A Hybrid Physics-Informed Deep Learning Framework for Robust Multivariate System Modelling Under Uncertainty – Copy

1. INTRODUCTION

Urban mobility systems produce large-scale sensor streams that are spatially dependent and temporally dynamic. A speed change at one freeway sensor may propagate to downstream sensors, while recurring daily patterns, incidents and capacity fluctuations alter the temporal profile of each node [1, 2, 3]. This makes traffic forecasting a graph time-series problem rather than an ordinary univariate prediction task. In this setting, a road network can be represented as G = (V, E, A), where V denotes the sensor nodes, E denotes the edges or relationships between sensors, and A denotes the adjacency matrix used to control spatial message passing [4].

Traditional recurrent and convolutional time-series models can learn temporal trends but often ignore the topology of the transportation network [5, 6]. Graph neural networks address this gap by allowing each sensor to aggregate information from other sensors through a graph structure. However, the graph used by many early models is fixed and is usually based on road distance, physical connectivity or expert assumptions. Such assumptions are useful but incomplete because real congestion propagation can change with demand, lane disruptions, time of day and non-recurring events [7, 8].

This paper presents an adaptive attention-driven architecture for graph-based traffic forecasting. The design combines four elements: static-adaptive graph fusion, local graph diffusion, global spatial attention and temporal self-attention. The static graph preserves stable sensor relationships, while the adaptive graph captures hidden sensor dependencies that are not directly encoded by a fixed topology. The local branch learns neighbourhood propagation, and the global branch allows non-neighbouring sensors to influence one another when their speed patterns become correlated. The temporal Transformer encoder then processes each node-specific sequence using self-attention rather than sequential recurrence.

The empirical design is based on METR-LA because it is a recognised benchmark in traffic forecasting. METR-LA contains traffic-speed observations from 207 loop-detector sensors in Los Angeles County at five-minute intervals and has been widely used in graph-based traffic forecasting studies [9, 10]. Unlike a small pilot file with only a few dozen time points, METR-LA provides enough temporal coverage to support a standard 12-step input and 12-step output forecasting setting. The paper therefore focuses on benchmark-ready validation rather than small-sample demonstration.

The main contribution of this study is a technically integrated spatio-temporal forecasting architecture for graph-based traffic-speed prediction. The contribution is defined as technical integration rather than the independent invention of graph neural networks, attention mechanisms, adaptive adjacency learning or Transformer modelling. The proposed framework combines static-adaptive graph fusion, local graph diffusion, global spatial attention and Transformer-based temporal encoding within one controlled forecasting pipeline. This integrated design is evaluated using chronological data splitting, training-only normalisation, fixed-seed repeated reporting, ablation analysis, saved-checkpoint evaluation and horizon-aware reporting at 15-, 30- and 60-minute forecasting horizons. The manuscript therefore positions the work as a controlled technical integration of existing graph-learning and attention-based forecasting principles rather than as a claim that any single component is entirely new.

2. LITERATURE REVIEW

2.1. Traffic Forecasting as Graph Time-Series Learning

Traffic forecasting has shifted from classical statistical modelling to graph-based deep learning because traffic sensors are not independent. Graph-based methods explicitly encode relationships among sensors and allow each node representation to be updated using nearby or correlated nodes. [11] reviewed the field and noted that graph neural networks are well suited to traffic systems because road networks naturally contain graph structures. Graph Transformers are another advancement, which instead of recurrent layers, involves self-attention mechanics to achieve more efficient long-range temporal learning. The study by [12] has proven the potential of the Transformer architectures in the graph-based tasks, where self-attention performs better than the RNN-based models to capture long-range dependencies on the spatio-temporal data. Transformer-based method is also advantageous in terms of parallelisation and scalability that makes it to be more applicable in large scale and real-time traffic forecasting techniques.

The ASTGCN (Attention-based Spatio-Temporal Graph Convolutional Network) of [13] involves the use of spatial and temporal attention to improve the model capacity to learn local and global dependencies. This model uses spatial attention to concentrate on local dependencies in traffic flow data and temporal attention to model long-term trends, which is effective than other methods to use in complex traffic networks to forecast. ASTGCN is a valuable advancement as it introduces multi-head attention to learn more spatio-temporal relationships [14].

Additionally, the Graph Multi-Attention Network (GMAN) by [15] uses multi-head attention to focus on local- and global-scale spatial dependencies at once. GMAN proposes multi-scale attention, which enhances the capacity of the model in containing the heterogeneous traffic characteristics that are very appropriate in real-time forecasting of traffic in the dynamic traffic scenario. Multi-scale attention enables GMAN to be more flexible to different traffic conditions, since it is able to capture the local relationships that are fine-tuned, as well as the large-scale system-wide impacts.

[16] introduced dynamic graph learning in STFGNN (Spatio-Temporal Fusion Graph Neural Network), which is a significant development. This network assumes a dual pathway model that incorporates graph convolutions of spatial and GRUs (Gated Recurrent Units) of temporal dependencies. The combination of space and time characteristics improves the development of intricate interactions of traffic flow data both in space and time and the overall forecasting outcomes [17]. The present study follows this graph time-series view and treats METR-LA as a dynamic multivariate signal defined over a sensor graph.

2.2. Fixed-Topology Spatio-Temporal Graph Models

Early spatio-temporal graph models such as STGCN and DCRNN established the usefulness of graph convolution for traffic prediction. STGCN used graph convolution and temporal convolution to avoid fully recurrent training, while DCRNN represented traffic propagation as a diffusion process over a directed graph [18]. These methods remain important baselines because they combine spatial graph learning with temporal forecasting. [19] suggest that foundational models such as ST-GCN and DCRNN combine graph convolutions with recurrent neural networks (RNNs) to integrate spatial topology and temporal sequences, but most of them use fixed adjacency matrices based on physical distance, road connectivity, or expert knowledge. These fixed topologies do not scale to the dynamism of traffic flows, in which spatial dependencies change over time due to varying patterns, incidents, external forces such as weather, and events. This rigidity causes inefficient representation of changing network forms, which causes reduced prediction performance, especially when there are non-recurring congestion or non-homogeneous urban conditions [20]. However, fixed topology can be restrictive when the true dependence between two sensors changes over time. Published METR-LA results show that fixed or semi-fixed graph baselines still perform competitively, but their errors increase at longer horizons.

2.3. Adaptive Graph Learning and Dynamic Dependency Modelling

Adaptive graph learning addresses the fixed-topology limitation by learning sensor relationships directly from data. [21] proposed AGCRN, which uses node-adaptive parameter learning and data-adaptive graph generation to infer hidden dependencies. [10] introduced D2STGNN, which separates diffusion and inherent components of traffic signals and includes dynamic graph learning. [22] proposed an evolutionary graph neural network that continuously updates a semantic adjacency matrix during training. These models have low structural interpretability, with learned representations being black boxes, making it difficult to analyse what spatial or temporal aspects a prediction is being driven by [23, 24]. These studies motivate the adaptive component of the present architecture, but this paper retains a static prior so that learned edges do not completely ignore physical or correlation-based structure.

2.4. Multi-Scale Spatial Attention

Single-scale aggregation may not be sufficient for traffic networks because congestion may be local in one period and network-wide in another. Multi-scale models attempt to capture neighbourhood patterns, regional trends and wider system effects. [5] proposed STGMS, a multi-scale spatio-temporal graph neural network that decomposes traffic features into multiple time scales and combines attention with graph convolution. [25] developed a long-term spatio-temporal graph attention network and evaluated it on METR-LA and PEMS-BAY. The present paper uses a dual spatial encoder: a local diffusion branch for graph-neighbourhood propagation and a global attention branch for long-range sensor interactions.

2.5. Transformer-Based Temporal Forecasting

Transformers have become influential in traffic forecasting because self-attention can connect distant time steps without recurrent recurrence. [26] proposed an adaptive graph spatial-temporal Transformer that models cross-spatial-temporal correlations. [22] showed that spatial-temporal Transformer networks can be used for traffic flow forecasting through carefully designed embeddings. The present method uses a Transformer encoder after spatial enrichment so that temporal attention is applied to node-wise hidden sequences rather than raw sensor values. This limits the attention burden and allows spatial encoding to shape temporal representations.

2.6. Benchmark Datasets and Reproducibility Requirements

Benchmark choice is central to research credibility. METR-LA and PEMS-BAY are widely used because they contain hundreds of sensors and tens of thousands of time steps. The Zenodo release provides METR-LA.csv and PEMS-BAY.csv in accessible CSV form, while LibCity documents METR-LA as a Los Angeles County loop-detector dataset with 207 sensors. A benchmark-ready study must preserve chronological order, avoid normalisation leakage, use repeated runs where feasible and report horizon-specific errors. Therefore, this manuscript adopts METR-LA, a 12-to-12 forecasting design and multiple reporting layers rather than relying on a very small traffic file.

2.7. Research Gap and Technical Positioning

The literature shows that adaptive graphs, attention mechanisms and Transformers are individually useful, but their integration must be carefully controlled (Table 1). A model that is too dynamic may overfit noisy relationships, while a model that is too static may miss time-varying propagation. Similarly, global attention improves flexibility but can become dense and difficult to interpret. The gap addressed here is the need for a unified architecture that combines static graph prior, learnable adaptive adjacency, local/global spatial attention and sparsity regularisation, with a validation protocol that is strong enough for benchmark-level assessment [5, 27].

Table 1. Recent high-quality literature informing the proposed architecture.

2.8. Critical Synthesis of 2020-2025 Studies

The 2020–2025 literature reveals four methodological movements in spatio-temporal traffic forecasting. The first is the transition from fixed spatial graphs to adaptive or learned dependency structures, as seen in AGCRN, D2STGNN and evolutionary graph-learning designs [10, 21, 22]. The second is the move from single receptive fields to multi-scale spatial or temporal encoders, as seen in MD-GCN, STGMS and long-term graph attention models [5, 25, 28]. The third is the increasing use of attention and Transformer structures to model non-local temporal and spatial interactions [22, 26]. The fourth is the recognition that reproducibility is part of methodological contribution: a model is not persuasive unless dataset choice, split strategy, missing-value handling, hyperparameter configuration, horizon-wise reporting and statistical interpretation are explicit [15, 25, 30–33].

Against this background, the present framework is positioned as a bounded-adaptivity architecture. It does not discard graph priors because stable structural or statistical relationships still contain useful information. It also does not rely only on static graphs because traffic conditions can create dependencies that fixed topology alone cannot express. The proposed graph learner therefore implements a fusion mechanism between a static training-derived graph and an adaptive learned graph, allowing the system to retain stable network structure while learning additional latent dependencies from speed observations.

2.9. Distinction from Closely Related Models

The proposed design differs from closely related traffic-forecasting models in the way it combines graph structure, spatial attention, temporal modelling and graph sparsity. DCRNN models traffic propagation as diffusion over a directed graph and uses recurrent sequence modelling for temporal dependency [9]. Graph WaveNet introduces an adaptive dependency matrix learned through node embeddings and combines it with dilated temporal convolution [34, 35]. AGCRN develops adaptive graph generation and node-adaptive parameter learning for traffic forecasting [21]. ASTGCN and GMAN use attention mechanisms to strengthen spatio-temporal dependency modelling [13, 15]. These studies provide important foundations for graph-based traffic prediction.

The present model is not claimed to be novel because it introduces attention, adaptive adjacency, graph convolution or Transformer modelling for the first time. Its contribution lies in the controlled technical integration of these ideas: static-adaptive graph fusion controls the topology, local graph diffusion preserves neighbourhood propagation, global spatial attention captures non-local sensor interactions, temporal self-attention models multi-step sequence dynamics, and L1 graph regularisation discourages dense uninterpretable adaptive connectivity. This distinction is important because it avoids alternating between different novelty claims. Throughout the manuscript, the novelty claim is therefore stated consistently as technical integration novelty.

3. METHODOLOGY

3.1. Problem Definition

Let in represent the traffic observation matrix at time t, where N is the number of sensors and F is the number of node features. For METR-LA, the primary feature is traffic speed, so F = 1 and N = 207. Given an input window with M = 12 historical observations, the task is to predict H = 12 future observations. Since METR-LA is sampled every five minutes, this corresponds to using one hour of historical speed data to forecast one hour ahead.

The prediction function is written as Equation (1):

Where, is the static graph prior and is the learned adaptive graph. The model parameters theta is optimised by minimising forecasting error while penalising unnecessarily dense adaptive edges.

3.2. METR-LA Dataset, Preprocessing and Chronological Split

This study uses METR-LA (Table 2) as the sole experimental dataset. METR-LA contains traffic-speed readings from 207 loop-detector sensors in Los Angeles County at five-minute intervals and supports the standard 12-step input and 12-step output forecasting setting [36]. Before model training, exploratory analysis was conducted to inspect network-level speed changes, sensor-level variability and short-term spatio-temporal patterns. These exploratory figures (Fig. 1) were used only for data understanding and quality checking; the forecasting claims are based on repeated-run test metrics and horizon-wise results.

Table 2. METR-LA experimental protocol.