Enabling arbitrary inference in spatio-temporal dynamic systems: A physics-inspired perspective
Authors: Yan Ge
Zhengyang Zhou
Qihe Huang
Yuxuan Liang
Yang Wang
Venue: ICLR 2026 Poster (2026)
Keywords: Neural operators
Spatio-temporal systems
Graph neural networks
Data mining
Primary area: learning on time series and dynamical systems

Abstract: Modern spatio-temporal learning techniques usually exploit sampled discrete observations to foresee the future. Actually, spatio-temporal dynamics are continuous and evolve continuously across time and space, thus modeling  spatio-temporal dynamics in a continuous space can be a long-standing challenge. Existing deep learning architectures often fail to generalize to unseen regions and new graph topologies, while many physics-driven approaches are confined to Euclidean grids and  poorly scale to complex graph structures. To address this gap, we propose PhySTA, a physics-inspired spatio-temporal learning framework designed for efficient and scalable arbitrary inference over graph-structured data. PhySTA integrates two key modules: (1) Continuous Operator-based Spectrum-Temporal Learning (CoSTL), which leverages a Graph-Time Fourier Neural Operator combined with Time-Gated Spectral Segmentation Perception to model continuous dynamics in operator space, and (2) Adaptive Multi-scale Interaction (AMI) that constructs multi-scale subgraphs and introduces node-edge coupled convolution to capture discrete interaction patterns and refine continuous predictions. By bridging operator learning with node-edge-graph interaction, PhySTA achieves both continuity-aware dynamic modeling and hierarchical interactive refinement. Extensive experiments across large-scale benchmarks demonstrate that PhySTA attains state-of-the-art accuracy while reducing computation cost and lowering parameter overhead.

Published as a conference paper at ICLR 2026
ENABLING ARBITRARY INFERENCE IN SPATIO- TEMPORAL DYNAMIC SYSTEMS: A PHYSICS-INSPIRED PERSPECTIVE
Yan Ge Zhengyang Zh, Qihe Huang', Yuxuan Liang", Yang Wang University of Science and Technology of China (USTC), Hefei, AnHui, China
Suzhou Institute for Advanced Research, USTC, Suzhou, Jiangsu, China
Hong Kong University of Science and Technology (Guangzhou), Guangzhou, GuangDong, China {geyan, hgh J @emall i ustc. edu. en, zzy0929 wew.eteeeeu.........e...e.a..... eduu yuxliang out com, angyan@uste. edu. cn
ABSTRACT
Modern spatio-temporal learning techniques usually exploit sampled discrete ob- servations foresee the future. Actually, spatio-temporal dynamics are continu- ous and evolve continuously across time and space, thus modeling spatio-temporal dynamics continuous space can be long-standing challenge. Existing deep learning architectures often fail to generalize to unseen regions and new graph topologies, while many physics-driven approaches are confined to Euclidean grids and poorly scale to complex graph structures. To address this gap, we propose PhySTA, a mhysics-inspired spatio-temporal learning framework designed for ef- ficient and scalable arbitrary inference over graph-structured data. PhySTA in- tegrates two key modules: (1) Continuous Operator-basee Spectrum-Tempore Learning (CoSTL), which leverages Graph-Time Fourier Neural Operator bined with Time-Gated Spectral Segmentation Perception to model continuous dynamics in operator space, and (2) Adaptive Multi-scale Interaction (AMI) that constructs multi-scale subgraphs and introduces node-edge coupled convolution to capture discrete interaction pattems and refine continuous predictions. By bridg- ing operator learning with node-edge-grajy interaction, PhySTA achieves both continuity-awen dynamic modeling and hierarchical interactive refinement. Ex- tensive experiments across large-scale benchmarks demonstrate that PhySTA at- tains state-of-the-a accuracy while reducing computation cost and lowering pa- rameter overhead.
INTRODUCTION
Spatio-temporal systems represent a class of complex dynamical processes whose states evolve jointly over space and time, underpimning critical applications such urban traffic network op- timization (Zhang et al. 2018), climate forecasting, and environmental monitoring (Chen et al.) 2023 A fundamental challenge in their modeling lies in the discrepancy between discrete sen- sor observations and the underlying continuous dynamics, shown in Fig. [ Due to the sparse deployment of sensors and limited sampling frequency. the available data are inherently discrete, whereas the dynamics of real-world systems unfold over continuous apatio-temport fields. This discrepancy complicates the reconstruction of system-level dynamics and restricts reliable inference in unobserved regions, ultimately limiting the applicability of spatio-temporal inference models.
From the perspective of spatial domains, apatio-temport systems can be broadly categorized into Fuelidean systems, exemplified by fluid velocity fields, and Mon-Fuclidean systems, exemplified by traffic flow graphs. Neural operators (Lu et al.; 2021 Li et al. 2021 Kovachki et al., (021) have shown universal approximation capabilities in learning operator mappings over function spaces, abling effective modeling of continuous dynamics. However, such methods are limited to Euclidean domains. In contrast, Graph Neural Networks (GNNs) have emerged the mainstream approach
"Zhengyang Zhou and Yang Wang the corresponding authors.

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Published as a conference paper at ICLR 2026
(n) Temporal Continity of Real World Signals
Spatial Continuity Traffic Flow
Sample Time
Temporally Discrete Observations
Spatially Diserete Traffie Limited
Figure I: Figures (a) and (b) illustrate continuous real-world signals, whereas (c) and (d) depict their temporally and spatially discrete observations, emphasizing the challenge of bridging continuous gaps in both dimensions.
for noc-Euclidean spatio-temporal systems (Li et al. 2018 encompassing static structural mod- els (Guo et al. 2019), diffusion-based methods, and dynamic graph models (Bai al. 2020 Wu et al. (2200). While these approaches capture node-level dynamics, they often struggic to explicitly model node-edge coupling within graph topologies and rely heavily on network depth capture multi-scale interactions, leading to inefficiency and performance bottlenecks.
Taken together, existing studies still fail to simultaneously address continuous evolution modeling and discrete interaction learning, thereby limiting predictive accuracy and generalization to unseen regions. To overcome this gap, we identify two core tasks for continuous spatio-temporal model- ing on graphs: (1) graph neural operator construction and spectral modeling, which extends the continuous modeling capacity of neural operators to graph-structus spectral-mone mode learning for dynamic representations, and (2) enhancement of graph neural interac- tions, which refines operator- driven inference by capturing discrete multi-scale node-edge dynamics while alleviating the computational burden associated with deep GNN stacking.
data and enables efficient
Inspired by the physical principles of operator theory for continuous dynamical modeling (Karni- adakis al. 2021 Lu et al. (02T) and multi-body gravitational interactions (Shabana 2020 Gold- stein et al. 2002), we propose PhySTA, a Physics-inspired for Arbitrary Inference. PhySTA instantiates non-Euclidean systems spatio-temporal graphs, and integrates the continuous modeling of neural operators with the discrete interaction adaptivity of graph neural networks, enabling efficient and generalizable arbitrary inferences.
Learning framework
Concretely, we first construct Graph-Time Fourier Neural Operator (GT-FNO) based on the mag- netic Laplacian, extending the neural operator paradigm directed graphs. To balance accuracy and efficiency, we introduce Time-Gated Spectral Segmentation Perception, which adaptively as- signs independent or shared parameters across spectral bands depending on stationarity, and applies time gating to dynamically reweight spectral components. For graph-structured interactions, we design Node-Edge Coupled Convolution to capture the joint effects of node and edge features dur- ing message passing. Further, inspired by multi-grid solvers, we build three-level coarse- coarse-mid-fing mid-fine subgraph hierarchy that enables efficient multi-scale interaction learning within a single layer.
Our main contributions are summarized as follows:
Continuous Learning for Spatio-Temporee Systems: GT-FNO extends neural operators to non-Euclidean domains, while Time-Gated Spectral Segmentation Perception learns spectral dynamics and temporal evolution, achieving universal approximation of graph- based continuous operators.
Node-Edge Coupled Multi-Scale Graph Convolution: A unified mechanism that cou- ples node and edge dynamics with multi-scale subgraphs, enabling single-layer modeling of cross-scale interactions and mitigating continuous inference errors.
Efficient Arbitrary Inference: Comprehensive evaluation on acage-scale traffic and air quality benchmarks demonstrates that PhySTA achieves tate-of-the-ar accuracy with sig- nificantly reduced computational cost (up to 74.6% fewer FLOPs) and parameters, effec- tively supporting decision-making under sparse sensor coverage.
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2 RELATED WORK
Euclidean Opatio-Temporis Modeling. Classical approaches solve spatio-temporal dynamics by discretizing continuous Euclidean domains and numerically integrating physical equations. Rep- resentative methods include the finite difference method (FDM) (Leue 2007), finite element (FEM) [Quarteroni] (008). finite volume (FVM), and spectral methods (Canuto et al. 2007), which enable high-fidelity simulation but suffer from high computational cost, complex meshing, and entangled numerical and modeling errors. With the rise of deep learning, CNN-based architec- tures (Zhang et al., 2017; Ye al. 2019) were proposed to capture spatial patterns on structured grids, reducing computational cost but remaining tied to regular meshes. Physics-informen
neural
networks (PINNs) (Raissi et al. 2019 Kashinath et al. 2021) embed PDE residuals into the train- ing objective, supporting both forward and inverse problems without explicit meshing, though they require expensive auto- citiooddfffeeetiation and retraining across PDE families. More recently, neural op- erators (Lu et al. 2021| Li et al. 2021: Kovachki et al. 2021 have been proposed to learn mappings between function spaces, enabling generalization across equations and resolutions. Representative examples include DeepONet, which employs branch-trunk architecture jointly encode func- tion inputs and query coordinates, and the Fourier Neural Operator (FNO), which parameterizes global convolution kernels in the spectral domain. Despite these advances, these approaches remain largely confined to Euclidean spaces and face challenges in handling non- non-Euclidean domains such as graphs, as well as in mitigating multi-scale accumulation errors in large-scale continuous modeling.
Modeling. In non-Euclidean domains, graph neural networks (GNNs) have become the dominant paradigm for capturing complex topological structures, with widespread applications in traffic forecasting and environmental monitoring. Early models, in- cluding STGCN (Yu et al. 2018), DCRNN (Li et al.] 2018), GWNet (Wu al.] 2019). and AST- GCN (Guo et al. 2019), combine static graph convolutions with temporal encoders (e.g., recurrent units, diffusion modules, or attention) to capture spatio-temporal dependencies. To address evolving topologies, dynamic graph models have been developed: DGCRN (Li et al.] (023) jointly learns latent graphs and node dynamics, DSTAGNN (Lan et al. 2022) infers varring adjacency ma- trices, and D2STGNN (Shao et al. 2022b) disentangles spatial and temporal components via hierar- chical modeling. More recent works emphasize temporal continuity and expressive power, such as AGCRN (Bai et al. 2020) with node-specific attention, STTN (Li & Zhu [021} combining Trans- formers with spatial convolution, and STG-ODE (Fang et al. 2021 reformulating graph convolution continuous-time ODEs. Beyond GNNs, alternative paradigms have emerged, including genera- tive modeling (Wen et al. 2023), multi-task learning (Yuan et al. 2024). domain transfer (Liu et al.) 2025), and large-scale self-supervised pretraining (Zha al. 2024 Ma et al. 2024). Nonetheless, they largely operate on discrete representations, leading to suboptimal performance in continuous non-Euclidean scenarios. Moreover, most graph-based methods focus on node propagation patterns while underrepresenting node node-edge interactions, which constrains generalization in dynamic hetero- geneous environments. This reliance on deep stacking for global pattern learning further hampers their compatibility with continuous modeling methods, exacerbating both accuracy and efficiency bottlenecks when addressing multi-scale error accumulation.
Non-E Non-Euclidean
METHODOLOGY
3.1 PRELIMINARY
Problem Statement. Given a spatiotemporal dataset, we extract a temporal window of length T,Xi. = (X.) where each X, RR- represents node features over a graph with adja- cency matrix A E NxN and C-dimensional variables per node. Let M denote the continuous spatial manifold that includes both observed and unobserved locations.
Our goal is to learn a continuous operator & that, given the historical observations X and graph structure A, predicts signal values at arbitrary spatial positions s E M and future timestamps (T + 1.......TT +
f(s, t) (
(1)
where (s,(t denotes the predicted signal at (possibly unobserved) location and future time f.
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3.2 THE OVERALL FRAMEWORK
In this section, we introduce PhySTA, a physics-inspired spatio-temporal learning framework de- signed for continuous modeling and arbitrary inference on graph-structured domains. At its core, PhySTA integrates three complementary components. First, the Graph Time Fourier Neural Op- erator (GT-FNO) employs Time-Gated Spectral Segmentation Perception approximate the so- lution operator of the target spatio-temporal system in the spectral domain. Second, the Adap- five Multi-scale Interaction (AMI) module, inspired by multi-body dynamies, captures discrete node edge coupled interactions across multiple scales to refine both global and local dependencies. Finally, Continity-Diseretennn
Interaction Module (CDIM) unifies CoSTL for continuous infer- ence and AMI for discrete refinement, supporting generalization to unseen regions and correcting long-horizon operator errors.
Continuous Spectrum- Temporal Learning (CoSTL) Graph Time Fourier Ncural Openator
Arbitrary continnous inference
Sparie-smppm postiennl eooding
Ciraph modee
Speztral Segromtiion Peroeption
GFT
FFT
IFFTRICFT
am
Time-aatc
YeaSTL
Comimuiy-
Discreteness
Time-miked Graph emboldiig
Mulliwcale Sabsrppt Pattrion
Modale
Yae
Mahti-naizal Aggpegztion
Distrate
Fire
mid
Coans
O4
Time-aryyngg graph prodict and inference
Adaptive Multi-scale Interaction (AMI)
Glildd Eiteaalle
Figure 2: Overview of the PhySTA framework. Taking historical node observations X and the adjacency matrix A inputs, the Continuity-Disereten Interaction Module (CDIM) fuses the intermediate outputs [YCSTTL, YAMI] to yield the final prediction Y.
3.3 CONTINUOUS SPECTRUM-TEMPOR MODELING
The paradigm of Fourier transforms with dimnessinnii mappings has been proven enable continuous operator approximation learning on Euclidean spaces. GT-FNO extends this paradigm to graph domains by transforming spatio-temporal signals into the spectral space. To capture dy- namics more effectively, time- time-gated gated spectral segmentation perception partitions modes into different frequency bands, while temporal gating modulates their evolution. The spatiotemporal embedding follows ST-FNO(Cao al. 2224),, explicitly providing high-frrquenncy multiscale spatiotemporal basis functions that help the network capture fine details and temporal evolution.
Graph-Time Joint Spectral Decomposition. Given graph signal XI:7s first perform the Graph Fourier Transform (GFT) on the graph domain using the complex eigenvectors (1y) of the magnetic Laplacian encode directional dependencies (for undirected graphs the vanilla Laplacian):
NV Σ (y(n) (n, 1=
X Xofelg,t)
(2)
This transformation maps the representation from the node domain (n. to the graph spectral do- main .) Subsequently, 1D Fast Fourier Transform (FFT) applied along the temporal axis for each graph mode:
T X Ngtftlg,w) w) Σ Xyfelg,t) - int
(3)
t=1
This step converts the representation from the time domain (q,t) to the temporal frequency do- main (q. Such a joint spectral embedding serves as the foundation for spatiotemporal operator construction.
Time-Gated Spectral Segmentation Perception. Let X E R denote the vector of Laplacian eigenvalues, where K is the number of retained spectral modes. Each eigenvalue-eigenvecoo pair
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corresponds to graph mode in the spectral domain. Spectral modes are first partitioned into negative and positive sets:
Ineg { A 0), Ipms [k A 2 03.
(4)
Within Ipa,, modes are further divided into low, mid, and high bands based on cumulative energy or predefined thresholds (so, 81 (e.g.. (0.1, yielding Tmid. and Inigh-
GT-FNO applies differentiated parameterization across frequency bands. For low frequencies Tow each mode is assigned an independent kernel W to preserve expressive capacity. Conversely, to capture fine-scale variations while controlling parameter complexity, modes within the negative, mid, and high frequency bands share a band-specific kernel (Waeg, W/mid and Whigh respectively) scaled by an additional learnable per-mode factor ag- For a detailed analysis of the relationships between dynamical components and spectral bands, see Appendix A.4
The spectral segmentation perception is expressed as:
Whow (gttfelk,w), Oz (Whand(kk) X (k))),
k E Ioo
Xeep w)
(5)
k E Imeg U Imii U Inigh:
where W/bud(() {W"eg, W Whigh corresponds the kernel assigned the specific frequency band containing mode k. This low-rank piecewise approximation preserves strong expressivity for long-term low-frequency trends while significantly reducing parameters in the mid and high fre- quencies, balancing stability and efficiency.
To enhance the modeling of non stationary dynamies, we introduce a time- sateeggtting mechanism on the spectral perception outputs. Let h(t) denote the absolute temporal embedding of the input sequence, which is transformed into the frequency domain h(w) via ID FFT for alignment. First, gating factor g(w) is generated based on this temporal embedding:
g(w) = (Wgh(w) + by).
where denotes the Sigmoid activation function, and Wg. by are learnable parameters. Subse- quently, this gating factor is applied uniformly across the spectral segments to modulate the features:
Xtgsep (ki w) = g(w) (k,
This allows adaptive modulation of each frequency band along temporal frequencies to better capture non-stationary evolution.
Continuous Reconstruction. After bandwise filtering and temporal gating, the processed spectral coefficients are mapped back to the spatiotemporal domain by sequential inverse transforms:
Yessn Σ>
Σ
Xigsop (k. (n) eint
k=l=l
where yok denotes the eigenvectors of the magnetic Laplacian that capture graph structural harmon- ics, and dhi represents the temporal Fourier basis modes.
The Fourier transform projects the input field onto continuous bases in the spectral domain, where a finiteddimensional truncation yields computable set of spectral coefficients. The learned operator is then mapped back to the spatiotemporal domain via inverse transforms. Owing to the completeness of Fourier bases in L spaces and their inherent ability to encode global dependencies, GT-FNO effectively learns continuous operator approximations, rather than discrete function mappings as in conventional neural networks. The approximation error primarily arises from spectral truncation and finite parameterization. To mitigate this, we introduce an Adaptive Multi-Scale Interaction (AMI) module, which models multi-scall discrete interactions and compensates for errors induced during continuous prediction. Detailed proofs of continuity and the algorithm process are provided in Appendices A.2 and A.3
3.4 ADAPTIVE MULTI-SCALE INTERACTION
To mitigate the cumulative errors introduced by operator learning in continuous inference, we in- corporate multi-scale discrete interactions through graph networks. Conventional graph convolu- tion fixes interaction weights by static edges, limiting the modeling of dynamic and heterogeneous
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dependencies. Inspired by gravitational physics, node edge coupled convolution jointly encodes node features and edge attributes to generate adaptive interaction weights. For long-range depen- dencies, multi-scale subgraph module constructs coarse- and fine-scale subgraphs and fuses their convolutions, enabling efficient global-local interaction modeling within a single layer. The time- mixed embedding follows STID (Shao et al. 2022a), offering joint spatiotemporal basis features that strengthen the model's ability to capture evolving cross-time correlations.
Adaptive Edge-Node Coupled Graph Convolution.
For each edge (i.j) E E with attribute eij and neighbor node feature By E IR:::
%
Big MLP ( E R" hig Biy] MLP ug));
Edge Altribule
MLP
(9)
O
By
O
Node Feature
Edgx-Nods Compter Weight
where Tig. By R are FiLM (Feature-wise Linear Modulation) coefficients for a node- conditioned setting. Each edge dynamically modulates messages based on both edge at- tributes and neighboring node features.
Py
Edge-Node
Modulation Feature
Way
Edge Nade Coupled Aggregation
Glagh Agaammcy
During message passing, neighbor features are modulated maj = Tij ④ My + BBj, and node states are updated via:
Figure 3: Edge-node coupled graph convolution. Edge attributes and node features jointly produce parameters [7ay,B3) to modulate messages mijs which are aggregated to update node representa- tions v..
// = Σ miy), JEN(()
(10)
where a is an activation function (c.g., ReLU).
Because By encodes local or historical context, the FiLM parameters "aj and B.y evolve dynamically, endowing the convolution with spatiotemporal adaptivity. This design preserves edge-driven influ- ences while incorporating node features, achieving heterogeneous, anisotropic, and context-sensitive message modulation.
Multi-scale Single-Layer Long-Range Modeling. The multi-scale subgraph module begins by applying community detection to partition the original graph. A coarse graph, composed of induced nodes (community-level abstractions), is used to capture cross-community interactions; the middle graph preserves local interactions; and fine graph, obtained by superimposing community edges onto the original graph, models global interactions. By integrating node-edge coupled convolutions across these coarse, middle, and fine subgraphs, the module enables simultaneous learning of global trends, regional dependencies, and local refinements within a single forward pass.
Let the original adjacency matrix be A E RNNN clipped to non-negative values for metric con- sistency. We maximize modularity, Q 1
2m Σis (A.s
Rake 5(c,, ey) via Louvain community 2m
detection and select the highest-degree node in each community an inducing point. This defines a mapping node2center u for every E C...
We capture global, local, and integrated propagation patterns via three-scale graph design. At the coarse scale, which models global trends, each community selects a high-order representative node. The original edge weights Aij are aggregated into induced edges Açcarse [u, v). and an edge-node dynamic convolution on this downsampled edge set Faowa produces the coarse-scale features N At the mid scale, designed to capture local interactions, we strictly preserve the original adjacency (Amia = A) and apply the same convolution on Emia to extract mid-scale features Xmia- Finally, at the fine scale, which provides integrated refinement, the coarse edge weights are redistributed back to the original node pairs. These weights are then sparsified via a Top-K operator and combined with the original adjacency to form Amne- A final convolution on Enne yields the fine-scale features X seamlessly merging global semantics with local structural details.
Multi-scale aggregation. Finally, we concatenate the original features with the representations from coarse, mid, and fine scales:
YAMI Concat( (1:3, Xeoure: Xmid, Xine) E (C+CHH)
(11)

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This unified representation captures comprehensive spectrum of graph signal behaviors from now-freuueccy global trends to high-freueeccy localized variations. The resulting three-scale struc- ture (Acoarses Amid, J. integrated with node-edge dynamic convolution, enables hierarchical, cross-scale, and tunable spatial modeling in a single layer. It down-aggregates long-range depen- dencies, smooths intermediate interactions, and up-refines global crrantics-offeriin strong utility for large-scale heterogeneous graph prediction.
3.5 ARBITRARY INFERENCE
Finally, our model integrates the two complementary outputs. The initial prediction from the Time- Gated Spectral Segmentation Perception, YCSTLL represents unconstrained extrapolation, while the correction signal from the AMI module, YAMI, encodes adaptive refinement based on known ob- servations. These outputs concatenated and fused via the proposed Continuity Continutty-Discreeeess Discreteesss Interaction Module (CDIM):
Ý = (YCOSL YAMI).
(12)
Specifically, CDIM is implemented a multi-layer perceptron, CDIM(2) W2 (Dropout(c(W b1))) + b2, where W1 E 1 ddea and W2 E Re xdau, ø is a nonlinear activation function, and 2 (CCSTL YAmi] concatenates the coarse predictions and adaptive refinements.
Unlike simple gating, CDIM provides higher flexibility by learning nonlinear correction functions (Hornik al.) 1989), thereby avoiding the fragmentation of predictions between continuous extrap- olation and discrete correction. Together, this formulation reflects a cohesive coarse-to-fine rea- soning path: CoSTL performs structure-aware spectral decomposition and inverse transformation to generalize across arbitrary target nodes in continuous space, while AMI adaptively refines these predictions through multiscale node-edge dynamic convolution over a hierarchical graph.
4 EXPERIMENTS
In this section, we present extensive experiments conducted on three real-world datasets to evaluate the proposed PhySTA across various levels of inference and prediction tasks.
4.1 EXPERIMENTAL SETTINGS
Datasets. We evaluate our model three real-world benchmarks for traffic and air quality forecast- ing: PEMS-BAY (Chen et al. 2001). SD (Liu et al. 2023), and Know (Wang et al. 2020). PEMS- BAYJcontains 325 sensor readings of highway speeds in the California Bay Area sampled every minutes, capturing rush hours, holidays, and incidents. SDJis subset of LargeST, spanning diverse urban regions with heterogeneous transportation structures (highways and local roads). KnowAif provides hourly air quality data from dense monitoring networks, including pollutants (e.g.. PM NO,) and meteorological covariates (e.g., wind, temperature). All datasets are chronologically split into training, validation, and test sets under a strict 6:2:2 ratio (Li et al. 2018), ensuring no temporal leakage.
Baselines. We compare PhySTA against representative approaches from three categories: (1) Static graph models: STGCN YYu al. 2018), GWNet (Wu al. 2019), ASTGCN (Guo et al. 2019: (2)
Dynamic graph models: DGCRN eL 2023), DSTAGNN (Lan et al. (022), D2STGNN (Shao
et al. 2022b); (3) ODE-lattention based models: AGCRN Bai et al. 2020), STTN (Li & Zhu
2021 and STG-ODE (Fang et al., 2021). These cover mainstream paradigms of spatio-temporal forecasting. We adopt the best-performing configurations and official implementations of each base- line for fair comparison.
Evaluation Metrics. Following prior work, we report Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Mean Absolute Percentage Error (MAPE). MAE captures average de- viation, RMSE emphasizes large errors, and MAPE provides percentage- based interpretability. To- gether, these metrics offer comprehensive evaluation of forecasting accuracy and robustness.
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Table 1: Inference performance across different datasets under varying missing rates. The best and second-best results are highlighted in bold and underlined, respectively.
Mask=D
Mask=D.3
Mask-0.5
Mask=0.77
Dataset
Method
RMSE
AGCRN
21.38
0.60
33.63
27.11
0.91
39.61
31.94
1.15
44.39
36.91
1.39
48.58
ASTGCN
21.67
34.96
42.63
31.47
DGCRN
42.04
1.27
59.21
39.53
1.33
52.26
43.69
1.58
57.16
$1.37
1.90
64.04
GWNET
20.70
0.57
33.19
24.41
0.75
37.63
27.45
0.90
40.12
30.87
1.07
43.94
KnowAir
LSTM
21.87
0.61
34.60
25.05
0.74
39.78
27.18
0.84
42.25
29.05
0.90
45.02
STGCN
20.60
0.55
32.97
24.69
0.76
37.51
27.36
0.89
40.87
29.75
0.97
44.23
STGODE
21.55
35.38
26.41
39.21
29.
STTN
20.99
0.49
35.30
23.68
0.67
37.56
26.84
0.86
40.10
29.37
1.01
41,91
PHYSTA
33.05
22.89
36.63
27.19
AGCRN
3.64
2.86
3.60
4.39
ASTGCN
189
0.04
4.03
3.86
0.09
7.40
6.53
0.13
10.23
5.44
0.12
8.79
DGCRN
1.85
4.08
2.97
6.13
3.70
7.19
4.40
GWNET
166
0.04
3.63
2.97
0.07
6.22
3.64
0.09
7.26
4.43
0.11
8.23
PEMS-BAY
LSTM
STGGN
188
0.04
4.13
3.91
2.98
0.07
6.27
6.37
4.09
3.78
0.09
7.39
4.98
4.54
0.11
0.12
8,36
STGODE
1,71
0.04
3.72
2.92
0,07
6.16
3.72
0.09
7.47
4.42
0.11
8.43
STTN
3.66
2,80
PHYSTA
1,66
0.04
3.61
2.75
0,07
5.85
3.52
0.09
7.04
4.25
0.11
8.19
AGCRN
21
0.17
34,90
64.24
1.11
110.73
87.96
1.54
130.37
116.23
2.15
154.52
ASTGCN
28.18
81.89
120.48
DGCRN
40.66
0.37
57.39
68. 08.80
0.76
101.63
82.56
0.89
117.54
110.09
1.19
145.97
GWNET
21.74
0.15
34.30
58.21
98.25
79.97
1.17
120.50
106.50
1.54
145.52
SD
LSTM
29.10
0.18
45.37
96.91
1.08
113.23
106
1.89
153,93
113.53
1.67
154.56
STGCN
22.33
0.17
36.61
60.58
0.80
103.38
86.03
1.21
130.27
111.05
1.67
152. 14
STGODE
23.41
37.34
62.87
134,08
117.69
1.38
STTN
22.40
0.15
34.85
55.73
0.76
92.03
82.95
1.34
122.23
108.46
1.75
144.34
PHYSTA
33.05
50.34
91.65
77.64
119.43
96.09
134.59
4.2 PERFORMANCE COMPARISON
To evaluate the inference ability of PhySTA, simulate unobserved regions during training by ran- domly masking a subset of node features. At test time, the same nodes are masked in the input while ground-truth labels remain fully available, enabling fair assessment of prediction under incomplete observations. Table rrppors the MAE, MAPE, and RMSE averaged over 12 forecasting horizons.
From Table T PhySTA consistently achieves the best or second-best results across datasets and missing ratios. On KnowAir, it attains the lowest MAE under all masking conditions, outperform- ing strong baselines such as STTN and STGCN, especially high sparsity (Mask = 0.7). On PEMS-BAY. PhySTA maintains stable accuracy even with severe missingness (Mask 0.3, MAE 2.75, RMSE = 5.85), surpassing advanced graph-based models including AGCRN and DGCRN. On the large-scale SD dataset, it demonstrates superior robustness at low-to-moderate missing ra- tios, reducing RMSE by over 5% compared to the next-best baseline while remaining competitive under extreme sparsity. In contrast, traditional models (c.g.. LSTM. STGCN) degrade substantially at higher missing ratios, and advanced GNN basclines (c.g., DGCRN, STGODE) show fluctuating performance under sparse inputs. Overall, PhySTA's physics-guide spectral operator and adaptive multi-scale interaction enable stable, accurate predictions across diverse spatiotemporal patterns and sparsity levels, confirming its robustness and generalization for arbitrary inference in real-world spa- tiotemporal systems.
Efficiency Analysis. We theoretically derive the computational complexity of PhySTA and com- pare parameter count and GPU memory con-
Table 2: Efficiency comparison of models.
Model
Complexity
Parameters
GPU memory (MB)
sumption against representative baselines. As
AGCRN
O(dnL)
760,580
11.140
shown in T the time complexity of PhySTA is I.) where d denotes the feature dimen- sion, n is the number of graph nodes, and represents the number of integration steps (typ- ically L < n). From Table 2 we observe that PhySTA uses only 123,474 parameters and re- quires 6,042MB GPU memory, achieving a fa- vorable trade-off between computational complexity and resource consumption. This demonstrates
ASTGCN
O(fnL)
2.153.034
11,028
DGCRN
O(PrL) 242,849
15.058
GWNET
O(PrL)
LSTM
OldLL)
97,932
9,864
STGEN
O(PaT)
507,532
STTN
O(d,P
113,740
18,864
STGODE
O(PaL.
729,228
18,864
PhySTA
O(PrL)
123,474
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that integrating boysics-inspired continuous modeling with multiscale message passing yields sub- stantial performance gains within an acceptable memory footprint.
4.3 ABLATION STUDY
We perform a component-wise ablation study on the PEMS-BAY dataset under two masking ratios (Mask = 0 and Mask = 0.5) to evaluate the contribution of key modules-Time-Gine Gated Spectral Segmentation Perception (TGSSP), GT-FNO. Edge-Node Coupling Convolution (ENCC), Multi-Scale Graph Convolution Net- work (MSGCN), and the Multi-scale Interac- tion Refinement module (AMI).
Table 3: Ablation study of prediction and inference(mask=0.5) ability on PEMS-BAY
Methods
Misk =
Mask 0.5
MAE
MAPE
RMSE
MAE
MAPE
RMSE
wof TGSSP
1.80
0.05
3.91
4.12
0.11
8.24
wol GTFNO
2,05
0.06
4.46
5,03
0.14
10.06
wéo ENCC
0.04
0.10
wob MSGCW
1.79
0.05
3.89
3.96
0.11
7.92
wo AMI
1.88
0.05
4.09
0.12
8.26
As shown in Table removing TGSSP (w/o
TGSSP) leads to minor drop: MAE increases from 1.66 to 1.80 and 3.52 to 4.12, with corre-
sponding MAPE rising from 0.04 to 0.05 and 0.09 to 0.11. Excluding GT-FNO (w/o GTFNO) causes more pronounced degradation, with MAE reaching 2.05 and 5.03, and MAPE 0.06 and 0.14, highlighting its role in capturing spatio-temporal dynamics. Without ENCC (w/o ENCC), MAE increases to 1.73 and 3.87, and MAPE 0.04 and 0.10, confirming the benefit of explicit edge-node coupling. Omitting MSGCN (w/ MSGCN) further reduces performance, particularly under Mask 0.5 (MAE 3.96, MAPE 0.11), showing the necessity of multi-scale sub-graph modeling.
Finally, removing AMI (w/o AMT) results in the largest drop, with MAE rising to 1.88 and 4. 4.,,, and MAPE to 0.05 and 0.12. demonstrating that adaptive multi-scale interaction refinement is cru- cial for robust prediction under incomplete data. A detailed hyperparameter sensitivity analysis for CoSTL and AMI (KnowAir, PEMS-BAN (AYY provided in Appendix A.7
4.4
CASE STUDY
Tup-5 Active Medes in HIGH Band
Leamod Gating Strslegy Frequaccy Band
075
Mak
Maki
09
Scalng Packrro
Mak:
Actiratton
2.70
300
150
300
353
300
Imainng Step
N80
LOW
Mil
MiGa
(a) HIGHH-band og dynamics (training).
(b) Band-wise gating distributions (evaluation).
Dynamic Response Burst (High Band)
Temporal Focus of Convolution Kemels
lopa Enege &o Activatie
3.5
10W
Special Energy fl the)
tneegg
3.0
IGG
Vabae
1.
10
03
Step
Dynamic gating response (evaluation).
(d) Temporal kemeel focus (evaluation).
Figure 4: Visualization of TGSSP's spectral specialization and temporal adaptation mechanisms.
Spectral Specialization and Temporal Adaptation. To dissect the internal learning mechanism of TGSSP, analyze its two key parameter cooups-mode-wice scaling factors (ogg) and temporal

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Published as a conference paper at ICLR 2026
gating activations- and examine how they jointly shape and inference. As shown in Figure a)), which visualizes trajectories during training, ag exhibits a progressive specialization effect: informative high-band modes are consistently amplified while less relevant ones decay, revealing a competitive "spectral selection" process. In contrast, Figures H d), derived from the evaluation phase, expose the model's s stabilized temporal modulation strategy- The gating distributions in Figure 4bb show that low-frequency modes adopt nearly uniform pass- through policy to preserve global trends, whereas high-frequency modes exhibit high-variance selective pattern tuned for local irregularities. This behavior aligns with the dynamic response in Figure [ where gate activations spike synchronously with transient spectral bursts and decay as the signal stabilizes, effectively functioning as a spectral-domain attention mechanism that enhances signal-noise contrast. Finally, the temporal kernel profiles Figure indicate differentiated tem- poral receptive fields across frequency bands, confirming that TGSSP learns band-specific temporal focus indispensable for capturing multi-scale, non-stationary dynamics.
behavior across training
(b) Arbirars Inference and Preditiio
(a) Hisurical inpuis makavwn moles colers speed
Mubad de
(d) Performance Exalluaion on Bath Known and Uskooma Locations
(c) Ground Truth
Figure 5: Visualization of PhySTA's spatiotemporal prediction and inference capabilities trans-
portation datasets.
Arbitrary Spatiotemporal Inference. To demonstrate the arbitrary inference capability of PhySTA, we randomly mask 30% of the sensors, requiring the model to infer and predict traffic flow at both observed and unobserved locations. Specifically, the input spans the time interval from 16:50 to 17:40, during which 30% of the nodes are masked and treated as unknown regions. These unobserved nodes are visualized as white circles on the map. The model is then tasked with forecast- ing traffic flow for the subsequent period, from 17:55 to 18:40. As evidenced by the visualization comparison and the quantitative MAE evaluation against the ground truth, PhySTA accurately cap- tures the underlying traffic dynamics and successfully performs arbitrary spatiotemporal inference over non-Euclidean domains, demonstrating strong generalization under partial observability.
5 CONCLUSION AND LIMITATIONS
In this work, we proposed PhySTA, a physics-inspired framework that integrates a Graph-Time Fourier Neural Operator with adaptive multi-scale interaction module. PhySTA enables con- tinuous dynamics modeling on graph-structured data and performs multi-scale error correction, ul- timately supporting arbitrary-region inference. Experiments on real-world benchmarks for traffic forecasting and air quality monitoring demonstrate that PhySTA not only outperforms prior meth- ods but also achieves significant improvements in runtime efficiency and memory usage. Even under highly sparse sensor deployments, PhySTA can perform cross regson arbitrary inference, making it particularly suitable for real-time prediction and decision support at both city and national scales.
Limitations and Future Work. Despite these promising results, the GT-FNO component still relies on spectral decompositions (e.g.. magnetic Laplacian). which can be computationally de- manding very large graphs Future work could explore graph-adaptive compression cross- modal alignment techniques to mitigate these issues. Further, the scalability and generalizability of PhySTA to broader spatio-temporal domains, such climate systems, power grids, or epidemiology. are highly required and remain as future work.
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6 ETHICS STATEMENT
This work adheres to the ICLR Code of Ethics (httss/icciaccaabbiccCadooCfhhh Our research focuses on the development of graph-time neural operators for spatiotemporal modeling and does not involve human subjects, sensitive personal data, or experiments with potential direct harm. All datasets used in our experiments are publicly available or have been anonymized, and no identifi- able information is included. We have considered potential biases in data and model evaluation, and all results are reported transparently. There are no conflicts of interest or external sponsorship that could influence the research outcomes. We encourage readers to consider ethical implications in the deployment of predictive models on real-world spatiotemporal data, particularly sensitive appli- cations such urban monitoring transportation, where fairness and privacy must be preserved.
7 REPRODUCIBILITY STATEMENT
We have made every effort to ensure the reproducibility of all experiments presented in this work. All model architectures, hyperparameters, and training procedures are described detail the main text and the appendix. All datasets used in our experiments publicly available, and provide links to their sources.
8 ACKNOWLEDGMENT
This paper is partially supported by the National Natural Science Foundation of China (Nos. 62502488, 12227901, and 62402414), the Natural Science Foundation of Jiangsu Province (No. BK20240460), the grant from the State Key Laboratory of Resources and Environmental In- formation System, and the Guangdong Basic and Applied Basic Research Foundation (No. 2025A151501 1994). The AI-driven experiments, simulations, and model training were performed on the robotic Al-Scientist platform of the Chinese Academy of Sciences.
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A APPENDIX
A.1 NOTATION
Table 4: Frequently Appeared Notations
Symbol
Description
N, N,T.d
Number of sensors (nodes), historical time steps, and latent width. Historical spatio-temporal observation tensor.
X1.r
g
Graph of spatial relationships.
Weighted adjacency matrix.
Continuous solution operator Graph spectral basis matrix.
y
Final fused spatio-temporal prediction.
sen: As
Graph Laplacian eigenvector and eigenvalue for mode k. Graph Fourier Transform representation. Joint graph graph-time time Fourier Transform representation. Negative and non-negative spectral mode sets. Low, mid, and high-frequency spectral bands. Band energy thresholds.
Nga tk
Xgin(ki, (ki w)
Ineg: Ipon Trww Imid, Thigh (80, 81) Wlow Whband()) an Xesp (kppki,w) w) h(t), h(w) g(w)
Mode-independen and band-shared learnable spectral weights. Mode-wise scaling factor for negative, mid, and high bands. Spectral segmentation perception output.
Temporal embedding and its frequency-domane representation. Time-gating factor.
X Xigusp( (ki, w) YCOSTL (n,
Time- gated spectral output.
Reconstructed continuous spatio-temporal output from CoSTL.
Cijy Egj
Raw edge attribute and its projected embedding.
Vajs Bij
FiLMM scaling and shifting coefficients for edge (i.j).
múj
Modulated message from neighbor node i.
X Xemase, Xmid, Xine
Coarse, mid, and fine-scale node features from subgraphs. Aggregated multi-scale interaction representation.
YAMI
A.2 GT-FNO CONTINUOUS OPERATOR APPROXIMATION ON GRAPHS
We provide formal justification for GT-FNO's ability to learn continuous operators on graph- structured spatio-temporal systems. Let X G x [0. 7' - RF denote a spatio-temporal graph signal defined on a directed graph G with N nodes, where [0, T) is the temporal domain. The target evolution operator is continuous mapping K. LG x [0, TJ) - L [0, We aim to show that GT-FNO can approximate K arbitrarily well.
1. Spectral Decomposition on Graphs. Let L denote the magnetic Laplacian of G and (xp)) its orthonormal complex eigenvectors with eigenvalues {As}- Any graph signal X(,,t) admits a spectral decomposition:
N X (n, Σ &(, (m),, &(k, = (X(,,),, sx)e.
Similarly, temporal signals expanded via Fourier series:
x(k,t)
Σ (kk,w)ell
WEnT
where 2r is the discretized temporal frequency set. Combining these, any graph-time signal admits a joint graph-time spectral decomposition:
X(n,t) - Σ Σ
(kk
k=1wefer
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2. Operator Approximation in the Spectral Domain. The target operator K. acts on X
(Kx (n,
Using the spectral decomposition, we have:
N ((n,t) Σ E Y(k,W)pa(n)e a(k,w)) (k.
where F[K]] denotes the induced action of in the joint graph-time frequency domain.
3. Low-Rank Piecewise Approximation via GT-FNO. GT-FNO approximates F(K] using piece- wise spectral mapping:
Wlow (kk on (W/band(k) (k, ((k,w)),
k: Too
(kk =
Too
where W/bund(k) is the shared kernel assigned to the specific frequency band (mid, high, negative) containing mode This is equivalent approximating F[K] by finite-dimensional linear oper- ator in each frequency band, which by standard results in functional analysis constitutes a spectral Galerkin approximation:
|F[ - -F([K[o[ooooola S <(Noow Nmid Nnigh):
where vanishes the number of modes in each band increases.
Universal Approximation Continuous Operators. Based on the universal approximation theorem for Fourier Neural Operators (Kovachki et al. 2021). let K. L2 x [0, - L' x [0, 7)) be a continuous operator. For any > O, there exists piecewise linear Fourier operator K.g such that
KKX <e, VX BR.
where BR is bounded subset of 22 Since GT-FNO implements exactly such a bandwise linear spectral operator, augmented by time-gating which is Lipschitz continuous, it inherits the universal approximation property graph-time signals:
VE O, 30 ||XX - GT-FNO < E.
5. Error Bounds. The total approximation error consists of three components:
|CX -GT-FNO;
|CX - KNN XI [K - Ko -Ko
Kg (X))N
spectral truncation
parameterization crror
time-gating approximation
All three terms can made arbitrarily small by increasing the number of graph/temporal modes and the capacity of the parameterized linear maps, guaranteeing continuous operator approximation on graph-time domains.
6. Conclusion. GT-FNO provides theoretically grounded continuous operator approximation framework on graphs, extending the universality of FNOs to graph- suaph-strutture spatio-temporal data. By leveraging spectral decomposition, bandwise linear parameterization, and Lipschitz continuous time gating. it can approximate any continuous spatio-temporal graph operator with an arbitrarily small 1,2 error.
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A.3 GT-FNO ALGORITHM AND ANALYSIS OF COMPUTATIONAL COMPLEXITY
Algorithm Graph-Timi Time Fourier Neural Operator (GT-FNO) Forward Pass
Require: Input graph signal X ER BBBxxxxXX7
Require:
Graph eigenvectors (+tx). Laplacian eigenvalues
Require:
Absolute temporal embedding h(t)
Require:
Model parameters: latent width d, spectral layers L, retained modes K
Ensure:
Continuous reconstruction output YCOSTI xB
1: Xpe +
2: Xnom - LayerNorm(XX (-Xp)
3: X ppnj f Conv2D (XX X norm. d)
b Lift to latent dimension d
4: X < SpectralConv
pmj- {vFt))
5: Xmlp < PointwiseFFN (attem)
6: X & [XXpoo + +Xmpp)
D a is a nonlinear activation Graph Fourier Transform to (k, t) domain ID FFT to joint spectral domain (k, w) Transform temporal embedding to frequency domain
7: Xgtt & GFT(Xiifico {spx))
8: Xgmn f FFFTime (Xga)
9: h + R+FFFTimm FFT (h)
for 1= to I. do
11:
Xuu t- -peetralSemeenaiinperccpttntn (Xgn. A)
Band-wise parameterization
12:
g+ *tooWghh
bg)
Generate gating factor Time-gated modulation
13:
Xussp f 9 @ Xssp
Xgttt Xissp
Update representations for the next layer
15: end for
Y
Yúue
IFFT lime (X (sss))
Inverse temporal FFT back to (k, t) domain
17: if { is complex then
18: Yhack f- IGFT((Ymme:...... ((xx))
D Inverse GFT back to continuous spatiotemporal domain
19: else
20:
Yback f Real(IGFT(Y time: {4pk
21: end if
22: YCOSTL < LinearProjection Ybuck, Cout, Town)
23: return YCOST.
Component Explanation: The forward pass of GT-FNO first encodes the input spatio-temporal signal using space-time positional encoding and normalizes it via layer normalization. A convo- lutional projection lifts the input to the desired latent width d, followed by a temporal spectral con- volution (SpectralConvTT) to capture initial correlations. pointwise feed-forward network (FFN) refines the latent features, which are residually combined with the projected features through an activation function. The transformed features then mapped the spectral domain via Graph Fourier Transform (GFT) using the graph eigenvectors &, and further processed using a ID FFT along the temporal dimension. Multiple spectral layers sequentially apply spatial spectral convolu- tion (SpectralConvS) and a time- time-gating mechanism to model cross-scale dynamics. The output is mapped back to the spatio-temporal domain using inverse FFT and inverse GFT. Finally, a linear projection slices and maps the features to the required output steps to produce the final prediction.
Computational and Model Complexity Analysis: The GT-FNO framework involves several key components that contribute to its computational and parameter complexity. Let B denote the batch size, N the number of spatial nodes, T the temporal length of input sequences, Cin/Cout the in- put/output feature channels, d the latent width, and K the number of spectral modes retained in FFT operations.
1. Positional Encoding and LayerNorm: The space-time positional encoding and layer normaliza- tion require O(B Cin TT) operations and introduce negligible additional parameters compared to the main network.
Convolutional Projection and Pointwise FFN: The initial convolutional projection from Cin to channels involves O(B d Cm N T) operations, with parameter count of d Oin A² for kernll size k. The subsequent pointwise feed-forward network, applied per node and per time step, costs O(B N. T. d²) operations and 2nd parameters per FFN layer.
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3. Spectral Transformations: Mapping features to the spectral domain using the GFT involves matrix multiplication with the graph eigenvector matrix of size N x N, yielding O(B d N2 .2.T) operations. Temporal FFT/IFFT operations contribute O(B d N Tlog T) complexity per layer, which is generally lower than the spatial spectral transform for large N.
4. SpectralConvS Layers: Each spectral convolution layer retains K modes, incurring O(B K-
T) operations for linear transformations in the spectral domain, and 2d K parameters per layer. Time- gatieggttng mechanisms introduce element-wise operations with O d N complexity and negligible parameters.
5. Overall Complexity: Combining all components, the total forward pass complexity is dominated by the spectral graph transform, yielding ----dN---++--22 B.C. -T) operations, where L is the number of spectral layers. The total parameter count approximately Cin d k2 -df... K + 2d per FFN, which grows linearly with the number of layers and quadratically with latent width d, but fundamentally independent of N due to mode truncation in the spectral layers.
Efficiency Comparison. By performing the graph Fourier transform via magnetic Laplacian cigendecomposition (complexity O(NTT) and ID FFT along the temporal axis (complexity O(NT log T)), we obtain a joint spatio-temporal spectrum. Applying the complex filter Y&(w)= = (A w) (w) has a complexity of (NTT)), but our four-band parameter sharing significantly reduces the parameter count. Although this slightly exceeds the standard FNO's O (NNd) pointwise cost, it profoundly enhances spectral sensitivity and interpretability while remaining far below the PINN's ON + 2 complexity. Thus, our approach extends neural operators to non- non-Euclidean graphs with only moderate computational overhead.
Furthermore, our proposed ndge FiLM convolution (aggregation cost O(EEld), combined with multiscale graph construction via Louvain clustering (O(N log N)) and hierarchical edge ag- gregation, effectively addresses discrete interaction modeling. This single-layer module extracts and fuses coarse, mid, and fine features at Fl d NdP) (where d2 < N). Compared to dynamic graph models with overall complexity O(N + NTd),, our method significantly reduces the computational burden while effectively capturing long-range dependencies. In summary, the com- bination of GT-FNO and the adaptive multiscale interaction graph alleviates traditional bottlenecks, simultaneously enhancing both efficiency and generalization in spatio-temporal modeling.
A.4 SPECTRAL ANALYSIS OF GRAPI-TIME DYNAMICS
Loncloepogy Origl
Levefogeen- Dapooe
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Figure 6: Visualization of graph-time dynamics across different spectral bands. The left panels (Graph View) display the spatial patterns of eigenvectors on the graph structure, while the right panels (Temporal Signal) illustrate their corresponding temporal evolution.
Consider an undirected graph with symmetric Laplacian L UAUT, T, where A diag(Ai,.... and 0 = A S ... S An Let the graph-time linear system be
d,=(1) (t,, [.)x(()) b(t).
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where x(t) RR, and (tt, L) commutes with L (e.g.. polynomial MM am (t) more
generally spectral function (L)). In the graph Fourier domain y(t) U x(t), the dynamics decouple completely:
(t) 1/u Un (t) (x),),
h- I...., n,
with (4) = ue b(t). For b(() = 0, the solution is
yı (1) = yn (0) exp (% ths (2o()e)
so each graph frequency evolves independently, and its temporal behavior fully determined by the corresponding eigenvalue ) through sha (At).
For typical diffusion/heat dynamics (A) (A) with monotone non-negative g(A), we have
yu (t) = (v)( exp (-fo ((s)s((e)dd))
Low-frequency modes (small Xx)) decay slowly, producing smooth, slowly varying graph signals. High- gegg-ffeeuency modes (large Aς) decay rapidly, and the presence observation noise or external input b(t), manifest as fast, jittery variations. Intermediate frequencies decay moderate rates but may exhibit oscillations in second-order or transport-type systems. For example, wave dynamics i+cc Læ 0 decouple as iı + 0 with solutions (t) =
Ag cos(cv /xt)+ Bg sin(c/Ax,t)
showing oscillation frequencies increasing with Vdg-
Discrete-time evolution Tt+1 = S, (L),,, with (L) = - I-)((() gives spectral amplification factors T2 = 1>7G()a)). hence (t) (0). Then: 0 T < corresponds monotone decay (stable low-frequency modes), -1 < rk < 0 to alternating damped oscillations (mid-fres meid-freuuency modes), and |rxl 1 divergence ""negative-frequene
behavior in the discrete-time signal). Note that L remains positive semidefinite; negative signs originate from the discrete-time stepping operator overly large explicit step sizes.
Intuitively, the graph Laplacian eigenvectors are ordered by the quadratic form x. T La: low- frequency modes vary little across edges (smooth patterns), while high-frequency modes fluctuate rapidly along edges. Consequently, diffusion-like dynamics suppress high-frequency modes and preserve now-frequency structure, wave/transport dynamics map frequencies faster oscillations, and discrete-timg stepping may induce phase-flipped or unstable trajectories.
Ti(L)a(0) with T((L) exp (Jo (L))s)) and assume (+x+)) is continuous (or piecewise Lipschitz) on [0, Amax]. Parti- tion [0, Amax] into bands { and upproximate with piecewise linear spectral multipliers TA)) such that
For operator learning, let the target evolution be ar(t)
sup Aee
Ta(A) (e)) S..
Then for any input X,
S 8||21
GT-FNO implements this idea with bandwise linear spectral operators and time gating, enabling controlled approximation of low-frequue wwwfrreuency decay, mid-frequency oscillations, high-frequency jitter, and discrete-time instability within bounded error.
In summary, each graph frequency determines the time scale and stability of its mode: decay rates scale with À under diffusion, oscillation frequencies scale with VA under wave dynamics, and discrete-time time stability is governed by 1 +G(A) Bandwise spectral operators provide principled mechanism to manipulate these modes, yielding interpretable and accurate graph-ting time dynamic representations.
A.5 DATASET DESCRIPTION
Datasets. We evaluate model three real-world datasets from traffic forecasting and air quality monitoring: PEMS-BAY. SD, and KnowAir. These datasets are widely used benchmarks in spatio- temporal learning, each introducing unique structural and temporal challenges. To ensure fairness
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Table 5: Summary of datasets.
Datasets
Range
Nodes
Time Interval
Length
Scene
SD
San Diego
716
35,040
mins
Traffic Prediction
PEMS-BAY
SF Bay Area
325
52,116
KnowAir
China
184
3 hours
11,667
Air quality Forecasting
and reproducibility, we follow the standard chronological segmentation protocol (Li et al. 2018), where the first 60% of the time series is used for training, the next 20% for validation, and the final 20% for testing, thus preventing temporal leakage.
PEMS-BAY is benchmark dataset for traffic speed forecasting in the California Bay Area. It contains 325 loop detector sensors deployed on major highways, recording average traffic speed every 5 minutes. The dataset spans from January to June 2017, covering more than 50,000 time steps in total. This dataset is challenging due to the high variability in traffic conditions, including rush hours, holidays, and unexpected incidents. The sensor graph is constructed using road network distances, resulting in weighted directed adjacency matrix where edge weights decay with geodesic distance.
SD (San Diego) a subset of the LTrgeST benchmark (Liu et al. 2023), focusing on traffic speed prediction in the San Diego region. Unlike PEMS-BAY, SD incorporates more heterogeneous road network, including both highways and local arterial roads, thus introducing a larger variation in spatial structures. The dataset consists of 716 sensors sampled every 5 minutes, spanning four consecutive months in 2021. Its scale and diversity make it suitable for evaluating models' general- ization capability across heterogeneous spatio-temporal patterns, particularly under non-stationary traffic dynamics.
KnowAir is large-scale urban air quality forecasting dataset encompassing 184 main cities across China. For each city, the dataset records the concentrations of multiple air pollutants (e.g., PM2s, NO₂, SO,, O₃) alongside meteorological factors such as wind speed, temperature, and humidity. The observations are sampled every 3 hours and span four-year period from 2015 to 2018, yield- ing over 11,600 time steps per city. Compared with traffic data, air quality signals evolve on slower temporal scales but are heavily influenced by complex physical processes (e.g., wind-driven dif- fusion, chemical transformation). This dataset is particularly challenging for graph-based models because the spatial correlations are highly dynamic and strongly influenced by external meteorolog- ical variables.
Preprocessing. All datasets are normalized using z-score normalization based training set statis- tics. Missing values, which occur sporadically in sensor readings, are imputed using temporal nearest-neighbor interpolation. The input horizon is fixed at 12 time steps (corresponding hour for PEMS-BA. BAYYSSD and 12 hours for KnowAir), and the prediction horizon is set to 12 time steps. consistent with prior work. The final data tensors are formatted as E BBxNxTXF where B is the batch size, N the number of sensors, T the temporal window length, and F' the number of features per node.
A.6 EVALUATION METRICS
To rigorously evaluate the performance the proposed model on both interpolation and extrapola- tion tasks, we adopt several task-specific regression metrics that comprehensively reflect the model's s ability to capture the underlying dynamics of irregular time series. The evaluation metrics are de- tailed as follows:
Mean Absolute Error (MAE). MAE measures the average magnitude of the absolute differences between predicted values and ground truth values, defined as:
N
T
MAE
Viee - Vi.el,
NT
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where N and T denote the number of sensors and time steps, respectively; Vs, is the predicted value, and Vi, is the ground truth. MAE robust to outliers and reflects the average absolute deviation.
Root Mean Squared Error (RMSE). RMSE quantifies the square root of the average squared differences between predicted and actual values, placing greater emphasis on larger errors:
RMSE
(U. vi,z)
NT
RMSE penalizes large deviations more than MAE, making it suitable for scenarios where high- magnitude errors are critical.
Mean A Absolute Percentage Error (MAPE). MAPE measures the average absolute percentage dif- ference between predictions and actual values:
MAPE 100%
ΣΣ
N
T
Me.t
Vi.e
NT
Yi.t
where the division is element-wise. MAPE provides an interpretable, percentage-based error metric but may become unstable when the ground truth values approach zero.
These metrics provide complementary perspectives on forecasting accuracy and robustness under various conditions.
A.7 HYPERPARAMETER SENSITIVITY ANALYSIS
To assess the robustness and adaptability of the proposed CoSTL and AMI modules, we per- form a sensitivity analysis on four key hyperparameters across the KnowAir and PEMS-BAY datasets: the GT-FNO hidden dimension (gfno_hidden) the spectral segmentation threshold (spectral.segmen) the number of global edges per node (topk_edge), and the num- ber of edge-node coupling convolution layers in the multi-scale dynamic graph convolution net- work(MSGCNN, denoted the ECC layer depth (ecc.layer).
MAT Reacrieity
(0000 KIOW 100R CALM BROW COCOT TROW CR0100
UR000 812000 X0000 US00
$2..00
Figure 7: Hyperparameter sensitivity analysis of PhySTA the PEMS-BAY and Know Air datasets.
As shown in Figure 7 larger gfnoshidden values improve MAE on PEMS-BAN BAY, while KnowAir exhibits greater variance, possibly due to overfitting or weaker spatial correlations. The spectral segmentation parameter (spectral segment) shows stable performance across wide range of
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threshold values, demonstrating AMI's robustness segmentation granularity, especially PEMS- BAY. Increasing the number of global edges per node (topk.edge) slightly degrades performance under high masking, which may result from noise amplification in over-connected structures. Fi- nally, stacking more graph convolution layers (i.e., a larger ecc layer) tends to increase MAE on AiowAir, likely due over-smoothing or gradient issues, whereas PEMS-BAY benefits moderately from increased depth.
These results confirm PhySTA's overall stability, while highlighting that architectural depth and global connectivity should be tuned carefully under sparse or irregular input conditions.
Scalability Analysis. The scalability of PhySTA is validated across datasets with diverse spatial scales, temporal resolutions, and domains, including traffic forecasting (PEMS-BA) -AY, SD) and air quality prediction (KnowAir), as summarized in Table 5 PhySTA consistently performs well across small (184 nodes in KnowAir), medium (325 nodes in PEMS-BAY and large-scale graphs (716 nodes in SD), demonstrating adaptability to increasing graph sizes and data volumes.
As shown in Table I PhySTA also maintains strong predictive performance under varying levels of node masking, simulating partial observability. On SD, achieves the lowest MAE of 20.64 with full data and competitive MAE of 96.09 at 70% masking. On KnowAir, PhySTA reaches MAEs of 22.89 and 27.19 at 30% and 70% masking, respectively, and similarly outperforms baselines on PEMS-BAS with scores of 2.75 and 4.25. These results show that PhySTA scales effectively with spatial and temporal complexity while remaining robust under data sparsity, confirming its suitability for real-world deployment in large and partially observable environments.
B USE OF LARGE LANGUAGE MODELS (LLMs)
In the preparation of this work, utilized large language models (LLMs) primarily general- purpose assistive tools for English language polishing, grammar and formatting checks, and for guidance in reviewing background knowledge related to operator learning. The LLM did not tribute to the core research ideas, algorithm design, theoretical derivations, or experimental analyses. All scientific content, claims, and results presented in this paper were generated and verified by the authors. We take full responsibility for the accuracy, originality, and integrity of the content, includ- ing any portions refined with the assistance of LLMs. No LLM was listed an author.
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Reviews:
[{"review_id": "0NUzDvo5nC", "reviewer": "Reviewer_sBzk", "summary": "The paper proposes PhySTA, a physics-inspired framework that unifies continuous operator learning with graph-based spatio-temporal modeling. It introduces two main modules:\n(1) a Graph–Time Fourier Neural Operator (GT-FNO) equipped with Time-Gated Spectral Segmentation Perception (TGSSP) for modeling continuous spectral dynamics on graphs, and\n(2) an Adaptive Multi-Scale Interaction (AMI) mechanism that captures multi-scale node–edge relationships via coupled convolution and hierarchical graph construction.\nA Continuity–Discreteness Interaction Module (CDIM) further fuses both continuous and discrete predictions for arbitrary inference in unobserved regions.\nExperiments on large-scale traffic and air-quality datasets demonstrate strong accuracy, robustness, and efficiency compared with several state-of-the-art baselines.", "questions": "See weaknesses.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Novel integration of physics-inspired operator learning and GNNs:\nThe proposed GT-FNO extends Fourier Neural Operators to non-Euclidean graphs, enabling continuous modeling over directed graphs—a clear conceptual innovation.\n\n2. Multi-scale and coupled graph design:\nThe AMI module effectively captures long-range, multi-level dependencies within a single layer, addressing over-smoothing and inefficiency issues seen in deep GNNs.\n\n3. Strong empirical performance and efficiency:\nPhySTA achieves consistent improvements across datasets with fewer parameters and memory cost (≈74% FLOP reduction), showing excellent trade-offs between accuracy and scalability.\n\n4. Comprehensive experiments and ablation analysis:\nThe inclusion of multiple datasets, mask ratios, and detailed component ablations provides good evidence of robustness and interpretability.", "weaknesses": "* Limited comparison to recent operator-based or physics-informed baselines:\nThe paper mainly compares against classical and GNN-based methods (STGCN, DGCRN, etc.), but omits recent neural operator or PDE-based baselines such as Graph Neural Operator (Li et al., 2023) or Geo-FNO (Li et al., 2024). These would strengthen the claim of operator generalization.\n\n* Writing quality and presentation:\nThe exposition is heavy and sometimes unclear, especially in the methodology section. Some mathematical notations are inconsistent, and figures (e.g., Fig. 2, Fig. 3) are not fully self-explanatory. The authors could simplify and streamline the presentation for readability.\n\n* Ablation and interpretability could be expanded:\nAlthough the ablation table is informative, qualitative insights on how each frequency band or subgraph level contributes to the final prediction are missing. Visualizations of spectral energy distribution or temporal gating behavior would enhance interpretability.\n\n* Scalability limitation not sufficiently addressed:\nThe reliance on magnetic Laplacian spectral decomposition may hinder scalability for very large graphs. While this is briefly mentioned in the limitations, empirical evaluation on larger graphs would make the claim more convincing.\n\n* Incomplete baseline coverage:\nSome recent transformer-based and neural-operator hybrid methods (e.g., Graphormer, SpaceTimeFormer) are missing from comparison, which may weaken the “state-of-the-art” claim.", "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": "j82CyaAxWS", "reviewer": "Reviewer_5Bv2", "summary": "This paper focuses on addressing the challenge of arbitrary inference in spatiotemporal dynamic systems, where existing methods either fail to generalize to unseen regions and complex graph structures or are confined to Euclidean grids. The proposed framework PhySTA integrates two innovative core modules: Continuous Operator-based Spectrum-Temporal Learning (CoSTL), which extends neural operators to non-Euclidean domains via a Graph-Time Fourier Neural Operator (GT-FNO) and Time-Gated Spectral Segmentation Perception for continuous dynamics modeling, and Adaptive Multi-scale Interaction (AMI) that constructs multiscale subgraphs and uses node-edge coupled convolution to capture discrete interactions. Theoretically, GT-FNO is proven to have universal approximation capability for continuous spatio-temporal graph operators with controllable \\(L^2\\) error. Experimentally, on traffic and air quality benchmarks (PEMS-BAY, SD, KnowAir), PhySTA achieves state-of-the-art accuracy across various missing data ratios, reduces computation cost significantly, and has fewer parameters and lower GPU memory consumption compared to baselines, demonstrating robust generalization for arbitrary inference even in sparse sensor scenarios.", "questions": "see details in weakness", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is well-structured, and its notations are generally clear.\n2. The proposed approach demonstrates performance improvements in most experimental settings compared to recent baseline methods.", "weaknesses": "1. In the motivation section, the authors propose that the modeling of continuous changes relies on graph spectral modeling, but the correlation between the two is not clearly established.\n2. It is not specified where the truncation operation is performed.\n3. In Equation 7, the graph is divided into three layers: coarse, mid, and fine. However, these subgraphs only exist on partial nodes. Therefore, it remains unclear whether the extracted features (X_coarse, X_mid, X_fine) have missing node features respectively.\n4. Issues with notations and formatting:\n   - In Line 256, there is a missing space before \"Inspired\"; in Line 264, there is a missing space before \"The\".\n   - In Equation 8, how are the two predicted values (y_costl and y_ami) obtained? Is the result of Equation 5 equivalent to Y_AMI?\n5. How do continuous spectrum-temporal modeling and adaptive multi-scale interaction handle dynamic topology changes respectively? At present, the study seems to treat the underlying graph (i.e., adjacency matrix A) as static, without corresponding discussions on dynamic scenarios.", "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": "toeAcec3Sl", "reviewer": "Reviewer_ucx5", "summary": "This paper proposes PhySTA (Physics-inspired Spatio-Temporal Learning for Arbitrary Inference), a framework that models continuous spatio-temporal dynamics on graph-structured data by integrating operator learning with multi-scale graph neural networks. The core idea is to bridge the gap between continuous real-world dynamics and discrete sensor observations, enabling reliable inference in unobserved regions.\n\nThe framework comprises two main modules:\nContinuous Operator-based Spectrum-Temporal Learning (CoSTL): Uses a Graph-Time Fourier Neural Operator (GT-FNO) and Time-Gated Spectral Segmentation Perception to model continuous dynamics in the joint spectral domain\n\nAdaptive Multi-scale Interaction (AMI): Employs a novel Node-Edge Coupled Convolution and Multi-scale Subgraph Partition (coarse-mid-fine hierarchy) to capture discrete multi-scale interactions and refine the continuous predictions.", "questions": "Addressing GFT Scalability (The Major Bottleneck): The paper acknowledges that the $O(N^2)$ complexity of the full Graph Fourier Transform (GFT) limits scalability. Since current spatio-temporal benchmarks scale up to 716 nodes (SD), the $N^2$ barrier remains a practical constraint for city-scale graphs ($N>5000$). Could the authors propose and experimentally validate a concrete, scalable approximation technique within the GT-FNO architecture (e.g., using a localized sparse spectral filter, Nyström approximation on the magnetic Laplacian, or spectral compression) to push the scalability beyond the current constraint?\n\n\nDifferentiability and Justification of Multi-scale Graph Construction: The Adaptive Multi-scale Interaction (AMI) relies on Louvain community detection to generate coarse and fine subgraphs. Since community detection is generally non-differentiable, how is the graph hierarchy generation process integrated into the end-to-end training pipeline? Furthermore, given the complexity of AMI, can the authors provide a side-by-side comparison (e.g., in the Appendix) of the performance gains versus a simpler, fully differentiable multi-scale strategy, such as one based on standard differentiable pooling (DiffPool/TopK)?\n\n\nDetailed Analysis of Spectral Component Contributions (TGSSP): The ablation study suggests that the specialized Time-Gated Spectral Segmentation Perception (TGSSP) module contributes only minor gains compared to other components. Given the module's complexity (bandwise parameterization, time-gating, complex eigenvalues/magnetic Laplacian), can the authors provide a more detailed visualization or analysis of what TGSSP learns? For example, showing how the learned per-mode scaling factor ($\\alpha_k$) and the time-gating factor ($g(\\omega)$) prioritize or suppress specific frequency bands over time would better justify the module's role in modeling non-stationary dynamics.\n\nMore applications: could it be applied to weather and climate forecasting?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Strong Theoretical and Architectural Novelty: The approach successfully extends the Fourier Neural Operator (FNO), a tool for continuous Euclidean domains, to non-Euclidean graph domains (GT-FNO) using the magnetic Laplacian and a joint graph-time spectral decomposition. This is a significant theoretical advance in spatio-temporal GNNs.\n\nRobustness to Data Sparsity: PhySTA consistently achieves top performance across all datasets under varying degrees of node masking (up to $70\\%$ sparsity). This directly validates its core claim of enabling robust \"arbitrary inference\" in unobserved regions\n\nfficiency and Low Overhead: PhySTA achieves state-of-the-art accuracy while using a lower number of parameters ($123,474$) and less GPU memory ($6,042 \\text{MB}$) compared to most deep GNN baselines like AGCRN and STG-ODE", "weaknesses": "1. Scalability Bottleneck of Spectral Decomposition: The paper acknowledges that the Graph Fourier Transform (GFT) step, which involves spectral decomposition (magnetic Laplacian), can be computationally demanding on very large graphs9. Since GFT's complexity is $O(N^2)$, this $N^2$ dependency limits the framework's scalability for massive real-world graphs (e.g., city-scale mobility or power grids with $N \\gg 1000$). It is crucial to discuss concrete techniques like Nyström approximation or sparse spectral methods to mitigate the $O(N^2)$ bottleneck\n\n2. Ambiguity of Multi-scale Aggregation: The AMI module's multi-scale single-layer modeling (coarse, mid, fine graphs) claims to capture complex interactions efficiently. However, the Louvain community detection used for subgraph creation is non-differentiable, potentially complicating end-to-end training. The authors should clarify how the node2center mapping and the three-level graph structure are integrated into the differentiable training pipeline.", "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)