Intelligent operation of thermal energy networks aims to improve energy efficiency, reliability, and operational flexibility through data-driven control, predictive optimization, and early fault detection. Achieving these goals relies on sufficient observability, requiring continuous and well-distributed monitoring of thermal and hydraulic states. However, district heating systems are typically sparsely instrumented and frequently affected by sensor faults, limiting monitoring. Virtual sensing offers a cost-effective means to enhance observability, yet its development and validation remain limited in practice. Existing data-driven methods generally assume dense synchronized data, while analytical models rely on simplified hydraulic and thermal assumptions that may not adequately capture the behavior of heterogeneous network topologies. Consequently, modeling the coupled nonlinear dependencies between pressure, flow, and temperature under realistic operating conditions remains challenging. In addition, the lack of publicly available benchmark datasets hinders systematic comparison of virtual sensing approaches. To address these challenges, we propose a heterogeneous spatial-temporal graph neural network (HSTGNN) for constructing virtual smart heat meters. The model incorporates the functional relationships inherent in district heating networks and employs dedicated branches to learn graph structures and temporal dynamics for flow, temperature, and pressure measurements, thereby enabling the joint modeling of cross-variable and spatial correlations. To support further research, we introduce a controlled laboratory dataset collected at the Aalborg Smart Water Infrastructure Laboratory, providing synchronized high-resolution measurements representative of real operating conditions. Extensive experiments demonstrate that the proposed approach significantly outperforms existing baselines.
Figures & tables
Figure 1: Schematic layout of the district heating network under study. The tree-structured graph illustrates the symmetrical relationship between the supply and return networks, where edge directions are inverted to reflect flow orientation.
Figure 2: Schematic block diagram of the experimental topology. The system is connected in a tree structure, linking the heat station to consumers.
Figure 3: Hardware implementation of the experimental setup at Aalborg SWIL.
Figure 4: Modules at Aalborg Smart Water Infrastructures Laboratory. For the pipe units, the following input–output pairs are considered: (I/O1_1, I/O1_2), (I/O2_1, I/O2_3), (I/O3_1, I/O3_2), and (I/O4_1, I/O4_3).
Component
Temperature
Pressure
Flow
Heat Station
Boiler_T1
Boiler_dP2
Boiler_Q2_1
Boiler_T2
Boiler_Q4_6
Boiler_T3
Boiler_T4
Boiler_T5
Boiler_T6
Table 1: Overview of the available system input sensors used as standard SCADA measurements
Variable
Range (5th – 95th)
Unit
Temperature
16.89 – 49.93
°C
Flow
0.09 – 12.59
lit/min
Pressure
3.27 – 10.71
bar
Table 2: System Operating Ranges (5th – 95th Quantile)
Figure 5: The Proposed HSTGNN Architecture for Virtual Sensing
Hyperparameter
Value
Window Size
{4,8,16,32,64}
Stride
1
Number of GNN Layers
{1,2,3,4}
Output Channels for GNN Layers
{4,8,16,32,64}
GNN Kernel Size
{1,2,3,4}
Node Embedding Size
{4,8,16,32,64}
Table 3: Summary of the hyperparameter search space and optimal configurations. Values selected via grid search on the primary dataset and utilized across all experiments are indicated in bold.
Sensor
Metric
Methods
GRU-GCN
LSTM
1D-CNN
DGC
GCN
HSTGNN
SM1
Flow Rate ( l/min )
RMSE
0.1969 ± 0.0017
0.1749 ± 0.0011
0.2416 ± 0.0022
0.1934 ± 0.0007
0.1940 ± 0.0013
0.1856 ± 0.0009
MAE
0.1157 ± 0.0035
0.1062 ± 0.0017
0.1561 ± 0.0006
0.1170 ± 0.0009
0.1180 ± 0.0026
0.1017 ± 0.0015
Inlet Temperature ( ∘C )
RMSE
0.1868 ± 0.0114
0.1723 ± 0.0093
0.1682 ± 0.0074
0.5080 ± 0.0124
0.5025 ± 0.0202
0.1243 ± 0.0045
MAE
0.1398 ± 0.0073
0.1203 ± 0.0051
0.1270 ± 0.0031
0.3978 ± 0.0080
0.3986 ± 0.0089
0.0941 ± 0.0022
Outlet Temperature ( ∘C )
RMSE
0.1976 ± 0.0051
0.2692 ± 0.0186
0.3301 ± 0.0281
0.6480 ± 0.0009
0.6478 ± 0.0111
0.1852 ± 0.0169
Table 4: Dataset 1: RMSE and MAE comparison across methods (mean ± std over 3 runs)
Sensor
Metric
Methods
GRU-GCN
LSTM
1D-CNN
DGC
GCN
HSTGNN
SM1
Flow Rate ( l/min )
RMSE
0.1561 ± 0.0030
0.1689 ± 0.0030
0.2487 ± 0.0014
0.1668 ± 0.0024
0.1646 ± 0.0020
0.1680 ± 0.0048
MAE
0.0806 ± 0.0046
0.0952 ± 0.0018
0.1493 ± 0.0012
0.0910 ± 0.0045
0.0869 ± 0.0027
0.0856 ± 0.0035
Inlet Temperature ( ∘C )
RMSE
0.3384 ± 0.0059
0.2024 ± 0.0256
0.2261 ± 0.0169
0.5713 ± 0.0011
0.5746 ± 0.0008
0.2003 ± 0.0135
MAE
0.2415 ± 0.0018
0.1512 ± 0.0201
0.1623 ± 0.0108
0.4463 ± 0.0071
0.4499 ± 0.0088
0.1471 ± 0.0132
Outlet Temperature ( ∘C )
RMSE
0.2351 ± 0.0125
0.2849 ± 0.0121
0.2848 ± 0.0053
0.5952 ± 0.0149
0.5957 ± 0.0050
0.1987 ± 0.0134
Table 5: Dataset 2: RMSE and MAE comparison across methods (mean ± std over 3 runs)
Sensor
Metric
Methods
GRU-GCN
LSTM
1D-CNN
DGC
GCN
HSTGNN
SM1
Flow Rate ( l/min )
RMSE
0.1447 ± 0.0010
0.2244 ± 0.0215
0.2450 ± 0.0090
0.1586 ± 0.0018
0.1572 ± 0.0068
0.1439 ± 0.0031
MAE
0.0762 ± 0.0005
0.1516 ± 0.0216
0.1641 ± 0.0112
0.0930 ± 0.0015
0.0903 ± 0.0095
0.0781 ± 0.0059
Inlet Temperature ( ∘C )
RMSE
0.2931 ± 0.0211
0.2465 ± 0.0112
0.2544 ± 0.0144
0.6284 ± 0.0258
0.6302 ± 0.0069
0.2247 ± 0.0132
MAE
0.2240 ± 0.0165
0.1834 ± 0.0146
0.1852 ± 0.0096
0.4372 ± 0.0213
0.4391 ± 0.0011
0.1565 ± 0.0021
Outlet Temperature ( ∘C )
RMSE
0.2369 ± 0.0118
0.2983 ± 0.0132
0.2661 ± 0.0139
0.4738 ± 0.0107
0.4543 ± 0.0254
0.2320 ± 0.0136
Table 6: Dataset 3: RMSE and MAE comparison across methods (mean ± std over 3 runs)
Sensor
Metric
Methods
GRU-GCN
LSTM
1D-CNN
DGC
GCN
HSTGNN
SM1
Flow Rate ( l/min )
RMSE
0.1934 ± 0.0006
0.1908 ± 0.0016
0.2424 ± 0.0009
0.2137 ± 0.0012
0.2127 ± 0.0005
0.1803 ± 0.0054
MAE
0.1018 ± 0.0002
0.1208 ± 0.0050
0.1492 ± 0.0013
0.1192 ± 0.0024
0.1187 ± 0.0004
0.0974 ± 0.0010
Inlet Temperature ( ∘C )
RMSE
0.2355 ± 0.0006
0.2478 ± 0.0020
0.2365 ± 0.0086
0.5202 ± 0.0235
0.5119 ± 0.0093
0.1722 ± 0.0126
MAE
0.1838 ± 0.0004
0.1976 ± 0.0035
0.1884 ± 0.0060
0.3986 ± 0.0251
0.3909 ± 0.0080
0.1292 ± 0.0102
Outlet Temperature ( ∘C )
RMSE
0.1846 ± 0.0029
0.2519 ± 0.0233
0.2407 ± 0.0298
0.4392 ± 0.0044
0.4400 ± 0.0044
0.1656 ± 0.0170
Table 7: Dataset 4: RMSE and MAE comparison across methods (mean ± std over 3 runs)
Figure 6: Outlet temperature prediction for SM 1, Dataset 1. HSTGNN closely follows the true signal; GRU-GCN is second best; graph-only models show larger deviations.
Figure 7: Inlet temperature for SM 1, Dataset 2. HSTGNN is most accurate; temporal-only and graph-only models deviate during transients.
Figure 8: Inlet temperature for SM 2, Dataset 3. HSTGNN tracks the measured signal well; GRU-GCN captures general trends.
Figure 9: Flow rate for SM 1, Dataset 4. HSTGNN and GRU-GCN perform best; graph-only models outperform temporal-only baselines.
Sensor
Metric
Ablation Variants
Full Model
w/o Pressure
w/o Flow
w/o Temperature
SM1
Flow Rate ( l/min )
RMSE
0.1856
0.1863
0.3439
0.1859
MAE
0.1017
0.1021
0.2162
0.1090
Inlet Temperature ( ∘C )
RMSE
0.1243
0.1598
0.2031
0.7013
MAE
0.0941
0.1247
0.1522
0.5139
Outlet Temperature ( ∘C )
RMSE
0.1852
0.2040
0.2760
0.7307
Table 8: Ablation study of HSTGNN by removing sensor groups (Dataset 1)
Sensor
Metric
Models
HSTGNN
Simplified
SM1
Flow Rate ( l/min )
RMSE
0.1803
0.2124
MAE
0.0974
0.1259
Inlet Temperature ( ∘C )
RMSE
0.1722
0.3734
MAE
0.1292
0.2809
Outlet Temperature ( ∘C )
RMSE
0.1656
0.3658
Table 9: Ablation study on Dataset 4: HSTGNN compared to a minimal variant without heterogeneous branching, sensor-type spatial modeling, or sensor-specific embeddings.
Figure 10: Learned top- K functional adjacency matrices for the temperature and pressure branches. Rows denote receiving sensors and columns denote source sensors. Non-zero entries correspond to top- 4 dependencies selected from the learned score matrices. The learned graphs are interpreted as data-driven functional dependency structures rather than direct reconstructions of the pipe topology.
Query / Key
Temperature
Pressure
Flow
Temperature
0.326
0.225
0.449
Pressure
0.270
0.454
0.276
Flow
0.306
0.299
0.395
Table 10: Global row-normalized type-level block-mean attention. Rows denote query sensor types and columns denote key sensor types. Values are computed after type-specific temporal and graph encoding and are interpreted as information-routing diagnostics rather than causal physical dependencies.
Fault location in distribution grids is critical for reliability and minimizing outage durations. Yet, it remains challenging due to partial observability, given sparse measurement infrastructure. Recent works show promising results by combining Recurrent Neural Networks (RNNs) and Graph Neural Networks (GNNs) for spatio-temporal learning. Still, many modern GNN architectures remain untested for this grid application, while existing GNN solutions have not explored GNN topology definitions beyond simply adopting the full grid topology to construct the GNN graph. We address these gaps by (i) systematically comparing a newly proposed graph-forming strategy (measured-only) to the traditional full-topology approach, and (ii) introducing STGNN (Spatio-temporal GNN) models based on GraphSAGE and an improved Graph Attention (GATv2), for distribution grid fault location; (iii) benchmarking them against state-of-the-art STGNN and RNN baselines on the IEEE 123-bus feeder. In our experiments, all evaluated STGNN variants achieve high performance and consistently outperform a pure RNN baseline, with improvements up to 11 percentage points F1. Among STGNN models, the newly explored RGATv2 and RGSAGE achieve only marginally higher F1 scores. Still, STGNNs demonstrate superior stability, with tight confidence intervals (within +/- 1.4%) compared to the RNN baseline (up to +/- 7.5%) across different experiment runs. Finally, our proposed reduced GNN topology (measured-only) shows clear benefits in both (i) model training time (6-fold reduction) and (ii) model performance (up to 11 points F1). This suggests that measured-only graphs offer a more practical, efficient, and robust framework for partially observable distribution grids.
Leak localization in water distribution networks is challenging when multiple leaks occur simultaneously, particularly under sparse monitoring. Existing data-driven methods often assume a single or predefined number of leaks or require extensive enumeration of leak combinations. This study proposes a graph neural network (GNN) model that localizes a variable number of simultaneous leaks without specifying the leak count in advance or exhaustively generating every possible combination of leak locations for training. The model is evaluated on two benchmark networks of different sizes, the 31-node Hanoi network and the larger 92-node NET3 network, under variable demand, sparse pressure-sensor coverage, variable leak magnitudes, and varying numbers of simultaneous leaks. The GNN outperforms multilayer perceptron, support vector machine, K-nearest neighbors, and random forest classifiers by up to 12% and 20% in accuracy respectively in Hanoi and NET3 networks. Consistent performance across both networks supports the scalability of the proposed approach.
Arvin Ajoodani, Sara Nazif, Pouria Ramazi
School of Civil Engineering, College of Engineering, University of Tehran, Tehran, Iran · Department of Biological Sciences and Department of Mathematics and Statistics, University of Calgary, Calgary, Canada
Hypergraphs are the natural description of higher-order interactions among objects, widely applied in social network analysis, cross-modal retrieval, etc. Hypergraph Neural Networks (HGNNs) have become the dominant solution for learning on hypergraphs. Traditional HGNNs are extended from message passing graph neural networks, following the homophily assumption, and thus struggle with the prevalent heterophilic hypergraphs that call for long-range dependence modeling. In this paper, we achieve heterophily-agnostic message passing through the lens of Riemannian geometry. The key insight lies in the connection between oversquashing and hypergraph bottleneck within the framework of Riemannian manifold heat flow. Building on this, we propose the novel idea of locally adapting the bottlenecks of different subhypergraphs. The core innovation of the proposed mechanism is the design of an adaptive local (heat) exchanger. Specifically, it captures the rich long-range dependencies via the Robin condition, and preserves the representation distinguishability via source terms, thereby enabling heterophily-agnostic message passing with theoretical guarantees. Based on this theoretical foundation, we present a novel Heat-Exchanger with Adaptive Locality for Hypergraph Neural Network (HealHGNN), designed as a node-hyperedge bidirectional systems with linear complexity in the number of nodes and hyperedges. Extensive experiments on both homophilic and heterophilic cases show that HealHGNN achieves the state-of-the-art performance.
Li Sun, Ming Zhang, Wenxin Jin +5
Beijing University of Posts and Telecommunications, Beijing, China · North China Electric Power University, Beijing, China · University of Kent, Canterbury, United Kingdom +2