Decompose Dynamics Before Learning Dependencies in Spatiotemporal Systems

1 INTRODUCTION

• Mixed local dynamics, edge- and system-dependent delays, and nonlocal temporal patterns make it challenging to understand how local processes evolve and relate across nodes.Learning from a single representation of these dynamics can obscure how processes evolve, propagate, and relate.Source: §1 · p. 1, Figure 1 · p. 2
• CANDOR decomposes each trajectory into background, accumulation, release, and shock before using those components to guide dependency learning.The components represent persistent patterns, gradual buildup, gradual decline, and abrupt changes, respectively.Source: §1 · pp. 1–2
• A delay-aware physical branch models propagation over directed topology, while a functional branch learns dependencies beyond physical connections.Adaptive fusion combines functional, forward-propagation, and reverse-support predictions; training objectives encourage branch specialization and semantic consistency.Source: §1 · pp. 1–2
• CANDOR is evaluated on urban traffic and river water quality, and the paper releases three water-quality datasets with over 1.8 million station-hour observations from 43 sites across five years.The evaluation spans human-driven traffic and naturally evolving environmental systems.Source: §1 · p. 2

2 RELATED WORKS

• Decomposition methods range from multiscale pattern extraction and component-specific supervision to modeling distinct sources of variation; CANDOR focuses on how components inform relationships between nodes.Its background, accumulation, release, and shock components guide physical propagation, while background representations inform the functional graph.Source: §2 · p. 2
• Graph-based forecasting combines structural priors and learned dependencies, while recent methods adapt graph structures or learn channel and patch-level relations.Existing methods generally learn a single dependency structure from entangled observations.Source: §2 · pp. 2–3
• CANDOR distinguishes itself by assigning different dynamical evidence to directed, delay-aware physical propagation and topology-free functional dependence.The resulting relations have complementary mechanistic roles.Source: §2 · p. 3

3.1 PROBLEM FORMULATION AND OVERVIEW

• CANDOR predicts future node states from historical observations and a directed physical network, first decomposing local dynamics and then learning complementary relations.The physical graph captures delayed interactions over directed connections; the functional graph uses background representations to infer dependencies beyond physical adjacency.Source: §3.1 · p. 3, Figure 2 · p. 3

3.2 EVOLUTION-STRUCTURED DYNAMICS DECOMPOSITION

• A shared background defines complementary residual views for accumulation and release, while changes in those deviations guide shock extraction.Accumulation and release use current and subsequent background references, respectively; shock captures abrupt deviations in both views.Source: §3.2 · p. 4, Figure 3 · p. 4

3.3 DECOMPOSITION-CONDITIONED DUAL-GRAPH LEARNING

• The physical branch combines delay-specific affinities shared across samples with history-conditioned affinities, and forward and reverse paths aggregate interactions over multiple spatial orders.The functional graph uses background state and evolution to select relations without a physical adjacency mask; all four components contribute to information exchange.Source: §3.3 · pp. 4–5

3.4 CONTEXT-ADAPTIVE FORECAST FUSION

• Fusion conditions the relative contributions of functional, forward-propagation, and reverse-support forecasts on node identity, target calendar, and forecast position.The shared prediction head decodes fused representations into patch-level predictions, which are averaged over overlaps.Source: §3.4 · p. 5

3.5 LEARNING OBJECTIVES

• Learning objectives constrain how components explain local dynamics and align with physical propagation states, in addition to supervising the final forecast.Reconstruction gives accumulation and release distinct targets based on their two background references, while role consistency aligns components with physical-path states.Source: §3.5 · p. 6

4 EXPERIMENTS

• Evaluation spans two traffic datasets and three five-year river datasets, with identical chronological splits and three seeds across 15 representative baselines.The water-quality data contain 1.88 million hourly observations from 43 sites; traffic forecasts 12 five-minute steps, while water-quality forecasts 240 hours.Source: §4 · pp. 6–7

4.1 MAIN RESULTS

• On METR-LA, CANDOR reduces Overall MAE from 2.982 to 2.854 (4.31%) versus the strongest baseline; on PEMS-BAY, it ranks first at every horizon.CANDOR also reduces METR-LA Overall RMSE from 6.136 to 5.860 (4.49%) and PEMS-BAY overall MAE and RMSE by 1.46% and 1.86%.Source: Table 1 · p. 7, §4.1 · p. 7
• Water-quality MSE improvements over the strongest baselines are 5.47%, 4.68%, and 5.29% across the three basins, with the lowest MSE and MAE at every reported horizon.CANDOR’s worst-seed result outperforms the best seed of every baseline in 222 of 225 water-quality comparisons.Source: Table 2 · p. 8, §4.1 · p. 8

4.2 ABLATION STUDIES

• Removing decomposition raises METR-LA MAE by 1.6% and BASIN180102 MSE by 5.8%; removing auxiliary objectives raises BASIN180102 MAE/MSE by 1.5%/2.2%.The ablation comparison evaluates seed-paired relative degradation; dots mark changes exceeding twice the full-model seed variation.Source: §4.2 · p. 8, Figure 4 · p. 8
• Removing physical propagation raises METR-LA MAE/MAPE by 9.5%/16.3%, while removing physical and functional branches raises BASIN180102 MSE by 10.5% and 12.8%, respectively.Replacing the learned operator with raw adjacency raises METR-LA MAPE by 3.5% and BASIN180102 MSE by 3.9%; single-direction variants also degrade performance.Source: §4.2 · p. 8
• Relation modeling and adaptive fusion contribute up to 3.4% and 2.4% in MSE in the complete matrix.Together with role alignments, the reported results support learning ordered physical propagation and complementary functional relations from evolution-specific component roles.Source: §4.2 · p. 9

4.3 MODEL ANALYSIS

• Figure 5 links release-associated residuals EΣ at time t and t + 1 to later incoming graph contribution oΣ at t + 2, while aligning R with EΣ and A with oΣ.Figure 6 shows nonlocal functional relations beyond physical adjacency and context-dependent fusion: METR-LA favors physical propagation, whereas PEMS-BAY and water-quality datasets assign more weight to functional relations.Source: Figure 5 · p. 9, §4.3 · p. 9, Figure 6 · p. 9

• Across human-driven traffic and natural river systems, CANDOR reduces traffic MAE by up to 4.31% and water-quality MSE by up to 5.47%.Water-quality results show that spatiotemporal models do not consistently outperform temporal forecasters under sparse topology and heterogeneous delays.Source: §5 · p. 9

• Patch embeddings pass through a causal temporal convolution and layer normalization to form the background; residual views compare observations with current and next-patch backgrounds.At the boundary, the final background is repeated, so both references remain within the observed window.Source: §A.1 · p. 12
• Recursive first- and second-order temporal differences guide feature-wise accumulation and release gates conditioned on node embeddings and calendar context.The two residual views use separate convolutional and normalization parameters.Source: §A.1 · pp. 12–13
• Separate scalar shock gates select pronounced residuals and rapid changes in both views, while projected residuals retain signed information.The gates weight original residuals and need not sum to one.Source: §A.1 · p. 13
• A residual-connected affine map integrates the four components; independent temporal and feature projections then produce component forecasts that are combined with calendar representations.The component roles are learned without requiring exact additive reconstruction or physical conservation.Source: §A.1 · p. 13

Delay-aware physical branch

• Physical propagation operators use receiving-node weights over directed topology, with affinities that vary across edges and temporal positions and include sample-dependent terms.The receiving-node convention makes entry (i, j) the weight for information from node j to node i; operators are computed once per observed window and reused during forecasting.Source: §A.2 · pp. 13–14
• Each physical direction builds temporal representations with a GRU and causal attention, then uses a learned backcast map to define propagation residuals across spatial orders.For a window ending at patch p, the latest Kt propagation states are transformed and ordered from oldest to newest before aggregation.Source: §A.2 · p. 14
• The graph output combines contributions across spatial orders with a local temporal summary; graph terms exclude diagonal contributions while the local summary retains same-node information.The forward and reverse predictions are retained separately, and reverse support supplies predictive context rather than asserting reverse physical transport.Source: §A.2 · p. 14

Functional relation graph

• The functional graph derives node descriptors from background trajectories, selects sparse node–relation pairs, and uses background descriptors for weights while retaining all four components in values.An empty candidate set yields zero weights, and multiple relation channels may connect the same node pair.Source: §A.2 · pp. 14–15

Functional branch and branch outputs

• The functional branch combines its context with the background state and node and calendar features to produce a forecast correction and sigmoid gate; functional, forward, and reverse outputs proceed to fusion.The functional graph’s background-based relation weights are distinct from its values, which retain information from all four components.Source: §A.2 · p. 15

Context construction and fusion

• The fusion context includes calendar embeddings for the target time and a positional encoding identifying the target patch within the forecast.The calendar lookups encode time of day and day of week; the positional encoding uses the zero-based target-patch position.Source: §A.3 · p. 15
• A fusion MLP produces three softmax-normalized scalar weights for each node, target patch, and branch, which are applied to the branch representations.The weights are broadcast across the representation features.Source: §A.3 · p. 15

Prediction and overlap recovery

• A shared two-layer head maps each fused representation to an output patch, and forecasts average the predictions from all target patches covering each step.Target patches continue the historical patch grid, and only patches whose start precedes the end of the forecast are included.Source: §A.3 · p. 16

Reconstruction and role consistency

• Reconstruction uses component outputs before layer normalization and applies smooth-L1 only where an observed successor exists; it is zero when P = 1.Gradients through the target are stopped, while component outputs remain differentiable.Source: §A.4 · p. 16
• Role consistency associates next-patch background and accumulation with temporal and graph contributions, and release with the current-patch residual available for propagation.These learned semantic constraints compare quantities within the same node and are not assertions of physical conservation; reconstruction and role consistency activate at a specified epoch.Source: §A.4 · p. 16

Prediction supervision and auxiliary regularization

• The final forecast is supervised with MAE, while an auxiliary prediction head supplies a direct prediction signal to each branch before adaptive fusion.The mean historical next-state prediction of the two physical paths is also supervised against the refined history.Source: §A.4 · p. 16

Auxiliary regularization

• Auxiliary regularization penalizes second differences in the background, encourages selective shock-gate activation, and discourages redundant temporal and propagation representations within each physical direction.The shock penalty does not enforce exact zeros in the normalized component.Source: §A.4 · p. 17

Traffic forecasting

• METR-LA and PEMS-BAY provide five-minute traffic speeds from 207 and 325 sensors, respectively, using 12 observations to predict the next 12 steps.The chronological split is 70%/10%/20%, and normalization statistics are estimated from training data only.Source: §A.5 · p. 17

Water-quality forecasting

• Three five-year hourly dissolved-oxygen datasets cover the Santee, Apalachicola, and Klamath basins, each with at least ten stations and no more than 10% missing observations per station.The datasets require at least N −1 directed river connections to ensure temporal coverage and network connectivity.Source: §A.5 · p. 17
• Water-quality series contain 43,824 hourly timestamps, use chronological 60%/20%/20% splits, and forecast 48–240 hours from the preceding 48 hours.After quality control, 1,837,295 of 1,884,432 station-hour entries remain observed; directed edges follow downstream reachability within 200 river kilometers.Source: §A.5 · p. 17

A.6 IMPLEMENTATION DETAILS

• On METR-LA, CANDOR achieves the lowest MAE of 2.854 with 0.90M parameters, while the shared (Kt, Ks, K) = (2, 2, 16) default is stable across tested sensitivity ranges.On BASIN180102, it achieves the lowest MSE of 1.542 with 0.88M parameters; the shared default avoids dataset-specific tuning.Source: §A.6 · p. 19
• All methods share forecasting targets, chronological splits, and no exogenous covariates; CANDOR uses a shared backbone with patch geometry adjusted to dataset temporal scales.Traffic uses 12 observed and 12 target steps; water-quality inputs span 48 hours, with horizons up to 240 hours.Source: §A.6 · p. 18, §A.7 · p. 20

A.8 ADDITIONAL EXPERIMENTS

• CANDOR achieves the best mean performance across both systems; traffic worst-seed Overall MAE beats every baseline’s best seed, and water-quality worst-to-best MSE separation holds in 222 of 225 comparisons.The three water-quality exceptions are all on BASIN030501.Source: §A.8 · pp. 20–21

A.9 ABLATION STUDIES

• Ablations show changing branch importance: removing physical propagation raises MAE by 9.5% on METR-LA, while removing functional relations raises MAE/MSE by 2.5%/6.5% on BASIN030501.On BASIN180102, removing physical and functional branches raises MSE by 10.5% and 12.8%, respectively.Source: §A.9 · p. 21
• Removing raw-adjacency replacement, decomposition, relation-specific modeling, or adaptive fusion worsens selected metrics, with ablation effects varying across datasets and structural regimes.Replacing the learned operator with raw adjacency raises METR-LA MAPE by 3.5% and BASIN180102 MSE by 3.9%; removing decomposition raises BASIN180102 MSE by 5.8%.Source: §A.9 · pp. 21, 24

A.10 DETAILED MECHANISM ANALYSIS

• Reconstruction and representation separation jointly indicate that the four components preserve local dynamics while acquiring differentiated roles.Reconstruction assesses collective sufficiency; separation assesses whether components avoid becoming interchangeable copies.Source: §A.10 · p. 25
• Under the two-patch window, localized states at t and t + 1 map to incoming graph contribution at t + 2, consistent with delayed transport; these alignments are not exact mass conservation.The reverse-support path is predictive context, not backward physical transport.Source: §A.10 · p. 25
• Functional relations include non-neighboring sites and vary in predictive weight across nodes, samples, and forecast positions, complementing physical support.They are inferred from background dynamics and should be interpreted as predictive relationships, not unobserved physical connections or causal transport pathways.Source: §A.10 · p. 26