Spectral-guided Physical Dynamics Distillation
Authors: Youjin Kim
Dagyeong Na
Jae Yong Lee
Junseok Kwon
Venue: ICLR 2026 Poster (2026)
Keywords: 3D Physical dynamics
Knowledge distillation
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: SGDD is a knowledge-distillation framework with future trajectories as privileged information and adaptive spectral weighting in a unified spatio-temporal space to enable accurate long-horizon trajectory prediction from initial states.

Abstract: The problem of physical dynamics, which involves predicting the 3D trajectories of particles, is a fundamental task with wide-ranging applications across science and engineering. However, accurately forecasting long-horizon trajectories from initial states remains challenging, due to complex particle interactions and entangled multi-scale dynamics involving both low- and high-frequency components. 
To address this, we propose a novel knowledge-distillation-based framework, SGDD (Spectral-Guided Dynamics Distillation), which integrates a spectral-guided enhancement to adaptively prioritize key frequency components within a unified spatio-temporal representation. Through knowledge distillation, SGDD leverages future trajectories as privileged information during training, guiding a teacher encoder to generate comprehensive dynamics representations while a student encoder approximates them using only the initial state. This enables the student to generate effective dynamics representations at inference, even without privileged information, thereby enabling accurate long-horizon trajectory prediction.
Experimental results on molecule, protein, and human motion datasets demonstrate that our method achieves more accurate and stable long-term predictions than previous physical dynamics models, successfully capturing the complex spatio-temporal structures of real-world systems.

Published as a conference paper at ICLR 2026
SPECTRAL-GUIDED PHYSICAL DYNAMICS DISTILLATION
Youjin Kim, Dagyeong Na, Jae yong Lee Junseok Kwon Chung-Ang University
Scoul 06974, Republic of Korea {ju9073, dakungl ovve, jaeyong, jskwon [akwonnecann eeau.eaauaeeeeeeee ac.k.
ABSTRACT
The problem of physical dynamies, which involves predicting the 3D trajecto- ries of particles, is fundamental task with wide-ranging applications across science and engineering. However, accurately forecasting long-horizon trajec- tories from initial states remains challenging, due to complex particle interac- tions and entangled multi-scale dynamics involving both low- and high-frey components. To address this, we propose novel chowledge-distllaatinn-- framework, SGDD (Spectral-Guue -uiided Dynamics Distillation), which integrates a spectral-guided enhancement to adaptively prioritize key frequency components within a unified spatio-temporal representation. Through knowledge distillation, SGDD leverages future trajectories privileged information during training, guiding a teacher encoder to generate comprehensive dynamics representations while a student encoder approximates them using only the initial state. This en- ables the student to generate effective dynamics representations at inference, even without privileged information, thereby enabling accurate ong-horizon trajectory prediction. Experimental results molecule, protein, and human motion datasets demonstrate that our method achieves more accurate and stable long-term predic- tions than previous physical dynamics models, successfully capturing the complex spatio-temporal structures of real-world systems.
INTRODUCTION
Physical dynamics refers to the problem of predicting and simulating the 3D trajectories of particles across sys- tems at various scales, such as molecules, proteins, and human joints. This problem is fundamental in a wide range of scientific and engineering applications, including drug design (Reddy et al. (007), protein engineering (Al- Lazikani et al.) (201), and robotics (Spong et al.) (2066). In recent years, has attracted substantial attention, with numerous studies proposing equivariant neural architec- tures to better capture the underlying symmetries of phys- ical systems (Satorras al. 2021 Wu al. 2023a Du
Baseline
Ours
GT Pred
et al. 2022 Xu al. 2024 Fuchs et al. 2020 Sun al.
Figure 1: wweFFruuenc--Dominnnnt Dynamies. Left: Baseline, Right: Ours.
2024
Despite recent progress, accurately forecasting long- horizon trajectories from initial states remains highly challenging (Lippe et al. 2023). This is be- cause long-horizon prediction amplifies the entanglement of global low-frequency trends and lo- calized high-frequency oscillations. Such entanglement poses significant difficulties for long-term forecasting due to two major contributors: (1) the low- and high-frequency components interplay in complex ways across space and time, and (2) the importance of these components varies across different systems.
"Corresponding authors.

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Published as a conference paper at ICLR 2026
Considering the first factor, while recent studies often incorporate frequency-awand techniques (Xu et al. 2024) (Sun et al. 2024), they model temporal and spatial structures separately. That is, they rely on spectral representations derived from either temporal spatial domains in isolation, with limited consideration of their integrated interaction. Consequently, frequency modeling based on a single dimension often struggles to fully capture the underlying physical processes that emerge from the interdependent dynamics of space and time.
For the second contributor, not all frequency components are equally important in long-term pre- diction. Specifically, low-frequency components capture stable, global patterns, whereas high- frequency components may contribute to instability and noise. Therefore, accurate prediction re- quires prioritizing low-frequency modes to ensure stability and long-term coherence, while comple- mentarily incorporating high-frequency details to enhance short-term precision.
These challenges are exemplified by Figure I where the baseline method struggles to track low- frequency dominant patterns. This underscores the need for frequency-award spatio-temporal mod- eling that jointly derives spectral representations from unified spatio-temporal domain and adap- tively emphasizes task-relevant frequency components. To effectively capture the spatio-temporal patterns and key frequency components embedded within entangled dynamics, leveraging trajec- tory data as privileged information through a knowledge distillation approach can provide direct and efficient guidance.
Building on these, we introduce SGDD (Spectral- Briicdd Dynamics Distillation), a novel sewwledg-ddisillltiio
dynamics representation learning framework. SGDD leverages future trajectories, which are used only during training as privileged information, to guide the learning of frequency-aware, spatio-temporal dynamics representation. In SGDD, a teacher encoder processes trajectories and a student encoder relies only on the initial state, with both producing a spatio- temporal dynamics representation. These representations are then refined through a spectral- guided enhancement module that adaptively emphasizes frequency components most relevant to the target trajectory via learnable weights. Through distillation, the student's ss enhanced dynamics representa- tion is aligned with the teacher's. s. This distilled representation serves as an inductive bias for the decoder, enabling accurate long-horizon predictions inference time, even in the absence of privi- leged information.
Our main contributions are summarized as follows:
We propose SGDD (Spectral-Guided Dynamics Distillation), novel knowledge distillation framework that extracts rich dynamics representations from privileged future trajectories in the spatio-temporal and spectral domains, and learns to approximate them using only the initial state.
We introduce a spectral-guided enhancement module that refines the dynamics representations by emphasizing key frequency components through learnable weights, thereby providing the de- coder with optimized inputs for accurate trajectory prediction.
We show the effectiveness SGDD on diverse multi-scale particle datasets (MD17, protein, and human motion), where it consistently outperforms strong baselines in trajectory prediction.
2 RELATED WORKS
Equivariant neural architectures have become essential tools for modeling physical dynamics. EGNN (Satorras et al., 202T) introduced an efficient B(n)-equivariant message passing scheme that jointly updates node features and coordinates. ClofNet (Du et al., 2022) ex- tended this approach by constructing complete local frames better capture higher-order geometric relations. Attention-base point clouds and graphs, while Radial Field (Köhler et al. 2019) developed equivariant normalizing flows to enable Boltzmann Generators for symmeetr--preserving sampling.
Physical Dynamics.
(Fuchs et al. 2020) ensured SE(3)--quivariance in
Recent efforts extend beyond spatial equivariance to explicitly address temporal evolution. ESTAG (Wu et al. 2023a) employed an Equivariant DFT together with spatio-temporal modules to cap- ture periodic and Mon-Markvvian behaviors. EGNO (Xu et al., 2024) formulated an Equivariant Graph Neural Operator that directly models trajectories via Fourier- temporal convolutions. GF-NODE (Sun et al. (202)) integrated Graph Fourier decomposition with Neural ODEs to couple local high-frequency and global low-frequency dynamics.
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In contrast to prior studies that emphasize either spatial equivariance temporal modeling, work advances them by directly modeling dynamics in the spatio-tempont domain and learning frequeney-aware representations that capture long-range structures through privileged supervision.
Knowledge Distillation using Privileged Knowledge. Knowledge distillation (KD) was initially introduced for model compression (Hinton et al.. 2015). where teacher network guides smaller student through soft targets and intermediate features (Tung & Mori] 2019 Shen et al. 2211 Cho & Hariharan 2019 Yang al. 2019). Privileged Knowledge (PK), following the Learning Using Privileged Information paradigm (Vapnik & Vashist 2009). refers to auxiliary signals available only during training, but inaccessible test time.
Recent studies have extended KD with PK. In human motion prediction, Sun et al. 2022 distilled future poses as PK through a two-step network, enabling the student exploit privileged supervision while relying solely on observed sequences. In Garming-to-rank, Yang et al. (2022) formalized priv- ileged feature distillation, where teacher model trained with both regular and privileged features transfers knowledge to a student restricted to regular features. Empirical and theoretical analyses demonstrate that KD with PK not only compresses models and improves generalization but also reveals the non-monotonic impact of highly predictive privileged features.
Graph Knowledge Distillation. Beyond PK, KD has also been actively studied in graph domains. Although graph neural networks (GNNs) excelled in representation learning, their message passing nature introduced scalability and latency challenges. Graph Knowledge Distillation (Graph KD) addressed this by distilling knowledge from large GNNs to smaller GNNs or lightweight MLPs.
Early GNN-to-GNN distillation methods (Zhang et al. 2020 Yan et al. 2020) reduced parameters but remained constrained by neighborhood-feteee fetching overhead. In contrast, GNN-to-MLD distillation removed explicit message passing, with Ziang et al. (2021) transferring node-level outputs to vanilla without explicit edges. More recently, Wu et al. (2023b) decomposed teacher knowledge into low- and high-frequency components and in- jected both into the student MLP, mitigating information-drownine problem and producing distilled MLPs that are both efficient and competitive with GNN teachers.
MLPs and Wu et al. (2023c) introducing
Together, these two lines of research KD with privileged knowledge and Graph KD highlight the versatility of distillation in leveraging auxiliary supervision or structural priors. Building on these insights, we develop a tmysics-inspiied framework that adapts graph- based distillation principles to effectively capture physical dynamics.
3 PROPOSED METHOD
3.1 TASK DEFINITION AND FRAMEWORK OVERVIEW
Problem Setting. We consider the task of multi-step trajectory prediction, which involves forecast- ing the 3D positions of particles over future time steps. At each time step t, the system state is rep- resented as a graph G. = (V, E, Zz, h). where V denotes the set of N particles (nodes). E represents physical connections between particles, Ze is tensor containing the 3D position x, and velocity Vr, and h encodes node features describing intrinsic physical properties. Given only the initial state Go = (V, E, Zo, h), the objective predict the sequence of future 3D positions {x1, X XT). ). Our framework performs this prediction by employing a physical dynamics model as the decoder:
X2,......... } Decoder(Go: 2), z = Encoder(Gu).
(I)
Here, denotes node-level dynamics representation, produced by the encoder from the initial state Go. summarizing the anticipated evolution of physical dynamics. The encoder and decoder are trained jointly in an end-to-end manner, ensuring that the learned representation captures the spatio-temporal patterns essential for accurate trajectory prediction.
Overview of Proposed Framework. The core idea of our framework is to construct dynamics rep- resentations that assist the decoder in accurately predicting future trajectories. As illustrated in Fig- ure two encoders are employed: the dynamics cs encoder Edyn. which extracts representations Zdya from the privileged future state sequence G1.7 and initial state Go. and the initial encoder Em which generates Zim solely from G.- Both representations are further refined through a spectral- guided enhancement module that leverages spatio-temporal graph basis to decompose them in
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9o
Einit
Vinit
Guided Enhaneement
Pinit
Decoder
&1:7
Teacher forcing
9o
Spectnl
Eayn
Zyyn
Guided
ddyn
Enhancement
91.T
Training only
Figure 2: Overall Framework SGDD.
the spectral domain and adaptively reweight components most relevant prediction. The resulting *9 is combined with the initial graph Go and passed to the physical dynamics decoder to forecast future trajectories. During training, knowledge distillation is employed that =0 learns mimic =dym thereby capturing rich dynamics representations derived from privileged information. At inference, the decoder effectively predicts future trajectories guided by = The entire framework trained end-to-end with a staged learning strategy, which ensures stable optimization and effective knowledge distillation.
spectral-guided representation (z or =dyn)
3.2 DYNAMICS REPRESENTATION EXTRACTION
To obtain informative dynamics representations, we construct apatio-temporal graph from the state sequence, which serves as the structural basis for both the dynamics encoder (Eaya), used only during training, and the initial encoder (Eini), used at inference. Both encoders operate on this graph to produce spatio-temporal representations of particle dynamics.
Spatio-temporal Graph. To effectively encode the spatio-temporal information of the state quence ( Gr}. shares the same node set and edge set, enabling the formation of a unified graph by linking states along the temporal axis. The resulting 9s: combines two independent graphs: a spatial graph Espatio capturing physical connectivity among particles, and temporal graph Stemp, encoding sequential dependencies across time steps. This spatio-tempora graph G.t serves as the foundation for repre- senting and manipulating dynamics in the spectral domain. For additional details the construction of the spatio-temporal graph, please refer to Appendix A.2
we construct a spatiollemporal graph Gac. Each state graph G. = (V, E, Z, h)
Dynamics Encoder (Faya). The dynami encoder Fdyn operates the spatio-temporal graph Gst, taking as input the ground-truth state sequence (G..... Gr} as well as Go- It pro- cesses the spatio-temporal signals extract dynamics representation Zdyn that captures both low- frequency components encoding long- mnn-termm trends and taneous variations:
components reflecting instan-
Zyyn Edyn (191. Gr).
Zdyn E NxTxx
(2)
where N denotes the number of nodes, the number of time steps, and d, the dimension of the dynamic representation, respectively.
Initial Encoder (Eiit). The initial encoder Emi derives a dynamics representation 2mi from the initial state Go. which contains only spatial edges. To embed Zintt in the same spatio- temporal space where Zayn defined, we construct an artificial spatio-temporal input by project- ing the initial node features through a fully connected layer and expanding them from Nxd to gg. xx This enables the initial encoder to produce representations aligned with the spatio- temporal structure of the dynamics encoder's output. The detailed formulation of the initial encoder is provided in Appendix C33.2
Finit Go)) Finit E NxTxd
3.3 SPATIO-TEMPORAL JOINT BASIS
We define a spatio-temporal joint basis to transform representations into the spectral domain. Let U. E NNN and U, E RxT be the eigenvector matrices of the normalized Laplacians I., and T.
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for the spatial and temporal graphs, respectively. Sepeeciilly.SSeccffcllllly L, U,A,U3 and L, U,A,T, where A, diag(Ae.1 Ag.2,..., Ne,N) ) and A, = diag(AA,1, 11., , are diagonal matrices containing the cigenvalues of the respective Laplacians, ordered in ascending order. These eigen- values represent the frequencies in the spectral domain, with smaller values corresponding to low- frequency (smooth) components and larger values to high-frequency (oseillatory) components. The spatio-temporal joint basis is then constructed via the Kronecker product:
= U. & Us. B E RRNTNT
(4)
The resulting basis enables projection of spatio-temporal representations into the spectral domain, disentangling complex spatial and temporal frequency components along orthogonal dimensions. Since using full set of N7 basis vectors is computationally expensive, reduce complexity by selecting the top K modes to form a truncated basis Bx E RNTXK This is achieved by retaining only the columns of B corresponding to the K smallest eigenvalues, defined as:
(5) where }b:} are the basis vectors associated with the lowest K eigenvalues. This truncation sup- presses high-variance, high-frequency content by excluding basis vectors tied larger eigenvalues. The truncated basis 3x is then employed to project dynamics representations into the spectral do- main and reconstruct them back into the spatio-temporal domain, serving an essential foundation for the -eeetral-guided enhancement module.
Bк [bi, b2. bx]: b, E R
3.4 SPECTRAL--GUDDE ENHANCEMENT
Both the dynamics encoder Edyn and the initial encoder Emi produce node-level dy- namics representations Adyn and 2mit which are then refined by the spectral-guided enhancement mod- ule to adaptively reweight frequency components as shown in Figure
53 Zyyn
5g Xinit
Spatio- lemporal
ēdyn
Anni
Before describing the enhancement process, we first note how representation into E Rd. (NT)) can be expressed in terms of the spatio-temporal joint basis Bx RNTTK Since BK is orthogonal (Appendix BB, the matrix P Bx Bx is the orthogonal projectionn onto span (Bx). Thus, any vector can be decomposed as:
B
B
R (reshaped
Adyn
Öjnit
Spectral
learnable weight
Odyn
Anpit
B
B
Spatio-
temporal
Zdyn
Zinit
= Pz + P)z, where P B B (6)
Here, Pz denotes the projection within the subspace spanned by Bx. while (1 P)= represents the resid- ual component that is orthogonal to (Appendix B). This decomposition allows Ps to capture the selected spectral modes, while P) preserves residual information outside the truncated sub- space. Having established this decomposition, we now describe the enhancement procedure. We first compute the spectral coefficients by projecting 2 onto the basis:
Figure 3: Spectral-Guidee Enhancement.
= Bx= = E _d:
These coefficients are then modulated by learnable, requency-speciffe weights w E R and pro- jected back into the spatio-temporal domain:
à a, = Bx Bx (w O BR²). Finally, the residual component is added to reconstruct representation:
(8)
2 = + P)z.
(9)
The resulting ~ integrates the dominant spectral components with residual information, yielding a richer and frequency-awen representation. This formulation provides direct and flexible control in the spectral domain, allowing the model to adaptively emphasize task-relevant frequency bands for improved prediction accuracy.
The outputs of both encoders are processed through this module, producing spectral-guided dynam- ies representations Zdyn Zini R xxxx,, In addition, the corresponding spectral coefficients prior to reconstruction are denoted Ödyn; amit which are later used for alignment in the spectral domain.
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3.5 DISTILLATION AND TRAINING STRATEGY
The core objective of our framework is to distill privileged dynamics information zdyn. which is obtained from the future state sequence (9I..... Gr... This information is transferred into Pinit which is derived solely from the initial state G.. This ensures that 2nit preserves dynamics-relevant information similar Zdyn+ even during inference.
To this end, we enforce dual-level alignment: in the spatio-temporal domain, between *g and iiiit and in the spectral domain, between their corresponding coefficients ödyn and Ginit This dual align- ment encourages the student representation to capture both oow-freuuency components that encode stable global trends and high-frequency components that reflect fine-scale variations, thereby im- proving generalization beyond naive position-level imitation.
The training process adopts staged learning strategy ensure stable convergence. In the initial pretraining phase, the teacher forcing ratio is set to 1.0 so that the decoder exclusively receives _gg as input. The total loss is defined as
ddyn
Lioral = (x + A
(10)
with the alignment term given by
Calign
Crep
18 Adyn
2g Pinit)
+ Lspoo (ūdyn (init).
(11)
Here, Lpred is the mean squared error (MSE) between the predicted and ground-truth trajectories. The alignment loss Latign consists of representation-level MSE (Crep) and spectral-level MSE (Cepes). Calign is weighted by the hyperparameter X. To prevent the teacher encoder Edyn from being influ- enced or distorted by the student encoder during alignment, gradients are detached from 28 and ddy
Ädyn when computing Latign
In the joint training phase, the teacher forcing ratio is set to 0.5, alternating the decoder inputs
_g between Sdy and 38 "mit" In this phase, the initial encoder Fini is optimized not only through alignment loss but also via direct supervision from the trajectory prediction loss. The overall loss
remains the same as in the pretraining stage.
EXPERIMENTS
We conducted experiments on molecular dynamics, human motion, and protein datasets to evaluate our SGDD framework. The results show that our method can effectively predict trajectories across diverse systems with different particle scales. addition, ablation studies confirm that the proposed framework is well-aligned and that individual components make complementary contributions.
Evaluation Metrics. Following (Xu et al. 2024), we evaluated performance using two metrics. State-to-State (S2S) evaluates only the final state at the last time step. The mean squared error (MSE) loss is computed as MSEs2s = [|x(tz) - x )(? where x(tr) denotes the predicted state and x(tr) is the ground-truth state at the final timestep tr. State-to-Ti Trajectory (S2T) evaluates the entire
trajectory by averaging the errors over all T discrete time steps. The loss is defined as MSE = E [|8(ta) - x(2+)112.
Baseline. For the state-to-state (S2S) evaluation, we used the following baselines: SEE()-
Transformer (Fuchs et al. 2020), Tensor Field Networks (Thomas et al., 2018). Message Passing Neural Network (MPNN) Gilmer et al.)
2017). Radial Field (RF) (Köhler al. 2019). EGNN
Satorras et al. 2021), EGNO (Xu et al. 2024), and GENODE (Sun et al. 2024 Our proposed
SGDD framework was instantiated with different decoder modules, specifically EGNO and GFN- ODE, resulting in two variants: SGDD-EGNO and SGDD-GFNODE. For the state-to-trajectory (S2T) evaluation, we compared against EGNN, EGNO, GFNODE, as well as additional temporal models including NDCN (Zang & Wang 2020 ITO (Diez al. 2024), and LG-ODE (Huang al. 2020
Implementation Details Our framework is implemented in PyTorch, and all experiments ducted on an NVIDIA A6000 GPU with CUDA 11.6. As the dynamics cs encoder, we employ STSGNN (Chen et al. 2025), which takes the spatio-temporal graph input, while the init iil

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Table 1: MSE (x10-2) on MD17 dataset. Upper part: State-to-State (S2S). Lower part: State-to- Trajectory (S2T). The best performance highlighted in bold, the second best is underlined, and performance gains (%) over baselines are shown beneath our SGDD variants.
$28
Aspirin
Benzene
Ethanol
Malonaldehyde
Naphihalene
Salicylic
Toluene
Uracil
RF
10.94:10:01
103. 72 1.29
4.641:0.01
13.031003
0.501:0:01
1.234:001
10.03:1000
0.64:0001
TPN
12.57410.18
58 48
98
4:0.4:0.04
13.02:10008
0.491:001
1.03::00
10.8911004
0.84:0.02
SB:3
11.124-0.06
68 +0067
4.744:0.15
13.89-40022
0.524-0.11
1.1344.2
10.88-0006
0.79-0.02
BONN
4-.41
02.40+-053
4.644-0.01
13.644001
0.474-0.22
1.0240022
11.784007
0.64-0001
EGNN.R R
1:.51
62.612077 20075
4.9200221
17.254005
0.824002
1.35+000
1 11:59:0004
1
EGNN.S S
9.50++0.0
66 66.45200.9
4.63--0.01
12.88000
0.452-001
1.004002
10.7840005
0.60-0.01
EGNO
9.18+0066
48 48.55-00555
4.62+-0.01
12.80+0022
0.374-001
0.86+40022
10.214:005
0.52-0-02
GFNODE
7,9320000
4.82::0000
3.92::0.00
12.87:0000
0.37:10000
0.80:10000
4.82:10.00
0.54:0.00
SGDD-EGNO
7.84::000
(412.5%)
12.97:00.00
(473.4%)
4.042:0.00
(-12.5%)
12:98:10:00
(1.4%)
0.36::0:00
142.75 78)
0.85:0000
(11115)
9.45#0011
(-74%)
0.53:00.00
(1.9%
SGDD-GFNODE
7,29
2.744:0.00
3.64--0.00
12.72+000
0,33-+0.00
0,794-0000
5.16++0.00
0.33-0..00
(+8.1%)
(+13.25%)
(+2.1%%)
(-129%)
(+10,816)
(+1,3%)
(7.1%)
(+1.9%)
S2T
NDCN
31.73++00400
56.21
+00 50
10.74=0:02
46.55.+-0228
2.25++0001
3.58+0.11
13.92-40.02
2. 2.38-00:00
ITO
20.56110.03
31.3 0.5.8
8 6.6020.227
28.44.10.33
18221:0.17
2.4810.34
12.47.1030
[ 1.3520.12
LG-ODE
19.36:2:011 12
51.92 1:32
7.08::0.01
24.41 1:000
1.73:1.000
3.822:0044
11. 18.10:0004
2.1 20002
EGNN
9.24::0007
57.85 4-
4.63-4-0.00
12.81 1:0.01
0.38:40001
0,85 4-05
10.4144004
0.56-00022
EGNN-R
12.07:4:0.11
23.75 0-0.0
3.44
13.38-4-00
0.63-1-0.01
1.15+-0.0
5.04-10002
0.89-0.01
9.49+0012
29
1.29=0.01
11:21-001
0.41-:0.01
1.36+0mm
4.85-2004
0.6-0:01
EGNO
7.37-0077
1.28-00:02
10.67-4001
0.32-2:001
0.7740001
4.58:2003
0.4=0001
GFNODE
6.02:::009
1.110007
2.74::0.01
9.43:10.02
0.24:1:002
0.63:10006
1.80:10.03
0.41:0002
SGDD-EGNO
6.20-10011
(+158%)
7,794-0.1
(46529)
2.88:0.01
(-12:19)
11.01:0005
63 (3.19))
0.33+:0:00
(-3,1%)
0.69::0:00
(+1033))
4.23:10.09
(-7.6%)
0.50=00.00
(-.7%)
SGDD-GFNODE
5.63++0001
1.36+:0011
2.674-0.01
10.95+0004
0.26+:0.00
0.60+0000
2. 2.39-40.03
0.44-0.00
(47.2%%
(+9,9%)
(+2.5%))
(-16.19)
(3.3%%
(+4.7%)
(1288%%)
(4.8%)
encoder is implemented using GAT (Veličković et al. 2017), which takes the initial state graph as input. The training procedure follows a two-stage strategy: pretraining is performed for approxi- mately one-third of the total epochs, after which joint training is applied for the remaining epochs. Teacher forcing ratios are fixed at 1.0 during pretraining and 0.5 during joint training. The align- ment loss weight À is set to 1.0 across all experiments. All models are optimized using the Adam optimizer. Dataset-specific settings (e.g., batch size, learning rate, weight decay, and model config- urations) and source code are included in Appendix and C.4
4.1 MOLECULAR DYNAMICS
Dataset. We used MD17 dataset (Chmiela et al., 2017). which provides molecular dynamics trajec- tories obtained from density functional theory (DFT) simulations. It contains eight small molecules of varying size. Following the same setting in (Xu et al. 2024), used 500/2000/2000 ran- dom sub-trajectories from the full trajectory of each molecule for training, validation, and testing, respectively. The prediction horizon consists of 8 uniformly spaced timesteps, with the final step cor- responding to 3000 frames. The number of atoms corresponding to nodes varies across molecules, typically around 10, and detailed statistics are provided in Appendix C.I
Result(Table [ and Figure 4). Our SGDD, instantiated as SGDD-EGNO and SGDD- GFNODE, achieved state-of-the- oh-the-art perfor- mance across all molecules in the S2S evalu- ation, demonstrating its effectiveness in long- horizon trajectory prediction. Notably, for Benzene, SGDD-EGNO shows a 72% per- formance improvement over EGNO, while exhibits a 54% improvement over GFNode. This case is significant because both EGNO and GFNODE exhibit high rors concentrated in the low-frequency range. In contrast, our SGDD framework learns dynamics representations that capture low-frequency mo- tion more effectively, thereby providing the decoder with a strong inductive bias and substantially enhancing prediction accuracy as illustrated in Figure 4 In the S2T evaluation, SGDD achieves state-of-thh--an results on four molecules (Aspirin, Benzene, Ethanol, Salicylic). For the remaining molecules, the cases where performance falls behind prior models can be attributed to the fact that our framework employs them decoders, while we cannot fully reproduce their reported results. For a direct comparison with our own implementations of baselines (EGNO, GFNODE), please refer to Appendix D
Freq-wise Loss (Benzene)
Freq-wise Loss (Benzene)
zONO
ahNide officde
0.34
23-
0.22
2.0
0.36
14-
0.51
05:
0.27
0.50
10
20
30
40
10
20
30
Frequency
Frequency
Figure 4: Frequency-wise MSE loss for Benzene.
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BONO
GFNode
r=
t=5
SGDD- EGNO
SGDD-
GFNode
(Ours)
(Ours)
GT Pred
Figure 5: Motion Capture(Run) Visualization.
4.2 MOTION CAPTURE
Dataset. We utilized the CMU Motion Capture dataset
(Carnegie Mellon University 2003), which consists of 3D skeletal sequences with 31 joints corresponding to nodes. Following the protocol (Xu et al., (0024), adopted two representative motion types: walk- ing and running. For the walk and run cases, we con- structed 200/600/600 and 200/240/240 uub-trajectories for training, validation, and testing. The prediction
Table MSE (x 2) on motion capture
dataset. Upper part: S2S. Lower part: S2T.
828
Walk
Run
MPNN
521.31133
THN
32.0211
56.6=1.7
SEG3--:
21+2211
61.242.3
EGAN
28. 28.7116
50.91099
horizon consists of 5 uniformly spaced timesteps, with
EGNN
90.7-124
816.7:1277
EGNN-S
26.4-:1.5
54.241199
the final step corresponding to 30 frames.
EGNO
8.121166
13.9--177
GENODE
9.3.1..00
44.910.1
Result (Table 2 Figure 5). In the S2S evalua- tion, our framework achieved state-of-the-art perfor- mance both walking and running motions. For these two cases, SGDD-EGNO improved upon EGNO by 17.3% and 16.8% respectively, while SGDD- GFNODE achieved nearly 30% gains in both cases. In the S2T evaluation, our framework also established new state-of-the-art results across all cases. Specifi- cally, SGDD-EGNO yields 8.6% and 9.4% improve- ments over EGNO for walking and running. respec- tively, while (GDD-GFNODE improves upon GENODE by 24.4% and 23.7%. The larger improve- ments in the S2T setting suggest that framework enables more reliable loni-norizon representa- tions, leading to stable predictions human motion dynamics. This further visually confirmed, where the gap between method and baselines becomes increasingly evident the prediction horizon extends.
SGDD-EGNO
28.210.00
SGDD-GFNODE
6.5=0..0
31.5+0.1
(820.8%)
S2T
EGNN-R
$2.0:116
2
EGNN-S
14.3-112
1.3
EGNO GENODE
3,5-00.5
149209
4.320.1
23.2=222
SGDD-EGNO
3,210.0
13,5
SGDD-GENODE
3.4:0..0
17.7-10
(623.79)
4.3 PROTEIN
Dataset. We used the Adk equilibrium trajectory dataset (Seyler & Beckstein 2017), which corresponds to the molecu- lar dynamics trajectory of apo adenylate kinase (Gowers al.) 2016). Following the standard setting, we divided the entire trajectory into 2481/827/878 sub-trajectories for training, val- idation, and testing. The prediction horizon consists of uni- formly spaced timesteps, with the final step corresponding to 15 frames. The nodes of the graph are defined as the backbone atoms of the amino acids, resulting
Table 3: S2S on ADk equilibrium trajectory dataset.
S2S
Linear MPNN BONN
5 84
1 32
1.73
EGNO
BOHN
1.03
2.23
total of 855 nodes.
EGHNO
1.80
Result (Table()). Our SGDD-EGNO achieved new state-of- the-art with an MSE loss of 1.75, corresponding to a 21.5% improvement over EGNO (2.23). Importantly, the protein dataset contains 855 backbone nodes, forming a relatively large-scale graph. These findings suggest that our framework is capable of learning spectral-guides dynamics representations that provide stable guidance to the decoder for accurate trajectory prediction, even when applied to large-scale spatial systems.
SGDD-EGNO
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4.4 ABLATION STUDIES
Table 4: Ablation study on frequency alignment, feature alignment, and weighting. Results are re- ported for the SGDD-EGNO model the MD17 and motion capture (Mocap) datasets. Numbers correspond to the S27 metric (x10 9). The best performance is highlighted in bold.
Preq Align
Feature Align
SGE
Ethanol
Malonaidehyde
Toluene
Mocup-Waik
Mocap-Run
2.84
2,95
12,98
2.90
11.04
4.18
4.04
12.61
2.89
11.11
4.86
3.30
13.01
2,85
11.06
4.65
3,26
14.37
The central goal of our framework construct dynamics representation that provides the decoder with an effective inductive bias for long-horizo prediction. In standard encoder-latent representa- tion-decoder pipeline where both inputs and outputs are full future trajectories, the encoder has ac- cess to rich dynamical information and can easily produce an informative latent representation. How- ever, in the setting we privileged supervision is no longer available. A natural way to bridge this gap is to let teacher encoder observe the future trajectory and let student encoder learn approximate the teacher's dynamics representation using only the initial state. This leads to a Sthowlege-Dissillaaionn formulation for learning the dynamics latent representation. Based on this idea, our framework introduces two key design components: (1) Dual Alignment, which aligns teacher-student repre- sentations in both the spectral (Freq Align) and spatio-temporal domains (Feature Align), and (2) Spectral- Guidee Enhancement (SGE), which adaptively reweights spectral components empha- size the most informative frequency modes. In the following subsections, we present ablation studies that analyze the contribution and design choices of framework. Section 4.4.1 evaluates the two core components, Dual Alignment and Spectral-Guided Enhancement. Section 4.4.2 examines the truncation parameter K in the Spectral-Guidet Enhancement module. Section 4..3 investigates the effect of encoder selection.
long-horizon trajectories from only the initial state this
4.4.1 ABLATION ON ALIGNMENT AND SPECTRAL GUIDED ENHANCEMENT.
We analyze the contribution of Dual Aligmment and the Spectral-Guie Guidedd Enhancement (SGE) mod- ule. As shown in Table[4 removing either component consistently degrades performance across both molecular and human motion datasets. Dual A Alignment operates in two complementary domains: (1) Feature Align, which matches studeet-ttudent representations in the spatio-temporal domain to pre- serve global structural patterns; and (2) Freq Align, which aligns spectral coefficients to capture multi-scale frequency behavior essential for long-horizon stability. There is no consistent superior- ity between the two alignments alone, but using both simultaneously yields the best performance overall. This suggests that the two aligmments operate in a mutually complementary manner, where peatio-tempori alignment provides robust inductive bias for overall structure, and spectral align- ment refines frequency prioritization to mitigate noise and instability. The effect of SGE is observed by comparing the first and second rows: learnable spectral weights through SGE improve perfor- mance in nearly all cases by emphasizing informative frequency modes during aligmment. Overall, the configuration that includes both alignments and SGE yields the best results, highlighting their complementary roles in forming robust dynamics representation.
4.4.2 ABLATION ON THE TRUNCATION PARAMETER K.
SGDD-EGNO (Walk-STT)
SGDD-EGNO (Run-S2T)
We further investigated the role of the trunca- tion parameter K in the spatio-temporal joint basis. As described in Section 3.3] we con- struct a truncated basis Вк by retaining only the K lowest-frequenc ual (1 (1- P) preserves information outside this subspace. We conducted experiments man Motion dataset with 31 joints and a pre- diction horizon of 5 timesteps, resulting in to- tal of 31 5 = 155 frequency components. As shown in Figure 6 the performance varies with
Loss (K31--)
modes, while the resid-
Hu-
200
1i5 Truncation Parameter K
Truncation Parameter K
Figure 6: Performance according truncation pa-
rameter K.

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the choice of the K. When K is too small, the enhancement focuses excessively only lowest- frequency modes. Although the residual term still preserves information outside the truncated sub- space, important frequency bands cannot be adequately emphasized. As K increases, the model can adaptively weight a richer set of spectral modes. However, the increase in controllable modes does not always lead monotonic loss reduction; therefore, it can be needed select appropriate K. Experiments on other datasets and discussions on selecting the best K are detailed in Appendix D.2
4.4.3 ABLATION ON ENCODERS
Table 5: MSE (x10-3) of SGDD-EGNO across different encoder combinations on MD17. (S2S)
Els
Edyn
Aspiria
Ethanol
Naphthalene
Saticylic
Uracil
GATI (42K)
STSGNN 397K)
7.75
4.00
0.36
0.85
0.52
GATT142K
STGGN(ISKK)
7.97
4.08
0.36
0.84
0.52
(1N6(136K)
STSCING397K)
7,85
4.21
0.36
0.86
0.53
GINE GiNE(336K)
STGGN
8.02
4.15
0. 1636
0.86
0.52
Transformen 541K)
STSGINN 397K)
8.90
4.12
0. 0.36
0.91
0.53
Transformet (41K)
STGGN(15IK)
8.69
4.27
0.36
0.94
0.53
We use STSGNN Edyn and GAT Einit in our main experiments, and detailed justifications for these choices are provided in the Appendix C.3 To examine how different encoder choices affect the SGDD framework, we additionally evaluate STGCN (Yan et al. 2018). spatio-tempont graph con- volutional network designed for dynamic skeleton-like structures, as Eayn and replace GATConv in Eini with either GINEConv(Hu al., 2019), an edge-enhanced variant of GIN, or Transformer- graph convolution layer, on the MD17 dataset. All experiments were conducted under identical training settings. Table reports the performance of different encoder combinations, showing that modifying Emi leads to larger performance variation than modifying Edyn. Replacing GATConv with GINEConv for Em yields similar results, whereas using TransformerConv leads to lower performance on several molecules. We attribute this to the substantially larger number of learnable parameters in TransformerConv, which leads to overfitting or underfitting under the same training configuration. Overall, SGDD shows moderate level of robustness to reasonable encoder substitutions such as GATConv + GINEConv or STSGNN <> STGCN- also indicating that heavier encoders may require different optimization settings or regularization strategies to realize their potential. This suggests that SGDD broadly applicable across encoder architectures, although appropriate training configurations remain important depend- ing on model capacity and dataset scale.
Conv(Shi et al., 2020). a transformer-
5 CONCLUSION
In this work, we introduced SGDD, a novel framework that leverages privileged supervision to learn rich dynamics representations. Our approach combines a spectral-guided enhancement mod- ule with a distillation scheme to disentangle multi-scale spatio-temporal dynamics and guide the decoder toward accurate Long-horizon trajectory prediction, even at inference time when privileged information is unavailable. Through experiments, we demonstrated that our framework consistently achieves state-of-ther -frt performance across particle dynamics at different scales. Our study con- tributes generalizable framework that unifies spectral representation learning with knowledge dis- tillation, paving the way for future research in robust and scalable physical dynamics modeling- However, our framework also has limitations. Since it relies existing physical-dynamics models as decoders, its overall performance can be influenced by the capacity of the chosen decoder. Fu- ture work could explore decoder-agnosti formulations or tighter integration between representation learning and prediction modules to further enhance robustness and scalability. In addition, SGDD has so far been applied only in settings with fixed spatio-temporal graph structures. Extending it to time-varying graphs-for for example, by updating the spatio-temporal basis as the topology evolves or by updating the spatio-temporal basis an online or adaptive manner- represents promising future direction for handling dynamic physical environments.
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ACKNOWLEDGMENTS
This work was supported in part by the National Research Foundation of Korea(NRF) grant funded by the Korea govermmen(MMSTT) & communications Technology Planning Evaluation (ITTP) grant funded by the Korea govem- ment(MSIT) (RS-2021-1121 1341, -5- RS-2025-25522680)
(022500211777), and in part by the Institute of Information
REFERENCES
Bissan Al-Lazikani, Joon Jung, Zhexin Xiang, and Barry Honig. Protein structure prediction. Cur- rent opinion in chemical biology, 5(1):51-56, 2001.
Carnegie Mellon University. Cmu motion capture database. http::/moces cs cmu edu
2003. Accessed: 7, 13.
Yuxin Chen, Fangru Lin, Jingyi Huo, and Hui Yan. Designing specialized two-dimensional graph spectral filters for spatial-temporal graph modeling. AAAI, pp. 11500-11508, 2025.
Stefan Chmiela, Alexandre Tkatchenko, Huziel E Sauceda, Igor Poltavsky, Kristof T Schütt, and Klaus-Robert Müller. Machine learning of accurate energy-conserving molecular force fields. Science advances, 1603015, 2017.
Jang Hyun Cho and Bharath Hariharan. On the efficacy of knowledge distillation. In ICCV, pp. 4794-4802, 2019.
Juan Viguera Diez, Mathias Schreiner, Ola Engkvist, and Simon Olsson. Boltzmann priors for implicit transfer operators. arXiv preprint arXiv:2410.10603 2024.
Weitao Du, Zhang, Yuanqi Du, Qi Meng, Wei Chen, Nanning Zheng, Bin Shao, and Tie- Yan Liu. Se (3) equivariant graph neural networks with complete local frames. In ICML, pp. 5583-5608, 2022.
Fabian Fuchs, Daniel Worrall, Volker Fischer, and Max Welling Se (3)-transformers: 3d roto- translation equivariant attention networks. NeurIPS, pp. 1970-1981, 2020.
Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. In ICML., pp. 1263-1272, 2017.
Richard J. Gowers, Max Linke, Jonathan Barnoud, Tyler J. E. Reddy, Manuel N. Melo, Sean L. Seyler, Jan Domañski, David L. Dotson, Sebastien Buchoux, Ian M. Kenney, and Oliver Beck- stein. Mdanalysis: A python package for the rapid analysis of molecular dynamics simulations. In SciPy, pp. 905, 2016.
Geoffrey Hinton, Oriol Vinyals, and Jeff Dean. Distilling the knowledge in neural network. arXiv preprint arXiv: 150.00253,, 2015.
Weihua Hu, Bowen Liu, Joseph Gomes, Marinka Zitnik, Percy Liang, Vijay Pande, and Jure Leskovec. Strategies for pre-training graph neural networks. arXiv preprint arXiv:1905.1 12265, 2019.
Zijie Huang, Yizhou Sun, and Wei Wang. Learning continuous system dynamics from irregularly- sampled partial observations. In NeurIPS, pp. 16177-16187, 2020.
Jonas Köhler, Leon Klein, and Frank Noé. Equivariant flows: sampling configurations for multi- body systems with symmetric energies. arXiv preprint arXiv: 1910.00733, 2019.
Phillip Lippe, Bastiaan S. Veeling, Paris Perdikaris, Richard E Turner, and Johannes Brandstetter. PDE-refiner: Achieving accurate long rollouts with neural PDE solvers. In NeurIPS, 2023.
A Srinivas Reddy, S Priyadarshini Pati, P Praveen Kumar, HN Pradcep, and G Narahari Sastry. Virtual screening in drug discovery-a computational perspective. Current protein and peptide science, 8(4):329-351) 2007.
11

---

Published as a conference paper at ICLR 2026
Victor Garcia Satorras, Emiel Hoogeboom, and Max Welling. E (n) equivariant graph neural net- works. In ICML, pp. 0323-9332, 2021.
Sean L. Seyler and Oliver Beckstein. Molecular dynamics trajectory for benchmarking md- analysis, 2017. URL htttp::/ wtt.aaaieeco//tteee/////// Molasuuaal oaw.a//eeeeeelleeaaaaaoaaaaaaaaa trajectory__ for_brrcwroorrreccmrremar. ng_
Zhiqiang Shen, Zhankui He, and Xiangyang Xue. Meal: Multi-model ensemble via adversarial learning. In AAAI, pp. 4886-4893, 2019.
Yunsheng Shi, Zhengjie Huang, Shikun Feng, Hui Zhong, Wenjin Wang, and Yu Sun. Masked label prediction: Unified message passing model for semi-supervised classification. arXiv preprint arXiv:2009.03509, 2020.
Mark W Spong, Seth Hutchinson, Mathukumalli Vidyasagar, et al. Robot modeling and control, volume 3. Wiley New York, 2006.
Fang Sun, Zijie Huang, Haixin Wang, Huacong Tang, Xiao Luo, Wei Wang, and Yizhou Sun. Graph fourier neural odes: Modeling spatial-temporal multi-scales in molecular dynamics. arXiv preprint arXiv:241 1.01600, 2024.
Xiaoning Sun, Qiongjie Cui, Huaijiang Sun, Bin Li, Weiqing Li, and Jianfeng Lu. Overlooked poses actually make sense: Distilling privileged knowledge for human motion prediction. ECCV, pp- 678-694, 2022.
Nathaniel Thomas, Tess Smidt, Steven Kearnes, Lusann Yang, Li Li, Kai Kohlhoff, and Patrick Riley. Tensor field networks: Rotation-and translation-equiarianl neural networks for 3d point clouds. arXiv preprint / 2018.
Frederick Tung and Greg Mori. 1374, 2019.
knowledge distillation. In CVPR, pp. 1365-
Vladimir Vapnik and Akshay Vashist. A new learning paradigm: Learning using privileged infor- mation. Neural networks, 22(5-6):544-555 -5744 2009.
Petar Veličković, Guillem Cucurull, Arantxa Casanova, Adriana Romero, Pietro Lio, and Yoshua Bengio. Graph attention networks. arXiv preprint arXiv:1710.1090. 2017.
Liming Wu, Zhichao Hou, Jirui Yuan, Yu Rong, and Wenbing Huang. Equivariant tempooltemporal attentive graph networks to simulate physical dynamics. In NeurlPS, pp. 45360-45380, 2023a.
Lirong Wu, Haitao Lin, Yufei Huang, Tianyu Fan, and Stan Z Li. Extracting www-thigh-frequeen knowledge from graph neural networks and injecting it into mips: An effective gnn-to-mip distil- lation framework. In AAAL, pp. 10351-10360, 2023b.
Taiqiang Wu, Zhe Zhao, Jiahao Wang, Xingyu Bai, Lei Wang, Ngai Wong, and Yujiu Yang. Edge- free but structure-awer aware: Prototype-gun guided knowledge distillation from gnns mlps. arXiv preprint arXiv:2303.1377 2023c.
Minkai Xu, Jiaqi Han, Aaron Lou, Jean Kossaifi, Arvind Ramanathan, Kamyar Azizzadenesheli, Jure Leskovec, Stefano Ermon, and Anima Anandkumar. Equivariant graph neural operator for modeling 3d dynamics. ICML, pp. 55015-55032, 2024.
Bencheng Yan, Chaokun Wang, Gaoyang Guo, and Yunkai Lou. Tinygnn: Learning efficient graph neural networks. In ACM SIGKDD, pp. 1848-1856, 2020.
Sijie Yan, Yuanjun Xiong, and Dahua Lin. Spatial temporal graph convolutional networks for skeleton-basee bused action recognition. In Proceedings of the AAAI conference on artificial intelli- gence, volume 32, 2018
Chenglin Yang. Lingxi Xie. Chi Su, and Alan L Yuille. Snapshot distillation: Teacher-studed opti- mization in generation. In CVPR, pp. 2859-2868, 2019.
12

---

Published as a conference paper at ICLR 2026
Shuo Yang, Sujay Sanghavi, Holakou Rahmanian, Jan Bakus, and Vishwanathan SVN. Toward understanding privileged features distillation learning-to to-rank. In NewrlPS, pp. 26658-26670, 2022.
Chengxi Zang and Fei Wang. Neural dynamics on complex networks. In ACM SIGKDD, pp. 892- 902, 2020.
Shichang Zhang, Yozen Liu, Yizhou Sun, and Neil Shah. Graph-less neural networks: Teaching old mlps new tricks via distillation. arXiv preprint 10000221 2021.
Wentao Zhang, Xupeng Miao, Yingxia Shao, Jiawei Jiang, Lei Chen, Olivier Ruas, and Bin Cui. Reliable data distillation graph convolutional network. In ACM SIGMOD, pp. 1399-1414, 2020.
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A NOTATIONS
A.1 GENERAL NOTATIONS
We summarize the notations used throughout the paper along with their descriptions.
Table A.1: Summary of Notations.
Notation
Description
V
Set of N particles (nodes).
E
Set of edges representing interactions connections between particles.
X E Nx3
3D positions all particles at time
R
3D velocities of all N particles time f.
7. = [x,,4]] E NR6
Node state tensor at time t, concatenating position and velocity.
h = (V, Z4,, h)
R Nxd
Time-invariant physical features of each particle.
Graph representation of the system time step f.
{1), X2,
} E R TTXN3
3D trajectory of N particles over T time steps. Future state sequence to be predicted given the initial state Go-
(9I....,TT
Zdyn: Pesi E IR Nxxxx
Opatio-temporal dynamics representations generated by the dynamics encoder and initial encoder, respectively.
Gdyn: Ginit Bd.xK
Spectral coefficients obtained by projecting the dynamics representations onto the joint basis A
E R
Learnble weights for each spectral mode.
adyn: Ginit E Rdxx
Frequency-adjuster spectral coefficients after applying the learned weights. Reconstructed representations the spatio-temporal domain obtained from the ad- justed spectral coefficients.
Edyn: Eair R2xXN7
Byn E RN
Final spectral-guiee dynamics representations incorporating residual components: Sdyn
Zdya (I Plaan Fini Amit + (I P) Aiii, where P = BB"
A.2 SPATIO-TEMPORAL GRAPH CONSTRUCTION
In our formulation, the spatio-temporal graph Gor describes both particle interactions and temporal evolution by linking particles within each time step as well as across adjacent time steps. This structure can be viewed as the combination of a spatial graph that captures particle connectivity and a temporal graph that captures sequence order, each admitting its own Laplacian and spectral basis.
Spatial graph. The spatial graph Gspatio (V, E.) is fixed across time and encodes the physical connectivity among particles. Its normalized Laplacian Ls yields the spatial eigenbasis U..
Temporal graph. The temporal graph Gtemp = (V,, F2)) is defined over the sequence of time indices V; = (I...., T). To model temporal continuity, we adopt the standard 1D chain construction in which each time step is connected to its previous and next steps:
E = { 1 T).
The normalized Laplacian L, produces the temporal eigenbasis Uy.
B PROOFS
To apply the Orthogonal Projection Theorem the spectral domain, we first establish that the trun- cated spatio-temporal joint basis B, constructed from the Laplacian eigenvectors, forms orthog- onal set.
Proposition B.1 (Orthogonality of the Joint Basis), Let U, E NxN and E TxT be orthogonal eigenvector matrices of the normalized Laplacians Lg and L,, respectively. Their Kronecker product
B = U, @ U, E NT
also orthogonal, i.e.,
B INT.
Consequently, any truncated basis By E NT obtained by selecting K columns of B forms a partial orthogonal basis and satisfies B3 Bx Tk-
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Proof. We use the property of Kronecker products:
(U, @ (U.))(Us U,) - UU) U1) & U..U.)
Since U2 and Us are orthogonal, U, U. = IT and Ug U. = IN. Therefore,
B = LT 8 1N INT.
Selecting a subset of K columns from B preserves orthogonality among the chosen columns, yield- ing BK Ig.
We next show that the decomposition used our spectral -gided enhancement indeed orthogonal. Lemma B.2 (Orthogonal Decomposition), Let Bx IR NT^ K be truncated orthogonal basis and define the projection matrix P Bx Bk. Then, for any E Rd:XNT, the following orthogonal
decomposition holds:
2 = Pz - P)= with (Pz, (1 P)2) =0
Proof. Since P is symmetric and idempotent (P P, p2 = P), it is an orthogonal projector. Thus any can be uniquely decomposed into its projection Pz and residual (1 P)z, which orthogonal because
(P))(1-P)a)=
= p²)2 = O.
C EXPERIMENT DETAILS
C.I DATASET DETAILS
MD17. We used the MD17 dataset (Chmiela al. 2017), which contains molecular dynamics tra- jectories of eight small molecules. In constructing the graphs, hydrogen atoms are conventionally excluded. and only heavy atoms are retained as nodes, resulting in the number of nodes reported in Table A.2 Edges are defined by extending the original molecular bonds to include up to 2-hop neighbors, following prior work. For each edge, the features are constructed by concatenating the hop type, the atomic types of the connected nodes, and the chemical bond type. Each trajectory is randomly split into train/validation/tes sets with 500/2000/2000 cltt-trajjcctoy pairs, respectively.
Table A.2: Statistics of MD17 dataset.
Name
Aspirin
Ethanol
Malenaldehyde
Naphthalene
Salicylic
Toluene
Uracil
#
Motion Capture. We used the CMU Motion Capture dataset (Carnegie Mellon University 2003
which contains 3D trajectories of various human motions. Following (Xu al. (024), we selected Walk and Run. The data were split into 200/600/600 trajectories for training/ haliii////vllaattoo/estsii in Walk, and 200/240/240 trajectories in Run. Each human skeleton is represented as a graph with 31 joints serving nodes. Edges are constructed based on the natural skeletal connec- tivity, joints directly connected in the human body.
provided in the MDAnalysis toolkit (Gowers et al. 2016), which contains the molecular dynamics trajectory graph where the nodes correspond to
Protein. We used the Adk equilibrium trajectory dataset (Seyler & Beckstein, 2017
of apo adenylate kinase. Protein structure is represented the backbone atoms (N, Ca, and of each amino acid), resulting in a total of 855 nodes for apo adenylate kinase. Edges constructed using cutoff strategy, where two atoms connected if their Euclidean distance is within 10 in the equilibrium structure, which follows the commonly adopted convention in protein graph construction.
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C.2 BASELINES
We provide an introduction to the two baseline models employed as decoders in our framework.
EGNO (Xu et al. 2024) Equivariant Graph Neural Operator (EGNO) is a method for modeling 3D dynamics of relational systems, directly modeling entire trajectory dynamics as temporal functions rather than next-step predictions. It formulates dynamics as function over time and learns neural operators approximate it, developing equivariant temporal convolutions in Fourier space stacked over equivariant networks to maintain SE(()--quivariance while capturing temporal correlations. In the spectral domain, EGNO performs equivariant temporal
terms of frequency-aware approaches convolutions in Fourier space to decompose and model frequency modes, ensuring equivariance in 3D space and handling multi-scale temporal evolution.
GFNODE (Sun al. 2024) Graph Fourier Neural ODEs (GF-NODE) is neural operator-based model designed to capture spatial-temporal multi-scale interactions molecular dynamics simula- tions. It addresses the challenge of predicting long-horizon trajectories by decomposing molecular configurations into spatial frequency modes using the graph Laplacian, evolving these modes in continuous time via Neural ODEs, and reconstructing the updated molecular geometry through an inverse graph Fourier transform. In terms of frequency-aware approaches in the spectral domain, GF-NODE decomposes the spatial structure into frequency components via Graph Fourier Trans- form, evolves each component temporally via Neural ODEs.
C.3 MODEL DETAILS
We employed fixed architectures for both the dynamics encoder and the initial encoder throughout all experiments, and this part, we present a short introduction and implementation details.
C.3.1 DYNAMICS ENCODER
Background We employed Shhn et al. 2025 as the backbone for the dynamics encoder. STSGNN takes spatio-temporal graphs input and performs filtering in the spatio-temporal joint spectral domain. Specifically, based on the cigendecomposition of the normalized Laplacian matri- ces of spatial and temporal graphs, introduces the 2-D Discrete Graph Fourier Transform (2-D) DGFT) to map input signals into the joint spectral domain. Filtering in this domain is then imple- mented via bivariate Bernstein polynomial approximation, which leverages learnable coefficients to construct 2-D filters with specialized spectral properties. This formulation enables STSGNN to capture spatial and temporal dependencies simultaneously, unlike conventional methods that rely separate spatial temporal spectral representations. Moreover, by exploiting the decoupling property of Bernstein bases, STSGNN can effectively preserve both low- and high-frequency infor- mation, while adaptively emphasizing the most task-relevant components. Consequently, it provides stable spatio-temporal joint representations that mitigate instability from high-frequency compo- nents while retaining global low-frequency patterns.
Justification for Choice STSGNN is a sophisticated spatio-temporal graph encoder that applies 2-D joint spectral filtering, enabling spatial and temporal dependencies to be captured in sin- gle spectral domain while maintaining stable propagation of both low- and high-frequency compo- nents. Such frequency-awand modeling aligns well with the nature of physical dynamics, where low- frequency structure governs long-term evolution and high-frequency variations capture short-term fluctuations. Therefore, we adopt STSGNN as our dynamics encoder, as it provides the frequency- resolved pempooltemporal representations required for modeling physical dynamics.
Implementation Detail
Input: -patooltemporal graph node features: Zorr E NxTX6 h Nxlxd
Output: Spatio-temporal dynamics representation Zdyn NxTxd.
The spatial Bernstein order and temporal Bernstein order were set to the same values in the original paper, namely 10 and 5, respectively, In addition, the model was configured with 2 layers, an output dimension of 32, a hidden dimension of 32, and a dropout rate of 0.1. The original model was implemented take multi-step inputs and
Detail:
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generate multi-step outputs. In our framework, retained only the feature extraction part for multi-step inputs and removed the output module designed for multi-step prediction.
We provide link to the source code for reproducibility. For specific details, please refer to the following repository
C.3.2 INITIAL ENCODER
Background For the initial encoder, we adopted the Graph Attention Network (GAT) layer (Veličković et al. 2017). GAT extends the idea of message passing in graph neural networks by incorporating a self-attention mechanism over graph neighborhoods. Specifically, instead treating all neighbors equally relying fixed weights, GAT computes attention coefficients that quantify the relative importance of each neighboring node when aggregating features. This achieved by applying shared linear transformation node features, followed by learnable attention kernel that operates on pairs of nodes. The coefficients are normalized using the softmax function, ensur- ing that the aggregated representation remains permutation-invariinn and adaptive the underlying graph structure. This formulation allows GAT to capture both local graph topology and feature rele- vance in data-driven manner. Compared to spectral approaches that rely on fixed graph filters, GAT provides greater flexibility in learning task-specific dependencies while maintaining computational efficiency.
Justification for Choice GAT is the simplest attention-based graph encoder capable of projecting an initial state into a higher-dimensional temporal representation through multi-head attention. In addition, its lightweight architecture and minimal inductive bias help avoid overfitting underfitting that may arise with heavier encoders, making GAT a suitable choice for initial encoder within our SGDD framework.
Implementation Detail
Input: initial state graph node features: Zo E Nx6 Nxd
Output: Spatio-temporal dynamics representation Zini NxTxxd
Detail: We set the number attention heads equal to the temporal dimension to extend the model along the time axis. The hidden dimension fixed at 32, and employed layers. The computation proceeds as follows. We begin with the initial node features.
To (Zo, E (x(+d))
For a GAT layer with T attention heads, the 1-th head computes
h 1 w F0,i
and performs neighborhood aggregation as
1
Σ
(0) (0)
JEN(6)
where aij.r denotes the attention coefficient for head Concatenating all head outputs yields
F1 1|=,5 (0) RTd.
Thus, the full output of the first layer satisfies
I1 E NxTTd.)
For a general I-th layer 2 0), the update rule is
Te+1.3 = |T_
Onge W xzs
jEN())
https::/ggithbb..commooujj//
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and therefore
J++1 E (Nx(dd)).
Finally, the output of the last layer,
FL. E (x(T))
is reshaped by interpreting the T heads temporal channels, yielding reshape(21.)
Finit E NxTxd.
C.4 IMPLEMENTATION DETAILS
Our framework is implemented to follow the training and architectural settings of EGNO as closely as possible, and we refer readers to the official EGNO implementation for additional details. For the decoder component, we strictly adopt the architectural configurations of both EGNO and GFNODE without introducing any additional capacity; only training-related hyperparameters are adjusted to accommodate the SGDD training procedure. This ensures that the observed improvements stem from our petiseentttiio-eeariin framework rather than increased model complexity in the baseline decoders.
MD17 We set the batch size to 100, the learning rate to 1 x 10 and the weight decay to x 10-15 Staged learning was adopted, with a maximum 5000 epochs and pretraining performed up to epoch 2000. The truncation parameter K was fixed at 64 by default. However, when the total number of basis vectors (N x T) for given molecule was smaller than 64, we set K = 0.5 x (N x T') for that molecule, ensuring that K < N x. T.
Motion Capture We set the batch size to 12, the learning rate to 5 x 10-4, and the weight decay to 1 10 The maximum number of epochs was 2000, with pretraining performed up epoch 500. For the pempoolt joint basis, a total of x 5 = 155 basis vectors were available. The truncation parameter K was fixed to 64.
Protein We set the batch size to 8, the learning rate to 5 x 10-5, and the weight decay to x 10-4 The maximum number of epochs was 15,000, with pretraining performed up to epoch 100. For the spatio-temporal joint basis, total of 855 x 4 basis vectors were available. In this case, the truncation parameter K fixed to 128.
D ADDITIONAL EXPERIMENTAL RESULTS
In this section we provide additional experimental results.
D.1 COMPARISON OF OUR FRAMEWORK WITH THE IMPLEMENTED BASELINE
We implemented EGNO and GFNODE baselines and employed them decoders within our framework. This allows us to compare the performance of the standalone EGNO and GFNODE implementations with their counterparts integrated into our framework. enabling more direct eval- uation. The comparison table on the MD17 dataset is shown in Table A.3] and the comparison table on the motion capture dataset is shown in Table A.4 For GFNODE, since the original model was designed only for the MD17 dataset, the results on the motion capture dataset are identical to those reported in Section 4.2
D.2 A ABLATION STUDIES
Ablation on Alignment and Spectral Guided Enhancement. We conducted ablation studies both the MD17 datasets using the SGDD-EGNO to examine the effects of different alignment strate- gies and the presence absence of learnable weights. The metrics were computed using both S2S and S2T. The results are summarized in Tables A.5 A.6
And for SGDD-GFNODE, able weights, and the experimental results on both the MD17 and motion capture datasets are re- ported in two tables, corresponding to the S2S and S2T metrics Tables A.bles/A77 and A.8
conducted only the performance comparison with and without learn-
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Table A.3: Comparison the MD17 dataset with own implementations. Upper part: S2S. Lower part: S2T. Best results are highlighted in bold and relative improvements (%) of SGDD variants over the corresponding baselines are shown below.
Aspirin
Malonaldehyde
Naphthalene
Salicylic
EGNO SGDD-EGNO
9.35
58.09
4,60
12.82
0.39
0.87
10.95
0.58
7,75
4.00
13.08
0.36
0.85
8.34
(+17.118)
(+76.5%%
(+13.0%)
1-2.0%)
(+7.0%))
(+2.256)
(+23.8%%
(+10.3%)
GENODE
7.64
4.89
3,92
12,86
0.37
0.80
5.00
0.54
SGDD-GENODE
7.38
2.23
3,64
12.72
0.34
0.79
4.97
0.52
(41.098)
(-8.19)
(+3.79)
EGNO
30.79
3.27
10.83
0.35
0.75
4.86
SGDD-EGNO
6.20
8.29
2,84
11.03
0.33
0.69
3,80
0.50
(+11.89))
(+73.09))
(+13.19)
(-1.8%)
(-5.7%8
(+8.04))
(+21.89%)
(47.4%)
GFNODE 6.18
2.26
2,86
11.03
0.32
0.66
2.33
0.43
SGDD-GENODE
2.66
10.91
0.27
2.32
(+8.416)
(+49.115)
(+6.9%)
(+1.0%)
(+15.696)
(+9,018)
(0.0%)
Table A.4: Comparison the motion capture dataset with implementations. Upper part:
$2S. Lower part: S27.
$2S
Walk
Run
EGNO
11.9
37.9
SGDD-EGNO
6.7
28.2
(+43.399)
S2T
EGNO SGDD-EGNO
5.5
3.2
13.5
(+41.59%)
(+23.298)
Table A.5: Ablation study on frequency alignment, feature alignment, and weighting. Results are reported for the SGDD-EGNO model on the MD17 datasets. Numbers correspond the S2T metric (x10-3)
Freq Align
Feature Align
Weight
Aspirin
Benzene
Ethanol
Malonaidchyde
Naphthslene
Salicylic
Toluene
Uracil
3.80
6.49
6.27
2.90
11.04
0.34
0.70
4.18
0.50
6.15
10.62
2,89
11.11
0.33
0.68
4.86
0.50
6.26
10.79
2.85
11,06
0.33
0.69
4.65
0.50
Table A.6: Ablation study frequency alignment, feature alignment, and weighting. Results reported for the (GDD-EGNO model on the MD17 datasets. Numbers correspond to the S2S metric (x10-2).
Freg Align
Align
Weight
Aspirin
Ethanol
Malonaldehyde
Naphthulene
Salicylic
Toluene
7.75
13.65
4.00
13.08
0.36
0.85
8,34
0.52
7.77
15.82
4.13
13.05
0.35
0.85
10.85
0.52
Table A.7: Ablation study on weighting. Results are reported for the SGDD-GFNODE model on the MD17 datasets and Motion Capture datasets. Numbers correspond to the S2T metric (x10-2).
Freg Align Feature
Weight) Aspirin Benzene Ethanol Malonaldehyde Naphthaleno Salicylie Toluene Uracil Mocap-Walk Mocap-Kum
3.66
1.15
2.66
10.91
0.60
2.32
0.43
7.04
1.70
2.71
10.94
0.25
0.6
2
0.43
4.02
19.66
Ablation on Truncation parameter K. (Table A.9 Here, to compare performance with respect to the truncation parameter K, trained and evaluated SGDD-EGNO across the MD17, motion capture, and protein datasets while varying K. Table reports the results on the motion capture dataset, Table A.10 p.esentsssss the results on Aspirin the molecule with the largest number of atoms the MD17 dataset-and Table A.II shows the results obtained on the protein dataset. Although performance varies with the choice of K, goal was not identify optimal value but rather
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Table A.8: Ablation study on weighting, Results are reported for the FEDD-GFNODE model on the MD17 datasets and Motion Capture datasets. Numbers correspond to the S2S metric (x 10 1)
Freq Align Feature Align Weight Aspirin Beazene Ethanol Malonaldehyde Naphthalene Salicylic Toluene Uracil Mocap-Wan Walk Mocap-Kin Run
4.97
8.46
3.50
3.61
12.73
0.32
0.80
4.70
0.53
7.71
36.77
select reasonable K based on the total number of available frequency components for each dataset (maximum 104 for MD17, 155 for Motion, and 3420 for Protein). Accordingly, we used K 64 for both MD17 and Motion, and K 128 for Protein. Interestingly, experiments conducted with different values of K show that values near 64 consistently yield the best performance across all datasets. This aligns with the fact that long-term behavior in physical dynamics tasks is generally dominated by low-frequency components. Moreover, this observation provides a practical guideline: for new datasets systems, initializing with K x 64 offers a robust starting point without requiring extensive tuning.
Table A.9: Comparison SGDD-EGNO on the motion capture dataset under different truncation level K of the basis B. Upper part: S2S. Lower part: S27.
K
Walk
Run
16
38.82.10.88
32
8.03-0.05
31.88-10.04
64
6.74=0.00
28.21#0.00
128
6.89=0.01
30.45=0.07
(ull(155)
7.26
28.18110.18
16
4.7::0.10
17.99=1.39
3.96::0.6
1..37.11.16
64
3.22=0.05
13.52:10.89
3.29=0.05
14.39::1.05
full(155)
3.06
13.39.10.89
Table 10: Comparison of SGDD-EGNO on MD17 (Aspirin) under different truncation levels K of the basis B. Upper part: S2S. Lower part: S2T.
Aspirin
6.33
32
6.32
6.20
Fulk(104)
8.04
32
8.06
7.84
fulk(104)
Table A.11: Comparison of SGDD-EGNO on the ADk equilibrium trajectory dataset under different truncation levels K of the basis B (S2S).
1.74
128
1.75
256
1.75
E ADDITIONAL VISUALIZATION
In addition, to demonstrate that the proposed SGDD framework achieves superior performance from a frequency perspective, we define frequency loss and present the results table. The frequency loss is computed by mapping the predicted and ground-truth trajectories into the spectral domain via the spatio-temporal joint basis B, and then calculating the loss for each frequency component.
E. MD17
Comparison between SGDD-EGNO and EGNO
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Frequency-wise Loss aspirin
0.6
0.5
0.4
Less
0.3
0.2
0.1
20
80
80
300
Frequency
Frequency-wise Loss benzene
BGND SCOD-RGMO
3.0
2.5
2.0
Lcas
3.5
2.0
0.5
0.0
Frequency
Frequency-wise Loss ethanol
RGMO
sommeran
0.30
0.25
Loss
0.30
0.05
20
Frequency
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Frequency-wise Loss malonaldehyde
0.8
Loss
0.4
0.2
10
23
25
30
35
Frequency
Frequency-wise Loss naphthalene
BGMD
SCOD-RGMO
0.07
0.05
LC29
0.04
0.02
0.00
Frequency
Frequency-wist Loss salicylic
scnn-rraan
Loas
0.04
0.03
0.02
0.01
50
Frequency
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Frequency-wise Loss toluene
12-
1.0
Loss
0.4
10
20
30
Requency
Trequency-wise Loss uracil
0.07
BENO SCND-AGMM
0.09
Lc3s
0.63
0.00
Frequency
Comparison between SGDD-GFNODE and GENODE
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Frequency-wise Loss aspirin
SGD0
0.4
Loss
0.7
0.1
20
80
80
300
Frequency
Frequency-wise Loss benzene
GPNece
9GDO--FMMcce
0.12
Lc35
0.08
0.06
0.00
Frequency
Frequency-wise Loss ethanol
GENece
0.20
0.15
Loss
030
0.05
Frequency
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Frequency-wise Loss malonaldehyde
0.8
Loss
0.4
0.2
10
23
25
30
35
Frequency
Frequency-wise Loss naphthalene
GPNece
SGDO-GFMecie
0.04
Lcas
0.01
0.00
Frequency
Frequency-wist Loss salicylic
GFNade scnn-Grknnd
Loss
50
Frequency
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Frequency-wise Loss toluene
0.4
Loss
0.3
0.
0.1
10
20
30
Requency
Frequency-wise Loss uracil
SG00 GFNccc
0.025
0.023
Less
0.035
0.020
0.005
Frequency
E.2 MOTION CAPTURE.
Comparison between SGDD-EGNO and EGNO
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Frequency-wise Loss walk
Loss
0.4
20
80
100
120
140
200
Frequency
Frequency-wise Loss run
50-
2.5
2.0
Los5
1.
3.0
0.5
20
80
100
120
140
200
Frequency
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Comparison between SGDD-GFNODE and GFNODE
Frequency-wise Loss walk
1.0
0.8
Loss
0.2
20
80
100
150
140
300
Frequency
Frequency-wise Loss run
Loss
20
60
80
100
120
140
300
Frequency
28


Reviews:
[{"review_id": "8u4OkSt20R", "reviewer": "Reviewer_nHYS", "summary": "This paper proposes a novel framework, Spectral-Guided Dynamics Distillation (SGDD), for long-horizon prediction of 3D particle trajectories. The method is designed to tackle the challenge of modeling complex spatio-temporal dynamics that involve a mix of low- and high-frequency components. The core of SGDD is a knowledge distillation setup where a \"teacher\" encoder has access to the full future trajectory (privileged information) during training, while a \"student\" encoder only sees the initial state. Both encoders produce a spatio-temporal representation, which is then refined by a \"spectral-guided enhancement\" module. This module projects the representation onto a joint spatio-temporal basis (derived from graph Laplacians), applies learnable weights to different frequency components, and reconstructs the representation. The student is trained to mimic the teacher's enhanced representation, effectively distilling the knowledge of future dynamics into a model that operates only on the initial state at inference time. The distilled representation is then fed to a decoder (e.g., EGNO or GFNode) to generate the trajectory. The authors demonstrate state-of-the-art performance on molecular dynamics, protein dynamics, and human motion datasets.", "questions": "1.  Could you please provide a precise mathematical definition of the temporal graph `G_t` and its Laplacian `L_t`? How are the temporal connections established?\n2.  Please provide a detailed explanation of how the representation `z_init`, derived from the single initial graph `G_0`, is prepared for projection onto the spatio-temporal basis `B_K`. What are the exact \"projection and expansion\" steps mentioned in lines 202-204?\n3.  Would it be possible to provide results for an ablation study where knowledge distillation is performed directly on the spatio-temporal representations (`z_dyn` and `z_init`) without the spectral-guided enhancement module? This seems crucial for proving that the spectral component is the key to your method's success.\n4.  The truncation parameter `K` is a key hyperparameter. The ablation in Figure 6 shows its impact, but how was `K` chosen for the main experiments on the MD17 and Protein datasets? Is it a fixed value, or was it tuned per dataset? How sensitive are the results to this choice?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1.  **Strong Empirical Performance:** The SGDD framework consistently achieves state-of-the-art results across multiple diverse and challenging benchmarks, demonstrating its effectiveness in practice.\n2.  **Effective Use of Privileged Information:** The knowledge distillation setup provides an elegant way to incorporate information from future trajectories during training to improve the representation learned from only the initial state.\n3.  **Addresses a Key Challenge:** The paper tackles the important and difficult problem of long-horizon forecasting in systems with complex, multi-scale dynamics.", "weaknesses": "1.  **Methodological Ambiguity:** The construction of the spatio-temporal graph and, more importantly, the process of projecting the initial-state representation onto the spatio-temporal basis are not clearly explained. This is a critical flaw that hinders understanding and reproducibility.\n2.  **Insufficient Ablation Studies:** The ablation studies do not adequately isolate the contribution of the core novelty—the spectral-guided enhancement. A baseline that uses the same distillation framework but omits the spectral module is essential to validate the paper's central claim.\n3.  **Limited Novelty of Components:** The constituent ideas (knowledge distillation with future-as-PK, graph spectral analysis) are not new. The contribution is in their combination, but the significance of this combination is not fully substantiated due to the aforementioned weaknesses.\n4.  **Complexity:** The proposed framework is highly complex, involving two separate encoders, a custom spatio-temporal basis construction, spectral projection, re-weighting, reconstruction, and a two-stage training process. This complexity makes it difficult to dissect the reasons for its success and may limit its practical adoption.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "OyZpsdj4BL", "reviewer": "Reviewer_Q8zS", "summary": "This paper addresses the challenging problem of long-horizon trajectory prediction for physical dynamics, where systems exhibit a complex interplay of low- and high-frequency components. The authors propose a novel framework, SGDD (Spectral-Guided Dynamics Distillation), which leverages privileged information (future trajectories) within a knowledge distillation setup. The core of SGDD is a spectral-guided enhancement module that operates on a unified spatio-temporal graph representation.", "questions": "1.  The spectral-guided enhancement relies on a truncated basis of the *K* lowest-frequency modes. The ablation in Figure 6 shows that performance is sensitive to *K*. How should one choose an optimal *K* in practice for a new dataset or system? Is there a more adaptive way to select the basis rather than a fixed low-pass filter?\n2.  The dual-level alignment (in both spatio-temporal and spectral domains) is shown to be crucial. Could you provide some intuition as to why aligning in both domains is superior to aligning in just one? Does one domain contribute more to the final performance than the other?\n3.  The paper focuses on distilling a representation. Have you considered also distilling the final output distribution (i.e., the predicted trajectory itself), as is common in classic knowledge distillation? Would this provide complementary benefits?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "- Addresses a Core Challenge: The paper directly tackles the critical challenge of disentangling and prioritizing multi-scale dynamics, which is a primary reason why long-horizon prediction is so difficult. The results on low-frequency error reduction are particularly impressive.\n- Strong and Consistent Performance: SGDD consistently outperforms very strong, recent baselines (EGNO, GFNode) across multiple diverse and challenging datasets (molecules, human motion, proteins).\n- Generality: The framework is designed to be modular. It can be instantiated with different equivariant GNNs as decoders (as shown with EGNO and GFNode), highlighting its flexibility and potential for broad impact.", "weaknesses": "- Complexity of the Framework: The overall system is quite complex, involving multiple encoders, a custom spectral module, and a staged training process. This might pose a challenge for reproducibility and for other researchers seeking to adopt the method.\n- Choice of Teacher/Student Architecture: The paper uses sophisticated models (STSGNN, GAT) for the teacher and student encoders. A deeper justification for these specific choices, and an analysis of how the performance depends on the capacity of these encoders, would be beneficial.\n- Dependence on a Good Basis: The method's success hinges on the quality of the spatio-temporal graph basis derived from the Laplacian. While this is standard, a discussion on potential limitations for systems where the graph structure is dynamic or poorly defined would be valuable.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "C6BaJCDEEr", "reviewer": "Reviewer_Podc", "summary": "The paper proposes SGDD, a “spectral-guided dynamics distillation” framework for long-horizon particle-dynamics forecasting that trains a teacher encoder on full future trajectories as privileged information and distills its spatio-temporal representation into a student that only sees the initial state at test time. Concretely, both teacher and student representations are mapped into a truncated joint spatio-temporal spectral basis built from temporal and spatial Laplacians. A learnable frequency-weighting module emphasizes low-frequency modes while preserving residual components, and training aligns the student and teacher both in representation space and in the spectral coefficients.", "questions": "1. Could you replicate all baselines within your codebase with matched search to rule out confounds from external reported numbers, and add a “decoder-capacity-matched” comparison where the decoder architecture and training schedule are identical with and without SGDD?\n2. The frequency weighting advocates prioritizing low-frequency modes. Can you provide frequency-conditioned error profiles across datasets (as in Fig. 4 for Benzene) and show at least one regime where high-frequency content is crucial? Do gains persist there, or does performance degrade as K grows?\n3. Beyond qualitative statements, it would be good to quantify decoder dependence by swapping in a weaker and a stronger decoder and plotting accuracy vs. decoder capacity to show that SGDD’s benefit is not merely a proxy for more parameters.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The joint spatio-temporal spectral basis with learnable per-mode weights provides a principled way to emphasize low-frequency global trends while preserving high-frequency residuals, which directly targets long-horizon stability.  \n2. The dual alignment together with gradient detaching for the teacher is a clean instantiation of privileged-information distillation that avoids trivial position-level imitation and back-door leakage during training.\n3. On MD17, the method demonstrates large, interpretable gains specifically where baseline errors concentrate in low frequencies.\n4. Ablations indicate that both spectral- and feature-space alignments contribute, and that the truncation K trades off emphasis breadth against noise, which is informative for practitioners.", "weaknesses": "1. Statistical rigor is uneven, with several results reported as single numbers or with “±0.0” and no confidence intervals, and with seeds unspecified for some tables, which limits conclusions about robustness.\n2. Fairness and tuning budgets are not fully documented. This is important because SGDD wraps baseline decoders; strong gains might partly arise from extra representation capacity or training length unless search spaces and early-stopping criteria are matched and reported. \n3. The method’s dependence on decoder capacity is acknowledged but not quantified.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "YeYtER89dr", "reviewer": "Reviewer_6vVc", "summary": "This work proposes Spectral-Guided Dynamics Distillation (SGDD) for long-horizon trajectory forecasting from only initial states. SGDD leverages full future trajectories as privileged information for a teacher encoder and distills that knowledge into a student encoder that must construct the full trajectory solely from the initial state. A spectral-guided enhancement module reweights joint spatiotemporal frequency components, enabling capture key low-frequency patterns.", "questions": "- I am a bit confused about the “spatial graph.” Each temporal graph $G_t$ already encodes the physical connections among particles. Why is an additional spatial graph $G_s$ mentioned?\n- Do the authors build the joint basis $B_{K} \\in \\mathbb{R}^{NT \\times K}$ from the training set and then **store** it for subsequent training and inference?  \n  - If **yes**, the storage and fixed-horizon issues in the Weakness section become significant.  \n  - If **no**, please clarify how $B_{K}$ is computed/updated and used in practice.\n- Are the frequency-specific weights $w$ **different** for the teacher and student encoders? My understanding is **yes**.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "- The paper is clearly written, well structured, and easy to follow.\n- The joint handling of spatial and temporal information, particularly the emphasis on capturing overall low-frequency patterns, is insightful.\n- The distillation strategy offers a practical way to train forecasting models that operate from initial conditions alone.\n- Aligning the student with the teacher both in representation space and in the spectral domain is conceptually sound.", "weaknesses": "- Forecasting long-horizon time series **solely** from initial states is often unrealistic. In many real-world settings, future trajectories depend on external inputs, or may be influenced by events. For example, predicting a patient’s blood glucose purely from the current state ignores forthcoming meals (e.g., salad vs. burger). Because SGDD predicts future trajectories from only the initial state, it appears unable to handle such exogenously driven series. This is the main reason I lean toward a **weak reject**.\n- As I understand it (see Question 2), the method relies on a spatiotemporal joint basis $B_{K} \\in \\mathbb{R}^{NT \\times K}$. While this enables joint spatiotemporal modeling, it introduces two concerns:\n  - Because the shape of $B_K$ explicitly depends on the number of time steps $T$, SGDD may only forecast a **fixed** number of steps at inference, limiting practical applicability.\n  - For very high-dimensional systems or very long sequences, storing $B_{K}$ could become memory intensive.\n- The primary experiments are in scientific domains, yet the method does not enforce physical constraints; the physical consistency of the forecasts is therefore uncertain.\n- Experimental results should report mean ± standard deviation across multiple random seeds.\n- If possible, evaluating SGDD with different encoders and decoders would better demonstrate the method’s robustness and generality.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)