of assumptions and complete (and correct) proof?
47

---

Answer: [Yes]
Justification: While our paper's primary contribution is architectural rather than theoretical, provide proof establishing the necessary conditions for our identifier assignment. Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proof' in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors encouraged provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix or supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: We provide pseudocode and hyperparameter tables for our model in the appendix section of our paper.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution a specific model and empirical evaluation, it may be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, other means that appropriate to the research performed.
also be provided via detailed
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how to reproduce that algorithm.
(b) the contribution is primarily new model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reprodueibility may be tricky in some cases, in which authors are welcome to describe the particular way they provide for reproducibility. In the case of closed-source models, it may be that access to the model is limited in some way (e.g., to registered users), but it should be possible for other researchers have some path reproducing or verifying the results.
5. Open access to data and code
48

---

Question: Does the paper provide open access to the data and code, with sufficient instruc- tions faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [Yes]
Justification: We provide full access to the source code to facilitate the production of experimental results. All used datasets have open access.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips. ec/
for more details.
While encourage the release of code and data, we understand that this might not be possible, so âÄIJNoâÄí is acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g.. for new open-source benchmark).
The instructions should contain the exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https
/nipsscc/ppll wttpcccpppppcceeee.ce..ec...ceee guides/ Codeeuomisssssiieec soiiooooccs for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (c.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary understand the results?
Answer: [Yes]
Justification: All chosen hyperparameters are discussed in the appendix section Experimental Details, and the splits for the dataset provided in the provided source code.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [No]
Justification: We omit error bars for model parameters due to our extensive experimentation across diverse domains, while reporting metries averaged across multiple sampling runs for selected experiments
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by emor bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
49

---

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g., Normally distributed errors).
It should be clear whether the error bar is the standard deviation or the standard error of the mean.
is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out range (e.g. negative еπτог rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [No]
Justification: We report aggregate computational resources for all experiments; we do not report the resources for each individual experiment.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make into the paper).
9. Code of ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics http:///hwerrrpppcpcpppcc///ataa
cc
Answer: [Yes]
Justification:
The research conforms in every aspect to the NeurlPS Code of Ethics.
Guidelines:
The answer NA means that the authors have not reviewed the NeurlPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., there a special consid- eration due to laws regulations in their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer: [No]
Justification: We do not anticipate direct negative social impact from our work, though it can serve as a foundational model for dynamical systems research.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
50

---

If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), faimess considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional unintentional) misuse of the technology.
If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification: The paper poses no such risks.
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards challenging, and many papers do not require this, but encourage authors to take this into account and make best. faith effort.
12. Licenses for existing assets
Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
Justification: We provide the source for all used datasets as URLs in the dataset section of the appendix.
Guidelines:
The answer NA means that the paper does existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset used and, possible, include a URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
51

---

If assets are released, the license, copyright information, and terms of use in the thcode has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
package should be provided. For popular datasets, paperswi
latasets.paperssit
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
If this information not available online, the authors are encouraged to reach out to the asset's s creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: The paper does not release new assets.
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/moode as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsoureing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification: Our paper does not involve crowdsourcing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included in the main paper.
According to the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or equivalent approval/review based the requirements of your country institution) were obtained?
Answer: [NA]
Justification: Our paper does not involve crowdsoureing nor research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing research with human subjects.
Depending the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
52

---

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors adhere the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.
16. Declaration of LLM usage
Does the paper describe the usage of LLMs if it is an important, original, or non-standard component of the core methods in this research? Note that if the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology. scientific rigorousness, or originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification: In our publication, LLM usage is not considered as an important part.
Guidelines:
The answer NA means that the method development in this research does not involve LLMs as any important, original, or non-standard components.
Please refer to our LLM policy battps://neuree h0tps//eecccoooeeeeee/////////// for what should or should be described.
53


Reviews:
[{"review_id": "2kyu3xMD3x", "reviewer": "Reviewer_fy2s", "summary": "This paper introduces a model for generative modeling of stochastic trajectories. The model bridges the traceability of individual entities and the latent system representation. It uses the identifiers (IDs) that allow for the retrieval of entity properties. The author conducted a comprehensive experiment to validate the model.", "questions": "1. Could the author provide more insights into the results on the ETH-UCY and the NBA dataset, where LAM-SLIDE only shows marginal or no improvement? \n2. Does the author have any insights on whether the number of particles in the system affects the model performance? \n3. Could the author provide some explanation on why LAM-SLIDE 2.1M seems to underperform GeoTDM 1.9M (all→each) in Table 13?", "limitations": "Yes.", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**\n1. The model introduces a novel approach by using identifiers (IDs) to allow traceability, closing the gap in previous generative modeling methods. \n2. The model has good performance across different domains, showing the general effectiveness of the approach.  \n\n**Weaknesses**\n1. The paper offers only limited discussion and ablation studies. More ablations on the design choices, such as identifiers (IDs), will be beneficial. \n2. The number of parameters for LAM-SLIDE is not indicated in Table 14.", "was_revised": true, "final_justification": "The author has addressed all my questions. The quality of the work meets the standard for a NeurIPS acceptance. However, I still believe further evaluation could be done on this work. Thus, I will retain my score as it is.", "rebuttal": "We greatly appreciate your constructive feedback, recognizing both our technical contributions (ID-based traceability and cross-domain performance) and opportunities to strengthen the presentation of our work. Below, the you can find the response to the raised concerns.\n\n### Weaknesses \n\n> The paper offers only limited discussion and ablation studies. More ablations on the design choices, such as identifiers (IDs), will be beneficial.\n\nWe agree that our ablation studies on identifiers (IDs) deserve more prominence in the main paper. Although we conducted extensive experiments (Appendix G), these findings were not effectively communicated in the main text. Our key results include the following, with the fourth point representing new analysis conducted for this rebuttal:\n\n- **ID pool size (Figure 7)**  \nReconstruction errors remain low (angstrom scale) across the ablated pool sizes. Since our ablation used the same number of training epochs across all pool sizes, larger pools naturally result in less frequent updates for the individual ID embeddings. So we can use much more possible IDs during training than entities, but because only a subset of the IDs are assigned in each update step, an increased pool size would benefit from longer training times.\n\n- **Latent vector count (Figure 8)**  \nThe reconstruction quality remains stable even under significant compression, where the number of latents $L$ is much smaller than the number of entities $N$. In the extreme case of $L=4$ performance degrades noticeably.\n\n- **Robustness of the ID assignment**  \nThe error bars in Figures 7 and 8 show the results of different random ID assignments, which produce nearly identical outcomes, even with large pool sizes (512 IDs). This indicates that the model learns ID-invariant representations rather than memorizing specific ID-to-entity mappings, a crucial property for generalization.\n\n- **Learned / Orthogonal ID embeddings**  \nWe compared fixed orthogonal ID embeddings as an alternative to learned embeddings. While orthogonal initialization also works, learned ID embeddings consistently deliver better results. Note that orthogonality is only approximate when the ID pool size exceeds the embedding dimension, as true orthogonality requires the number of IDs ≤ embedding dimension.  t-SNE visualization shows learned IDs distribute evenly across the embedding space.\n \nIn response to your feedback, we moved these ablation studies from Appendix G to a new Section 4.7 (\"Analysis of Design Choices\") in the main paper. This section includes our existing analyses of identifier pool size and assignment strategies, augmented with an additional ablation conducted in the response of reviewer PyHh.\n\n> The number of parameters for LAM-SLIDE is not indicated in Table 14.\n\nYou are right there there is an inconsistency in notation. While the parameter count appears in the second-to-last column, we agree it should be positioned next to the method name as in the other tables. We updated Table 14 in our manuscript and provide a markdown version of the updated table below. The 100ns method is a reference simulation acting as lower bound and does not have parameters.\n\n | Method | Torsions |  |  | TICA |  | MSM | Time |\n |--------|---------|---------|---------|---------|---------|---------|---------|\n |  | BB | SC | All | 0 | 0,1 joint |  | (M) |\n | 100 ns | .103 | .055 | .076 | .201 | .268 | .208 | ~ 3h |\n | MDGen 34M | .130 | **.093** | **.109** | .230 | .316 | .235 | ~ 60s |\n | LAM-SLIDE 28M | **.128** | 0.122 | 0.125 | **.227** | **.315** | **.224** | ~ 53s |\n | LAM-SLIDE 11M | .152 | .151 | .152 | .239 | .331 | .226 |  |\n | LAM-SLIDE 7M | .183 | .191 | .187 | .26 | .356 | .235 |  |\n | LAM-SLIDE 4M | .284 | .331 | .311 | .339 | .461 | .237 |  |\n\n### Questions\n\n> Q1: Could the author provide more insights into the results on the ETH-UCY and the NBA dataset, where LAM-SLIDE only shows marginal or no improvement?\n\nWe agree, that the improvements on the ETH-UCY dataset are not as strong as on the other datasets. However, we would like to note that the baseline methods are specifically designed for this type of task and often rely on prior knowledge about relationships between entities, such as inter-entity distances. Additionally, they additionally incorporate extra features like velocity and acceleration. In contrast, our method does not make such assumptions or require these additional features. While our model could also benefit from task-specific feature engineering, we leave such ablations to future work.\n\n> Q2: Does the author have any insights on whether the number of particles in the system affects the model performance?\n\nWe investigated this using the MD17 dataset and found no clear correlation between particle count and model performance. Instead, molecular flexibility appears to be the dominant factor here. As an intuitive example, despite having fewer atoms, Ethanol (9 atoms) shows higher prediction error than Naphthalene (18 atoms) due to its greater conformational flexibility. This pattern is evident in Table 4 and visualized in Figure 11, where Ethanol's dynamic hydroxyl group creates more complex trajectories than Naphthalene's rigid aromatic structure, suggesting system complexity, rather than raw particle count, drives prediction difficulty.\n\n> Q3: Could the author provide some explanation on why LAM-SLIDE 2.1M seems to underperform GeoTDM 1.9M (all→each) in Table 13?\n\nWe would appreciate clarification on this observation, as we do not immediately see where LAM-SLIDE 2.1M underperforms GeoTDM 1.9M (all→each) in Table 13. We believe a potential misinterpretation could stem from our underlining approach. In Table 13 we underlined the second-best non-LAM-SLIDE method rather than the overall second-best method, which we recognize could lead to confusion. To address this, we have updated the table in our manuscript by adding two additional columns showing the ADE/FDE mean for each row, and present the last five rows of the updated below. We hope these changes make the results more accessible for the reader.\n\n|                          | Aspirin |        | Benzene |        | Ethanol |        | Malonaldehyde |        | Naphthalene |        | Salicylic |        | Toluene |        | Uracil |        | Mean     | Mean     |\n|--------------------------|---------|--------|---------|--------|---------|--------|---------------|--------|-------------|--------|-----------|--------|---------|--------|--------|--------|----------|----------|\n|                          | ADE     | FDE    | ADE     | FDE    | ADE     | FDE    | ADE           | FDE    | ADE         | FDE    | ADE       | FDE    | ADE     | FDE    | ADE    | FDE    | ADE      | FDE      |\n| GeoTDM 1.9M              | 0.107   | 0.193  | 0.023   | 0.039  | 0.115   | 0.209  | 0.107         | 0.176  | 0.064       | 0.087  | 0.083     | 0.120  | 0.083   | 0.121  | 0.074  | 0.099  | 0.082    | 0.131    |\n| GeoTDM 1.9M (all→each)   | 0.091   | 0.164  | 0.024   | 0.040  | 0.104   | 0.191  | 0.097         | 0.164  | 0.061       | 0.092  | 0.074     | 0.114  | 0.073   | 0.112  | 0.070  | 0.102  | 0.074    | 0.122    |\n| LAM-SLIDE 2.5M           | 0.059   | 0.098  | 0.021   | 0.032  | 0.087   | 0.167  | 0.073         | 0.124  | 0.037       | 0.058  | 0.047     | 0.074  | 0.045   | 0.075  | 0.050  | 0.074  | 0.052    | 0.088    |\n| LAM-SLIDE 2.1M           | 0.064   | 0.104  | 0.023   | 0.033  | 0.097   | 0.182  | 0.084         | 0.141  | 0.044       | 0.067  | 0.053     | 0.081  | 0.054   | 0.086  | 0.054  | 0.079  | 0.059    | 0.097    |\n| LAM-SLIDE 1.7M           | 0.074   | 0.117  | 0.025   | 0.037  | 0.110   | 0.195  | 0.097         | 0.159  | 0.053       | 0.074  | 0.063     | 0.091  | 0.064   | 0.094  | 0.064  | 0.089  | 0.069    | 0.107    |"}, {"review_id": "cuUxKny7rP", "reviewer": "Reviewer_PyHh", "summary": "This paper introduces a novel approach that replaces GNNs for modeling spatial dynamical systems, using modern latent space generative modeling techniques instead. The key innovation is an identifier (ID) system where each entity is assigned a unique ID with learnable embeddings. These IDs are concatenated with entity positions and features, then encoded via cross-attention to a fixed number of latent tokens. A flow-matching model operates in this latent space for temporal dynamics, and a decoder uses ID-based queries to retrieve entity-specific information. The method is evaluated across diverse domains including molecular dynamics (MD17, Tetrapeptides), human motion (pedestrian, basketball), and N-body systems, achieving state-of-the-art or competitive performance while requiring 10-100x fewer function evaluations than baseline methods.", "questions": "Please see the weakness part above,", "limitations": "Please see the weakness part above,", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "======\n\nStrengths:\n\n*Eliminates graph construction overhead: No need to learn to cluster and build the graph, as in GNN; learns to encode directly into fixed-size latent space, avoiding k-NN computations and edge message passing.\n\n*Enables modern architectures: The fixed-size latent representation enables usage of SOTA scalable generative models (flow matching, diffusion models) that have driven recent advances in image/video generation.\n\n*Superior performance: Achieves better performance than GNN-based methods across all tested domains, notably beating equivariant GNNs on MD17 without using domain-specific symmetries.\n\n*Unified architecture: Single model can handle molecules of different sizes (9-21 atoms in MD17), demonstrating better generalization than molecule-specific training.\n\n=====\n\nWeakness\n\n*Hyperparameter sensitivity: As mentioned in the paper, the pool size for potential entities matters. But tweaking this hyperparameter requires trial-and-error or domain expertise. More importantly, is it necessary to retrain the model for different pool sizes? Can models learned with different pool sizes share common knowledge such that we don't have to learn from scratch?\n\n\n*Dynamic entity handling is unclear: How does the method handle changes in entity count during dynamics? What if there are new entities entering the system? How are entity mergers or departures handled?\n\n\n*Tracking dependency: The method relies on perfect temporal tracking being provided by the dataset. How sensitive is the approach to tracking errors? What happens with identity switches or occlusions?\n\n*Semantic analysis missing: The paper needs more discussion on the semantic meaning of assigned entities given N entities and L latent vectors:\nFor N<L (overparameterization), do multiple latents correspond to the same entity or capture different aspects?\nFor N≥L (compression), does this lead to semantically meaningful clustering in latent space?\nI appreciate the brief discussion and results in Appendix G.3 on number of latent vectors, G.4 on pool size effects , though more extensive ablations would indeed clarify this.\n\n*(minor)Incomplete ablations: What drives the performance gains - the ID system, the flow matching model, or simply the larger model capacity? More ablations comparing to fixed-size GNN baselines would strengthen claims.\n\n\nIn summary,\nthis paper presents an innovative approach to modeling spatial dynamical systems by replacing the dominant GNN paradigm with fixed-size latent representations. The core contribution - the ID-based encoding/decoding system - elegantly solves the problem of maintaining entity identity while leveraging modern generative architectures. The experimental results are impressive, showing consistent improvements across diverse domains. So I'm leaning towards the acceptance of the paper.\n\nHaving said that, \n several limitations temper my enthusiasm. The requirement to pre-specify maximum entity counts and retrain for different pool sizes significantly limits the \"foundation model\" potential. The dependence on perfect tracking and inability to handle dynamic entity counts restrict real-world applicability. The paper would benefit from deeper analysis of what the model learns (e.g., embedding structure) and more thorough ablations to isolate the sources of improvement.", "was_revised": true, "final_justification": "The rebuttal has addressed all my concerns. Based on the effectiveness of this method on modeling dynamics, and the overall positive rating of this paper, I would like keep my rating as WA.", "rebuttal": "We appreciate your comprehensive assessment and recognition of our key contributions: eliminating graph construction overhead, leveraging modern scaleable architectures, and achieving strong performance without domain-specific inductive biases. We are grateful for the opportunity to strengthen our work through addressing the concerns raised. Your suggestion for semantic analysis is particularly valuable, leading to new insights about our model's internal representations that we had not previously explored.\nWe also agree, that \"foundational architecture\" may overstate our current scope and we adjusted our wording accordingly. Our intention was to highlight the flexibility of ID-based approaches across different dynamical systems. While limitations like pre-specified entity counts exist, we believe this work contributes a novel paradigm that opens promising research directions.\n\n\n\n### Weaknesses\n\n> Hyperparameter sensitivity: As mentioned in the paper, the pool size for potential entities matters. But tweaking this hyperparameter requires trial-and-error or domain expertise. More importantly, is it necessary to retrain the model for different pool sizes? Can models learned with different pool sizes share common knowledge such that we don't have to learn from scratch?\n\nWe agree pool size represents an important hyperparameter requiring consideration. However, Figure 7 indicates this limitation is less restrictive than initially apparent, where reconstruction errors increase marginally when pool size exceeds natural dataset requirements (MD17's 21 entities). The performance degradation appears to result from computational constraints (reduced embedding update frequency) rather than model capacity limitations, suggesting that extended training or transfer learning strategies could enable effective knowledge sharing across different pool configurations without requiring complete retraining. To summarize, we can safely use too many IDs, it just takes longer to train.\n\n> Dynamic entity handling is unclear: How does the method handle changes in entity count during dynamics? What if there are new entities entering the system? How are entity mergers or departures handled?\n\nWhile the scope of current experiments did not explicitly cover scenarios with changing entity counts or new entities entering the system, we think our architecture is flexible enough to handle these cases.\n\nCurrently, entity attributes (such as atom type in molecular dynamics or team membership in basketball) remain static throughout the dynamics. However, our framework could incorporate such behavior by introducing time-varying properties such as \"active/inactive\" or \"groups assignment\" to the entities. This would enable modeling of:\n\n - Entity entry and exit events\n - Entity mergers and splits\n\nWe acknowledge that this as a valuable direction for future research and added it to the discussion section of our manuscript.\n\n\n> Tracking dependency: The method relies on perfect temporal tracking being provided by the dataset. How sensitive is the approach to tracking errors? What happens with identity switches or occlusions?\n\nWhile we have not explicitly tested our method's sensitivity to tracking errors, we acknowledge this as a crucial consideration for real-world applications. Our current approach assumes perfect tracking, which is reasonable for simulation-based datasets (e.g., molecular dynamics). However, we recognize that real-world tracking systems are prone to sensor errors and occlusions. Future work could address this limitation by introducing synthetic tracking errors during training, such as randomly dropping entities to improve model robustness.\n\n\n> Semantic analysis missing: The paper needs more discussion on the semantic meaning of assigned entities given N entities and L latent vectors: For N<L (overparameterization), do multiple latents correspond to the same entity or capture different aspects? For N≥L (compression), does this lead to semantically meaningful clustering in latent space? I appreciate the brief discussion and results in Appendix G.3 on number of latent vectors, G.4 on pool size effects , though more extensive ablations would indeed clarify this.\n\nFor a better understanding of the semantic structure of the latent space, we conducted additional experiments.\n\n **Experimental Setup:**\nWe analyzed decoder attention weights to understand entity-latent correspondence in both overparameterized (N<L) and compressed (N≥L) regimes. Using two first-stage models (L=16 and L=32) trained on MD17, we examined how they encode aspirin molecules (N=21 atoms) across different conformations. Due to NeurIPS rebuttal constraints, we present our findings textually here but included the visualizations in the revised manuscript.\n\n **Key Findings:**\n - **Consistent attention patterns:** When encoding different aspirin conformations with consistent atom IDs, we observe identical attention patterns per ID, which is distributed over the latent vectors. This suggests that each ID learns a unique \"addressing scheme\" for retrieving its information from the latent space (independent of the stored information).\n - **Adaptive attention head utilization:** The models exhibit different strategies based on compression ratio:\n   - **L=32 (overparameterized):** Leverages only a single attention head, we think the ID based retrieval acts like a hashing table and there is no need for the model to utilize both heads if the latent space is large enough\n   - **L=16 (compressed):** In the compressed setting we saw unique attention patterns in both heads, suggesting the model needs to avoid potential collisions with other IDs. \n\n\n\n> Incomplete ablations: What drives the performance gains - the ID system, the flow matching model, or simply the larger model capacity? More ablations comparing to fixed-size GNN baselines would strengthen claims.\n\n\nOur approach builds on recent progress in latent-space image and video modeling [1,2,3], and we aimed to apply these concepts to dynamical systems with a varying number of entities, a setting currently dominated by GNNs. We would be happy to include specific comparisons to fixed-size GNNs in the manuscript if you have a particular suggestion.\n\n**References**\n\n[1] A. Polyak et al., Movie Gen: A Cast of Media Foundation Models  \n[2] P. Esser et al., Scaling Rectified Flow Transformers for High-Resolution Image Synthesis  \n[3] R. Rombach et al., High-Resolution Image Synthesis with Latent Diffusion Models"}, {"review_id": "EU01kZ5JbQ", "reviewer": "Reviewer_4WJU", "summary": "This paper proposes LaM-SLidE, a novel framework for modeling spatial dynamical systems using latent space generative modeling with traceable individual entities. The method combines encoder–approximator–decoder architecture with identifier-based latent representations, allowing traceability of individual particles even when modeling is performed in a compressed latent space. A core feature is the ability to train and generalize across systems with variable numbers of entities, enabling parameter sharing and representation unification. The authors evaluate the method on a range of tasks, including pedestrian dynamics, basketball plays, molecular dynamics, and N-body systems, showing competitive or superior performance against a broad set of baselines.", "questions": "1. Lines 104–106 introduce $M^t$, but immediately state it is time-independent. Can this symbol be omitted in favor of a single static matrix $M$? Similarly, equations (1) and (2) formalize the Markov property, but this property is not used anywhere in the model. Would the paper be clearer without these?\n\n2. The paper defines the stochastic interpolation path in Eq. (6), with boundary conditions $\\alpha_1 = \\sigma_0 = 0$, $\\alpha_0 = \\sigma_1 = 1$. Shouldn't this be the opposite? Please clarify.\n\n3. The description of the approximator starting at line 180 is difficult to follow. Even after consulting Appendix B.4 and C, the architectural details and training procedure are unclear. Could the authors provide a more complete and explicit description of the model, perhaps with a step-by-step example or pseudocode?\n\n4. How does the model adapt to variable-length conditioning ($T_o$)? The framework appears to handle different observation lengths, but it is not clear how the approximator handles this in training or inference. Is $T_o$ used as an explicit condition, or is the model trained over varying $T_o$ lengths? This point deserves clarification.\n\n5. Why not use a step-wise (Markov) prediction? Given that the dynamics are assumed to be Markovian (as implied by Equations 1 and 2), why is the model trained to predict the entire future trajectory at once, rather than iteratively predicting the next step? A step-wise approach may allow for simpler and more interpretable modeling.\n\n6. For systems of identical particles (e.g., molecules with symmetric atoms), how does LaM-SLidE handle permutation invariance? The current identifier mechanism seems to break symmetry. Could the authors discuss the implications and whether permutation equivariance could be incorporated into this framework?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "Strengths:\n\n1. The paper tackles an important and nontrivial challenge—how to apply latent generative models to physical systems with variable-sized, traceable entities. The proposed approach is both technically novel and practically impactful.\n2. One of the most compelling aspects is the ability to jointly model systems of varying particle counts in a shared latent space, a feature often lacking in prior GNN- or diffusion-based methods.\n3. Experiments are comprehensive and cover diverse domains.\n4. The proposed framework is modular and general enough to be applied across domains.\n\nWeaknesses:\n\n1. Some symbols are defined but immediately undermined, or unused in the actual modeling (e.g., $M^t$ and equations (1–2)), leading to unnecessary conceptual load.\n2. The description of the latent dynamics approximator (especially from line 180 onwards) is vague and scattered. Even after reading Appendix B.4 and C, it remains unclear how the approximator operates in detail.\n3. While the model appears to support different input lengths $T_o$, it is unclear how this is explicitly handled.\n4. The paper does not explain why the modeling is not done step-wise in time if the system is Markovian. Such an autoregressive approach might be simpler and more interpretable.\n5. The paper lacks discussion on how the model handles or violates permutation invariance among identical particles, a property crucial in many physical systems.", "was_revised": true, "final_justification": "Thank authors for their responses and discussions. I believe they have addressed my concerns very well, and I support the acceptance of this paper to the conference.", "rebuttal": "We appreciate that you acknowledged the technical novelty and practical impact of our approach, particularly highlighting our ability to jointly model systems with varying particle counts in a shared latent space, a capability you noted is often missing in prior methods. We also appreciate your recognition of our comprehensive experimental validation and the framework’s general applicability. We hope our responses below address the concerns you raised about our work.\n\n### Weaknesses / Questions\n\n> Lines 104–106 introduce , but immediately state it is time-independent. Can this symbol be omitted in favor of a single static matrix ? Similarly, equations (1) and (2) ...\n\n    \nRegarding $M^t$: While we could simplify to a static matrix M, we chose to retain the time-dependent notation $M^t$ to emphasize the generality of our framework. This notation makes explicit that our approach could naturally extend to systems with time-varying entitiy properties except positions, this aligns with the flexibility discussed in response to question 2 of reviewer PyHh. \n\nRegarding Equations (1) and (2): Our aim was to provide readers, especially those unfamiliar with dynamical systems, with a conceptual introduction to the topic. While we understand the your concern, we respectfully disagree in this particular point with the assertion that the Markov property is entirely absent. As the your correctly noted elsewhere, our approach does not involve single-step trajectory predictions. Instead, we employ a block-wise rollout strategy, where entire trajectory segments are predicted step-by-step. This process can be interpreted as Markovian at the level of each block. Predicting multiple time steps also offers computational advantages, which we discuss below.\n\n\n> The paper defines the stochastic interpolation path in Eq. (6), with boundary conditions , . Shouldn't this be the opposite? Please clarify.\n\nYou are correct, this was a typo. The correct boundary conditions should be $\\alpha_0=\\sigma_1=0$ and $\\alpha_1=\\sigma_0=1$. We corrected it in the revised manuscript.\n\n> The description of the approximator starting at line 180 is difficult to follow. Even after consulting Appendix B.4 and C, the architectural details and training procedure are unclear. Could the authors provide a more complete and explicit description of the model, perhaps with a step-by-step example or pseudocode?\n \nWe apologize for any confusion and have included Python pseudocode below for better clarity. To clarify our contributions: while our latent approximator uses alternating attention blocks across temporal and spatial dimensions, a commonly used approach in latent video modeling, our primary contribution is the flexible fixed-size latent space that allows us to effectively utilize these architectural components.\n\n\n```python\ndef latent_layer(z, t, z_cond):\n    ## Latent layer of the approximator (high level)\n    z = z + embed_cond(z_cond)\n    t_embed = embed_time(t)\n    \n    z = attention_spatial(z, t_embed)     # T x L x Dz, attention is calculated over L    | ref [4,5]\n    z = rearrange(z, \"T L Dz -> L T Dz\")\n    z = attention_temporal(z, t_embed)    # L x T x Dz, attention is calculated over T    | ref [4,5]\n    z = rearrange(z, \"L T Dz -> T L Dz\")\n    \n    return z\n\n\n## Training (Python Pseudocode)\n## Assume a trained encoder/decoder\n## For easier illustration we omit the batch dimension and show the training procedure for a single samplel\n\n##################\nx                                     # T x N x Dn (N=number of entities, D=[x,y,z,atom_type]) ... e.g. Aspirin 21x4\nids                                   # N e.g. Aspirin 21\nmask                                  # T binary mask [1, 0, 0, ....], e.g. one conditioning frame\n##################\n\n# Encoding and conditioning setup\nz1 = encode(x, ids)                   # T x L x Dz Encoder state into latent vector \nz_cond = mask_frames(z1, mask)        # T x L x Dz Contains only the conditioning frames, the other frames are masked out and set to 0.\n\n# SiT path\nz0 = torch.randn_like(z1)             # Sample z0 from a normal distribtion\nt = torch.uniform(0, 1)               # Sample t uniformly\nzt = t * z1 + (1 - t) * z0            # Create interpolation\n\n# Loss\nz1_hat = approximator(zt, t, z_cond)\nloss = torch.mean((z1_hat - z1)**2)\nloss.backward()\n\n\n## Inference \n# We use the identiy from ref [11] Table 1, \n# to transform a data prediction model into a velocity model, and use\n# torchdiffeq (https://github.com/rtqichen/torchdiffeq) for integration\n \n##################\nx_cond                                                   # T x N x Dn, but non-conditioning frames are zero\n##################\n\nvelocity_model = data_to_velocity_model(approximator)    # Transform model trained on data prediction into velocity prediction model.\n\nz_cond = encode(x_cond, ids)                             # T x L x Dz, contains only the conditioning frames/ missing frames are set to 0       \nz0 = torch.randn_like(z1)\nz1_pred = solver(velocity_model, z0, z_cond)             # Solve ODE\n\nx_hat = decoder(z1_pred)\n```\n\nWe've also recognized, that some figures may not render properly on Mac systems, particularly Figure 5 which depicts the overall flow of our approach. These figures display correctly when viewed in Firefox browser on Mac systems.\n\n> How does the model adapt to variable-length conditioning ($T_0$)? The framework appears to handle different observation lengths, but it is not clear how the approximator handles this in training or inference. Is it used as an explicit condition, or is the model trained over varying  lengths? This point deserves clarification.\n\nYou are correct, that we use explicit conditioning with fixed lengths for each task during training and inference. So we handle different observation lengths, but explicitly trained for each task. Our current experiments use the following number of conditioning frames (see also Table 7 in our manuscript):\n\n| Experiment | Conditioning Frames |\n|------------|-------------------|\n| Pedestrian trajectory forecasting (ETH-UCY) | 8 |\n| Basketball (NBA) | 8 |\n| N-Body | 10 |\n| Molecular Dynamics (MD17) | 10 |\n| Molecular Dynamics - Tetrapeptides (4AA) | 1 |\n\nWe acknowledge that variable-length conditioning represents a promising extension. Recent video modeling work demonstrates the benefits of this approach: [7] trains with varying conditioning lengths with different noise levels, while [10] employs random frame masking for video modeling. Futher, [9] showed strong results modeling PDEs, while randomly masking conditioning frames during training. While we have not yet explored variable-length conditioning, we think it could enhance both performance and flexibility.\n\n\n> Why not use a step-wise (Markov) prediction? Given that the dynamics are assumed to be Markovian (as implied by Equations 1 and 2), why is the model trained to predict the entire ...\n\nWhile the underlying dynamics are Markovian (as reflected in Equations 1 and 2), we opt for non-autoregressive, full-trajectory prediction following recent advances in video generation and simulation modeling [6,7]. These works have demonstrated that predicting multiple future steps jointly can lead to improved performance in terms of both accuracy and stability, especially for long-range forecasting tasks. Recently [9] used a mutli-step  prediction approach for PDEs, showing remarkable results.\n\nFrom a practical perspective, full-trajectory prediction also brings substantial computational benefits. For instance, in the Tetrapeptides dataset, we predict 1000 future timesteps in parallel, resulting in a significant speedup compared to autoregressive rollouts, which require sequential evaluation of each step.\n\n> For systems of identical particles (e.g., molecules with symmetric atoms), how does LaM-SLidE handle permutation invariance? The current identifier mechanism seems to break symmetry. Could the authors discuss the implications and whether permutation equivariance could be incorporated into this framework?\n\nIn molecular simulations, each atom is individually tracked throughout the simulation, maintaining consistent identity across the entire trajectory. This means that even for molecules with permutation symmetries, the identity of each atom remains fixed, effectively sidestepping the permutation equivariance/invariance issue. While we acknowledge that permutation equivariance/invariance is an important property in certain contexts [6,8], we have opted to omit it in favor of simplicity and computational efficiency, following also recent trends [1,2,3] not explicitly incorporating these inductive biases into our architecture. We discuss our data augmentation approach in Appendix Section E.3, which partially addresses symmetry considerations. We think, that incorporating permutation equivariance/invariance or other inductive biases remains an interesting direction for future work and added a discussion on it into our manuscript.\n\n**References:**\n\n[1] J. Brehmer et al., Does equivariance matter at scale?  \n[2] K. Joshi et al., All-atom Diffusion Transformers: Unified generative modelling of molecules and materials  \n[3] J. Abramson et al.,  Accurate structure prediction of biomolecular interactions with AlphaFold 3  \n[4] W. Peebles et al., Scalable Diffusion Models with Transformers  \n[5] M. Dehghani et al., Scaling Vision Transformers to 22 Billion Parameters  \n[6] B. Jing et al., Generative Modeling of Molecular Dynamics Trajectories  \n[7] B. Chen et al., Diffusion Forcing: Next-token Prediction Meets Full-Sequence Diffusion   \n[8] D. Knigge et al., Space-Time Continuous PDE Forecasting using Equivariant Neural Fields     \n[9] F. Rozet et al., Lost in Latent Space: An Empirical Study of Latent Diffusion Models for Physics Emulation  \n[10] V. Voleti et al., MCVD: Masked Conditional Video Diffusion for Prediction, Generation, and Interpolation  \n[11] Y. Lipman et al., Flow Matching Guide and Code."}, {"review_id": "i6kqTQR3hO", "reviewer": "Reviewer_8R2U", "summary": "The goal of the paper is to bring the advantages of latent modeling from image generation models into dynamical systems models. One part of the method focuses on modeling the IDs of trajectories so that idintifiable reconstruction is possible after latent space.", "questions": "**Question 1**\n>In order to leverage a latent systems representation, we need to be able to trace individual entities of the system\n\nwhy? can you elaborate further. why needs to differ from image generators? \n> my interpretation: because you are tracing elements, trying to train a generative model in latent space. \n\n**Question 2**\nSince this work is in latent space for dynamical system, how does it compare with methods such as: \"Latent ODEs for Irregularly-Sampled Time Series\" (NeurIPS 2019)? and why that type of work do not require IDs?\n\n**Question 3**\n> aspirin consists of 21 atoms, needs to have at least 21 unique identifiers\n\nwhy? elaborate", "limitations": "yes", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 1, "strengths_and_weaknesses": "**Weaknesses**\n\nW1: GeoTDM comparison. While GeoTDM is basically the most important baseline and all experiments are taken from there, the differences with this method are not discussed in any section.\n\nW2: Missing baselines. At this link https://paperswithcode.com/sota/trajectory-prediction-on-ethucy the first top 10 methods have better performance than the method proposed by this paper for the ETH-UCY dataset. The task is from Table 2 from the submitted paper.\n\nW3: the improvement over GeoTDM is marginal.", "was_revised": true, "final_justification": "The authors clarified that the method requires much less compute compared to GeoTDM. However, in my opinion, they should still discuss, and to what is possible, compare with, the related works that achieve better results. I previously put the link to 10 methods getting better performance.", "rebuttal": "Thank you for reviewing our submission. While your assessment was critical, we appreciate your suggestion to elaborate on the relationship to image generation, specifically why traceability does not appear to be an issue in this context. Specifically, our work presents novel ID representation that are where not applied in dynamical system modeling before, offering an alternative to GNN based approaches and leveraging recent progress in latent space image/ video modeling. We are looking forward to the discussion period and hope we can address open concerns.\n\n\n### Weaknesses\n\n> GeoTDM comparison. While GeoTDM is basically the most important baseline and all experiments are taken from there, the differences with this method are not discussed in any section.\n\n\nWhile GeoTDM serves as an important baseline (in 3/5 experiments), we also want to highlight some conceptual differences between LaM-SLidE and GeoTDM. In particular, LaM-SLidE introduces a novel ID-based entity tracking mechanism which allows tracking entities in a fixed size latent space. To enable this, we relax certain equivariance properties [3] for our models — unlike GeoTDM. This design choice aligns with recent architectural trends, such as AlphaFold 3 [2], where similar relaxations have been argued to contribute to performance improvements. In response to this review we added a more detailed discussion in the results section. We also want to underline an advantage of GeoTDM: as shown in Table 15, LaM-SLidE requires significantly fewer function evaluations.\n\nNumber of function evaluations (NFEs):\n|          | N-Body | MD17 | ETH |\n|----------|--------|------|-----|\n| GeoTDM | 1000   | 1000 | 100 |\n| LAM-SLIDE    | **10**     | **10**   |**10**  |\n\n> Missing baselines. At this link ... the first top 10 methods have better performance than the method proposed by this paper for the ETH-UCY dataset. The task is from Table 2 from the submitted paper.\n\nWe appreciate your comment regarding the performance of recent methods on the ETH-UCY dataset. It is indeed true that several methods listed on the leaderboard achieve higher accuracy. However, we would like to mention in order to have a valuable baseline we relied on the evaluation pipeline introduced in GeoTDM, which could be different from the one used on this leaderboard. Further many of these approaches are highly specialized for human trajectory prediction, incorporating domain-specific inductive biases and handcrafted features. For example, the current top-performing method on the ETH-UCY dataset [1] models repulsion and attraction forces derived from the environment, using semantic segmentation to distinguish walkable areas, learnable goal positions, and other features specifically designed for human motion forecasting. We want to note that, in general, our model could also process additional node features, but we aimed for a proof-of-concept using the simplest version of the dataset to validate whether the ID-based representations can be leveraged across different domains.\n\n\n> The improvement over GeoTDM is marginal\n\n**Performance Comparison with GeoTDM**\nLaM-SLidE demonstrates clear performance advantages across multiple benchmarks. On the MD17 benchmark, LaM-SLidE achieves superior results compared to GeoTDM (Table 4). Similarly, in N-body system dynamics tasks, LaM-SLidE consistently outperforms GeoTDM, with the exception of scenarios where all methods achieve comparably low error rates.\nThe ETH-UCY dataset presents more nuanced results, with less definitive performance differences between GeoTDM and LaM-SLidE. Given that pedestrian forecasting is inherently subject to behavioral stochasticity, the observed marginal performance variations may be attributable to random factors rather than fundamental algorithmic differences. Notably, LaM-SLidE offers significant computational speedup compared to GeoTDM.\n\n**Comparison with MDGen**\nWe also encourage you to consider our comparison against MDGen shown below (Table 5). MDGen represents a highly specialized approach and an equivariant architecture operating in torsion space specifically designed for peptide molecules. Despite MDGen's domain-specific optimizations and restricted applicability to other domains, our general-purpose method achieves comparable performance, demonstrating the effectiveness of our approach across diverse domains.\n\n|           | Torsions |      |     | TICA |        | MSM  | Time   |\n|-----------|----------|------|-----|------|--------|------|--------|\n|           | BB       | SC   | All | 0    | 0.1 joint |      |        |\n| 100ns     | .103     | .055 | .076| .201 | .268   | .208 | ~ 3h   |\n| MDGen     | .130     | **.093** | **.109**| .230 | .316   | .235 | ~ 60s  |\n| LAM-SLIDE | **.128**     | .122 | .125| **.227** | **.315**   | **.224** | ~ 53s  |\n\nLaM-SLidE achieves competitive performance across most metrics while maintaining competitive computational efficiency to MDGen (53s vs 60s). This also highlights the strong potential of deep learning–based surrogate models in the simulation domain.\n\n> In order to leverage a latent systems representation, we need to be able to trace individual entities of the system. Why? can you elaborate further. why needs to differ from image generators? my interpretation: because you are tracing elements, trying to train a generative model in latent space.\n\nImage generators compute pixel values for each location within a structured grid, where every pixel is inherently tied to a fixed coordinate. In such representations, pixels act as entities whose positions are implicitly defined by their location in the grid. The pixel values are typically derived from a latent image representation, and the underlying neural network architecture dictates how each specific grid position (i.e., pixel value) is computed from the latent space. The relationship between latent features and pixels is fixed by design.\n\nIn contrast, the dynamical systems considered here, represent states using a set of discrete entities that are not bound to static spatial positions, and the number of entities is not fixed. These entities are free to move, and their positions can evolve over time. Without identifiers, the system would compress all entities into a shared latent representation that no longer has links to individual entities. Because of this loss of direct association, unique identifiers are required to maintain reference to each entity.\nThis identity-based, set-oriented approach is especially effective when entities are sparsely distributed across space. In such cases, tracking entities by their identifiers is significantly more efficient than scanning a dense grid volume to detect their presence.\n\n\n**References**\n\n[1] J. Yue et al., Human Trajectory Prediction via Neural Social Physics.  \n[2] J. Abramson et al.,  Accurate structure prediction of biomolecular interactions with AlphaFold 3.  \n[3] J. Brehmer et al., Does equivariance matter at scale?"}]

Author rebuttal:

Meta-review: This submission proposes LaM-SLidE, a novel framework for modeling spatial dynamical systems using latent generative modeling with identifier (ID)-based entity traceability. The core contribution is an encoder–latent approximator–decoder architecture that incorporates learnable IDs to ensure entity traceability while leveraging fixed-size latent representations. This allows modern scalable generative models (flow matching, diffusion) to be applied to systems with variable numbers of entities. The approach is validated across multiple domains (molecular dynamics, N-body systems, pedestrian trajectories, basketball plays), showing competitive or superior results compared to baselines such as GeoTDM and specialized GNN methods, while requiring substantially fewer function evaluations.

Overall, this is a technically solid and novel paper that makes a meaningful contribution to generative modeling of dynamical systems. While the ETH-UCY results remain less competitive than specialized trajectory forecasting methods, the strength of this work lies in its generalizable architecture that unifies modeling across diverse domains while significantly improving efficiency. The authors have been highly responsive in addressing concerns, adding ablations, clarifying methodology, and expanding the discussion of limitations and related work.

The main weaknesses (clarity, limited ablations, and missing baseline discussion) have been sufficiently mitigated through the rebuttal and revisions. Concerns about scope (tracking, dynamic entities, permutation invariance) are valid but appropriately acknowledged as future directions rather than fatal flaws.
Decision: Accept (poster)