A Multi-Task Temporal Fusion Framework for 48 h Ahead Joint Prediction of Dam Crack Responses and Rebar Stress from Multi-Source Monitoring Data

Abstract

Crack opening and reinforcement stress are two complementary indicators of the service state of reinforced concrete hydraulic structures, yet they are often predicted separately. This study develops a data-driven multi-task temporal fusion framework for joint 48 h ahead prediction of dam crack responses and rebar stress using multi-source monitoring data. The measured data comprise five crack-monitoring series, five rebar stress series, local temperature channels, reservoir water level, antecedent rainfall, and an auxiliary environmental signal over approximately four years. Target responses are aligned only at common measured timestamps; no synthetic target observations are introduced. A simplified engineering layout and plan-based crack–rebar distances are further used to examine whether an explicit spatial prior can strengthen the shared temporal representation without introducing synthetic target values. A residual multi-task temporal fusion network (MTTF-Net) is proposed with a shared Transformer encoder, attention pooling, task-specific decoders, and a response-continuity regularization term. The model is compared with persistence, Ridge regression, random forest, Extra Trees, XGBoost, and GRU baselines under a chronological train/validation/test split. For the independent test period, Ridge regression obtains the lowest overall RMSE (2.2968), whereas MTTF-Net provides the lowest crack RMSE (0.0141), the lowest overall MAE (1.0035), and the second-best overall RMSE (2.3813). Distance-informed ablation, denoted as MTTF-Net-S, remains close to MTTF-Net in macro-averaged $R^2$ but is not superior in the overall test metrics, indicating that the available horizontal distances are valuable engineering metadata but cannot replace richer three-dimensional structural connectivity. These results indicate that the monitoring data contain a strong linear autoregressive component, while multi-task temporal fusion improves nonlinear crack response prediction and remains competitive for stress forecasting. The source code is prepared as a public implementation package, whereas the measured monitoring dataset is subject to data owner restrictions.

1. Introduction

Long-term structural health monitoring (SHM) provides the empirical basis for evaluating the service condition of dams, retaining structures, bridges, and reinforced concrete infrastructure. Recent review work on concrete dam monitoring has emphasized the transition from isolated instrument interpretation toward integrated data processing, behavior modeling, and safety decision support [1]. Deng et al. organized dam health monitoring studies from data preprocessing to behavior assessment, showing that long-term monitoring systems increasingly depend on reliable data-driven modeling pipelines [2]. For displacement-based concrete dam SHM, Wang et al. summarized recent observation data-modeling practices and highlighted the importance of environmental actions such as water level, temperature, and rainfall [3]. Plevris and Papazafeiropoulos reviewed the role of artificial intelligence in SHM-oriented maintenance and safety, emphasizing that data-driven decision support is becoming a central component of infrastructure management [4]. Vijayan et al. discussed intelligent technologies and IoT-enabled SHM for civil engineering structures [5], while Boratto et al. combined agglomerative clustering with unsupervised feature selection for SHM data interpretation [6]. These studies collectively indicate that modern SHM research is moving toward multi-source sensing, automated data management, and interpretable predictive models.

1.1. Research Status

Dam prediction studies have mainly focused on deformation, displacement, seepage, or stress indicators. Madiniyeti et al. combined sparrow search optimization with LSTM for concrete dam deformation prediction, showing the utility of optimized recurrent networks for monitoring sequences [7]. Zhang et al. proposed a DenseNet-LSTM deformation model with feature selection for concrete gravity dams, which strengthened nonlinear feature extraction before temporal prediction [8]. He and Li used measured prototype temperature data to improve the interpretability of dam deformation forecasting, emphasizing that environmental variables should be treated as physical drivers rather than merely auxiliary inputs [9]. Yin and Wu developed a separate modeling technique for concrete dam deformation, reflecting the long-standing idea that different monitored components may undergo different response mechanisms [10]. Wen et al. introduced a combined model based on multiple regression and stacked GRUs for concrete dam deformation prediction [11], while Li et al. proposed DRLSTM as a dual-stage deep learning model driven by raw dam monitoring data [12]. Wei et al. used a Pearson K-means multi-head attention model for deformation prediction during the first impoundment of super-high dams [13]. Wang et al. recently proposed an attention-enhanced LSTM sequence-to-sequence model for dam deformation prediction, further illustrating the value of attention mechanisms in dam-oriented monitoring sequences [14]. For stress-related dam behavior, Tao et al. estimated in-service concrete dam stress from deformation data using a hybrid SIE-APSO-CNN-LSTM framework [15]. These studies provide strong methodological references, but most of them predict one response type at a time.

Crack monitoring has developed along a partly separate line. Goszczynska et al. experimentally analyzed crack width development in reinforced concrete beams, providing a mechanics-oriented reference for crack evolution under load [16]. Cramer et al. simulated crack propagation in reinforced concrete elements, showing how crack behavior can be connected to structural response mechanisms [17]. Ganasan et al. used machine learning models to predict crack width in reinforced concrete beam–column joints under lateral cyclic loading [18]. Razavi Tosee et al. proposed a hybrid grey wolf optimizer neural network for crack width prediction in CFRP-strengthened RC slabs [19]. Rao et al. developed an attention recurrent residual U-Net for pixel-level concrete crack width prediction from images [20]. Hui et al. developed a computer vision concrete crack identification method using MobileNetV2 and adaptive thresholding, representing a recent example of AI-assisted crack inspection [21]. Li et al. further reported deep-learning-based fine-grained quantitative detection of defects in submerged concrete, which is relevant to hydraulic structure inspection but remains image-oriented rather than monitoring time-series forecasting [22]. Laxman et al. combined automated crack detection with crack depth prediction for reinforced concrete structures using deep learning [23]. These works confirm the importance of crack indicators, but most focus on laboratory members or image data rather than long-term multi-source monitoring series.

Another related stream concerns sensing, data quality, and AI-assisted SHM workflows. Malekloo et al. reviewed machine learning for SHM, with attention to emerging high-dimensional data sources [24]. Mishra et al. discussed IoT-based SHM for civil engineering structures, which is relevant to database-oriented monitoring systems [25]. Jayawickrema et al. reviewed fibre optic sensing and deep learning for civil structure SHM [26], and Hassani and Dackermann surveyed advanced sensor technologies for nondestructive testing and SHM [27]. Luleci et al. reviewed generative adversarial networks in civil SHM, mainly for data generation, anomaly handling, and data restoration tasks [28]. Recent SHM studies also show that sensor data quality, clustering, and feature relevance must be treated carefully before a monitoring model is deployed [6]. For trustworthy deployment, Sun et al. reviewed explainable and human-in-the-loop bridge SHM and risk prognosis [29], while Xu et al. discussed few-shot learning for civil infrastructure structural health diagnosis [30]. These studies motivate a conservative experimental design in which target values are not fabricated and chronological validation is preferred.

Forecasting methodology has also benefitted from recent general time-series research. Zhou et al. proposed Informer for efficient long-sequence time-series forecasting with Transformer-style attention [31]. Deng et al. developed a multi-view multi-task learning framework for multivariate time-series forecasting, which is conceptually close to joint prediction of multiple structural responses [32]. Song et al. reviewed deep-learning-based time-series forecasting and summarized the development from recurrent networks to attention-based models [33]. Cini et al. reviewed graph deep learning for time-series forecasting, emphasizing relational modeling when multiple sensors interact [34]. These methods suggest that crack and rebar stress responses should not necessarily be modeled as independent scalar targets when they are driven by shared environmental and structural states.

1.2. Research Gap and Contributions

Despite this progress, a gap remains in the joint prediction of dam crack responses and rebar stress using field monitoring data. Crack opening reflects local concrete damage and joint movement, whereas rebar stress reflects internal force redistribution in reinforced components. The two response groups are physically related, and this study uses a simplified plan layout to identify representative crack–rebar associations and horizontal distances. However, the available information is still not a complete three-dimensional coordinate table or finite-element connectivity model. It is therefore used as a transparent spatial prior for ablation rather than as a forced one-to-one target-pairing rule. In addition, field monitoring data are not laboratory data: sampling frequencies differ, environmental variables may be sampled more frequently than response variables, and target interpolation can create false supervision. A defensible joint model must therefore align measured target timestamps conservatively, use environmental variables only as observed contextual inputs, and compare the proposed model with strong persistence and linear autoregressive baselines.

This study therefore builds a reproducible joint crack–stress prediction framework for infrastructure monitoring. All numerical results, figures, and tables are generated from measured monitoring data and reproducible code rather than from synthetic target observations. The main contributions are summarized as follows:

• A conservative multi-source dataset is constructed from measured crack, rebar stress, temperature, reservoir water level, rainfall, and auxiliary environmental records, with common target timestamps retained and no synthetic target responses introduced.
• A residual multi-task temporal fusion network is formulated for 48 h ahead joint crack–stress prediction, and plan-distance-informed ablation (MTTF-Net-S) is designed to test the value and limits of the available crack–rebar spatial metadata.
• Persistence, Ridge, random forest, Extra Trees, XGBoost, GRU, MTTF-Net, and MTTF-Net-S are evaluated under the same chronological protocol, with reproducible code, equations, figures, tables, and symbol definitions prepared for external inspection.

2. Materials and Data Construction

Monitoring Variables

The local monitoring archive contains five crack response files, five rebar stress files, as well as reservoir water level, rainfall, and auxiliary environmental measurements. Each crack and rebar file has three columns: the timestamp, local temperature, and measured response. The reservoir water level and rainfall files were sampled at a higher frequency than the response files. The raw material was organized in the project directory under 01_data/raw. The patent document and legacy database access scripts were retained as references but are not used to generate any synthetic observations.

The monitored object was an in-service reinforced concrete hydraulic structure instrumented for crack response, internal rebar stress, local temperature, reservoir water level, rainfall, and auxiliary environmental conditions. For confidentiality, the project name, exact coordinates, and detailed structural drawings are not disclosed in full. Based on the engineering layout of the monitored structure, this revision includes a simplified plan layout and a plan-based crack–rebar association table so that the structural locations of the main J-series crack gauges and R-series rebar stress gauges are explicit. The reported distances are horizontal layout distances; they provide a defensible spatial prior for ablation but should not be interpreted as complete three-dimensional mechanical transfer paths.

The aligned response period extended from 11 March 2021 07:00:00 to 6 March 2025 07:00:00. All crack and rebar target responses were merged using their common measured timestamps. This inner-join strategy is conservative: it reduces the number of usable records but avoids fabricating target values. Environmental variables were then aligned to the response timestamps by nearest or backward as of lookup because those variables were sampled more frequently. The final aligned table contains 4281 timestamps and produces 4255 supervised sequences when using a 21-sample historical window and a six-step forecast horizon. With an 8 h response interval, this corresponds to approximately 7 days of input history and 48 h ahead prediction.

Table 1. Dataset construction settings and common timestamp alignment summary.

Item	Value
Aligned timestamps	4281
Supervised sequences	4255
Input features	29
Prediction targets	10
Input window L	21 samples (approximately 7 days)
Forecast horizon H	6 samples (approximately 48 h)
Raw target observations before alignment	43,565
Target observations retained after alignment	42,810
Target observations removed by alignment	755 (1.73%)

and

Table 2. Monitoring response variables used in aligned joint-prediction dataset, including records removed by conservative common timestamp alignment. All listed sensors cover the same monitoring window from 11 March 2021 to 6 March 2025; therefore, repeated start and end dates are not listed by row.

Group	Sensor	Raw	Aligned	Discarded	Discarded (%)
crack	J01-01	4367	4281	86	1.97
crack	J02-05	4367	4281	86	1.97
crack	J10-01	4343	4281	62	1.43
crack	J13-01	4345	4281	64	1.47
crack	J15-07	4342	4281	61	1.40
rebar_stress	R02-02	4367	4281	86	1.97
rebar_stress	R02-03	4364	4281	83	1.90
rebar_stress	R03-01	4364	4281	83	1.90
rebar_stress	R03-02	4364	4281	83	1.90
rebar_stress	R15-01	4342	4281	61	1.40

summarize the retained variables and the conservative alignment loss. The common timestamp alignment retains 42,810 target observations and removes 755 target observations from 43,565 raw target observations, corresponding to a discarded proportion of 1.73%.

After the dataset summary, Figure 1 outlines the measured data workflow used in this study, from raw monitoring records to conservative alignment, sequence construction, chronological splitting, model comparison, and reproducible output generation.

The simplified project plan in Figure 2 provides the engineering context for the crack–rebar associations. The corresponding horizontal distances are then listed in

Table 3. The plan-based crack–rebar spatial associations used to construct the distance-informed ablation. The distances are horizontal layout distances interpreted from the engineering plan of the monitored structure and do not represent full three-dimensional force-transfer paths.

Crack Gauge	Rebar Stress Gauge	Horizontal Distance (m)
J01-01	R02-02	15.0
J02-05	R02-02	5.0
J02-05	R02-03	5.2
J02-05	R03-01	7.5
J02-05	R03-02	7.8
J10-01	R15-01	105.0
J13-01	R15-01	44.0
J15-07	R15-01	4.5

; presenting the figure before the table allows the distance values to be read as structured monitoring metadata rather than arbitrary target pairing. The locations shown in the simplified plan and the listed distances should be regarded as engineering approximations derived from the available layout information, and they are used only to evaluate the influence of spatial relationship information on prediction performance rather than as exact survey coordinates.

The public implementation package for MTTF-Net is available at https://github.com/ArthurCode-prod/MTFF-Net (accessed on 11 June 2026). The repository provides the reusable model, data reading interfaces, training and evaluation scripts, and a toy smoke-test example; the measured monitoring dataset used for the reported experiments is not included in the public repository.

Figure 3 provides the temporal context of the main environmental and structural response variables after conservative alignment.

Figure 4 gives an exploratory view of the linear correlation structure among the aligned target and environmental variables.

3. Mathematical Formulation

Table 4. Main mathematical symbols used in proposed formulation.

Symbol	Definition	Unit/Range
t	Response timestamp index	dimensionless
L	Length of historical input window	samples
H	Forecast horizon	samples
P	Number of input features	dimensionless
$Q_c$	Number of crack targets	dimensionless
$Q_s$	Number of rebar stress targets	dimensionless
${\mathbf{x}}_t$	Input feature vector at timestamp t	${\mathbb{R}}^P$
${\mathbf{X}}_{t-L+1:t}$	Historical input matrix ending at timestamp t	${\mathbb{R}}^{L\times P}$
${\mathbf{c}}_{t+H}$	Crack response vector to be forecast	${\mathbb{R}}^{Q_c}$
${\mathbf{s}}_{t+H}$	Rebar stress response vector to be forecast	${\mathbb{R}}^{Q_s}$
${\widehat{\mathbf{c}}}_{t+H}$	Predicted crack vector	${\mathbb{R}}^{Q_c}$
${\widehat{\mathbf{s}}}_{t+H}$	Predicted rebar stress vector	${\mathbb{R}}^{Q_s}$
${\mathbf{h}}_t$	Shared latent representation from MTTF-Net	${\mathbb{R}}^d$
$\mathrm{\Theta }$	Trainable model parameters	–
$\mathcal{L}$	Joint training loss	–
$\mathcal{P}$	Set of plan-based crack–rebar sensor associations	–
${\delta }_{ij}$	Horizontal distance between crack gauge i and rebar-stress gauge j	m
${\eta }_{ij}$	Inverse distance score for associated crack–rebar pair	m−1
${\rho }_{ij}$	Row-normalized distance weight used to form rebar context for crack gauge i	dimensionless
${\kappa }_{ij}$	Column-normalized distance weight used to form crack context for rebar stress gauge j	dimensionless
${\mathbf{u}}_t^{\left(s\right)}$	Distance-weighted rebar stress context vector for crack gauges at timestamp t	${\mathbb{R}}^{Q_c}$
${\mathbf{v}}_t^{\left(c\right)}$	Distance-weighted crack context vector for rebar-stress gauges at timestamp t	${\mathbb{R}}^{Q_s}$
${\mathbf{x}}_t^{\mathrm{sp}}$	Distance-augmented input feature vector at timestamp t	${\mathbb{R}}^{P+Q_c+Q_s}$

lists the main symbols used in the formulation, and each symbol has a single meaning throughout this manuscript.

Let t denote the index of an aligned response timestamp. The crack response vector and rebar stress response vector are denoted by

$${\mathbf{c}}_t={\left[c_{t,1},c_{t,2},\dots ,c_{t,Q_c}\right]}^{\top }, {\mathbf{s}}_t={\left[s_{t,1},s_{t,2},\dots ,s_{t,Q_s}\right]}^{\top },$$

(1)

where $Q_c=5$, and $Q_s=5$ in this study. The stacked response vector is

$${\mathbf{r}}_t=\left[\begin{array}{c}{\mathbf{c}}_t\\ {\mathbf{s}}_t\end{array}\right]\in {\mathbb{R}}^{Q_c+Q_s}.$$

(2)

At timestamp t, the input feature vector ${\mathbf{x}}_t\in {\mathbb{R}}^P$ contains historical crack and rebar responses, local temperatures, reservoir water level, antecedent rainfall features, the auxiliary environmental signal, and calendar encodings. For a look-back window of length L, the input matrix is

$${\mathbf{X}}_{t-L+1:t}=\left[\begin{array}{c}{\mathbf{x}}_{t-L+1}^{\top }\\ {\mathbf{x}}_{t-L+2}^{\top }\\ ⋮\\ {\mathbf{x}}_t^{\top }\end{array}\right]\in {\mathbb{R}}^{L\times P}.$$

(3)

The supervised learning set is defined as

$$\mathcal{D}={\left\{\left({\mathbf{X}}_{t-L+1:t},{\mathbf{c}}_{t+H},{\mathbf{s}}_{t+H}\right)\right\}}_{t=L}^{T-H},$$

(4)

where $H=6$ is the forecast horizon. For any scalar variable z, standardization is performed using parameters estimated from the training split only:

$$\tilde{z}={\displaystyle \frac{z-{\mu }_z}{{\sigma }_z+ϵ}},$$

(5)

where ${\mu }_z$ is the training set’s mean, ${\sigma }_z$ is the training set’s standard deviation, and $ϵ$ is a small numerical constant.

Plan-Based Distance Prior

To examine whether the available spatial information can inform the joint model, the plan-based crack–rebar associations in Table 3 are converted into an inverse-distance prior. Let $\mathcal{P}$ denote the set of associated crack–rebar gauge pairs, and let ${\delta }_{ij}$ be the horizontal distance between crack gauge i and rebar stress gauge j. The unnormalized inverse distance score is

$${\eta }_{ij}=\left\{\begin{array}{cc}{\delta }_{ij}^{-1},\hfill & (i,j)\in \mathcal{P},\hfill \\ 0,\hfill & (i,j)\notin \mathcal{P}.\hfill \end{array}\right.$$

(6)

The row-normalized and column-normalized spatial weights are

$${\rho }_{ij}={\displaystyle \frac{{\eta }_{ij}}{{\sum }_{j^{\prime }=1}^{Q_s}{\eta }_{i{j}^{\prime }}}}, {\kappa }_{ij}={\displaystyle \frac{{\eta }_{ij}}{{\sum }_{i^{\prime }=1}^{Q_c}{\eta }_{i^{\prime }j}}},$$

(7)

where ${\rho }_{ij}$ forms a rebar stress context for each crack gauge, and ${\kappa }_{ij}$ forms a crack response context for each rebar stress gauge. At timestamp t, the distance-weighted contextual variables are

$$u_{t,i}^{\left(s\right)}={\displaystyle \sum _{j=1}^{Q_s}}{\rho }_{ij}{s}_{t,j}, v_{t,j}^{\left(c\right)}={\displaystyle \sum _{i=1}^{Q_c}}{\kappa }_{ij}{c}_{t,i}.$$

(8)

The distance-augmented input vector used in the ablation model is

$${\mathbf{x}}_t^{\mathrm{sp}}={\left[{\mathbf{x}}_t^{\top },{\left({\mathbf{u}}_t^{\left(s\right)}\right)}^{\top },{\left({\mathbf{v}}_t^{\left(c\right)}\right)}^{\top }\right]}^{\top }\in {\mathbb{R}}^{P+Q_c+Q_s}.$$

(9)

These features are computed only from measured historical crack and rebar responses within the input window; no target interpolation, synthetic crack values, or synthetic stress values are introduced.

4. Proposed MTTF-Net

4.1. Shared Temporal Encoder

MTTF-Net uses a shared temporal encoder for both target groups. The shared encoder is used to learn common temporal forcing patterns induced by reservoir water level, rainfall, local thermal actions, structural inertia, and response memory, whereas the crack and stress output heads retain task-specific response characteristics. In the main MTTF-Net model, the standardized input matrix in Equation (3) is used directly. In the spatial ablation model, denoted as MTTF-Net-S, the same encoder and decoder structure are trained with the distance-augmented input in Equation (9). This design allows the plan-based spatial prior to be tested explicitly without forcing one-to-one crack–stress target pairing. The standardized input matrix is projected into a d-dimensional latent space:

$${\mathbf{Z}}_t^{\left(0\right)}={\tilde{\mathbf{X}}}_{t-L+1:t}{\mathbf{W}}_e+{\mathbf{E}}_{\mathrm{pos}},$$

(10)

where ${\mathbf{W}}_e\in {\mathbb{R}}^{P\times d}$ is the input projection matrix, and ${\mathbf{E}}_{\mathrm{pos}}\in {\mathbb{R}}^{L\times d}$ is the learnable positional embedding.

For a Transformer encoder layer, multi-head attention is computed as

$$MHA(\mathbf{Q},\mathbf{K},\mathbf{V})=Concat\left({\mathbf{O}}_1,\dots ,{\mathbf{O}}_M\right){\mathbf{W}}^O,$$

(11)

with

$${\mathbf{O}}_m=softmax\left({\displaystyle \frac{\mathbf{Q}{\mathbf{W}}_m^Q{\left(\mathbf{K}{\mathbf{W}}_m^K\right)}^{\top }}{\sqrt{d_k}}}\right)\mathbf{V}{\mathbf{W}}_m^V,$$

(12)

where M is the number of attention heads, $d_k$ is the key dimension, and ${\mathbf{W}}_m^Q$, ${\mathbf{W}}_m^K$, ${\mathbf{W}}_m^V$, and ${\mathbf{W}}^O$ are trainable matrices. A lightweight one-dimensional convolutional residual block is applied after the Transformer encoder to enhance local temporal variation. Attention pooling then maps the encoded sequence to a shared state:

$${\alpha }_{\mathsf{\ell }}={\displaystyle \frac{exp\left({\mathbf{u}}^{\top }tanh\left({\mathbf{W}}_a{\mathbf{z}}_{t,\mathsf{\ell }}+{\mathbf{b}}_a\right)\right)}{{\sum }_{j=1}^{L}exp\left({\mathbf{u}}^{\top }tanh\left({\mathbf{W}}_a{\mathbf{z}}_{t,j}+{\mathbf{b}}_a\right)\right)}}, {\mathbf{h}}_t={\displaystyle \sum _{\mathsf{\ell }=1}^L}{\alpha }_{\mathsf{\ell }}{\mathbf{z}}_{t,\mathsf{\ell }},$$

(13)

where ${\mathbf{z}}_{t,\mathsf{\ell }}$ is the encoded latent vector at the ℓth position in the input window.

4.2. Residual Multi-Task Decoders

Because the monitoring responses are highly continuous over time, MTTF-Net predicts response increments relative to the last observed standardized response. The standardized prediction is

$$\left[\begin{array}{c}{\widehat{\tilde{\mathbf{c}}}}_{t+H}\\ {\widehat{\tilde{\mathbf{s}}}}_{t+H}\end{array}\right]=\left[\begin{array}{c}{\tilde{\mathbf{c}}}_t\\ {\tilde{\mathbf{s}}}_t\end{array}\right]+\left[\begin{array}{c}{g}_c({\mathbf{h}}_t;{\mathrm{\Theta }}_c)\\ g_s({\mathbf{h}}_t;{\mathrm{\Theta }}_s)\end{array}\right],$$

(14)

where $g_c(·)$ and $g_s(·)$ are task-specific fully connected decoders for crack and stress responses. All trainable weights and biases in the embedding, encoder, pooling module, and two decoders are collectively denoted by $\mathrm{\Theta }$.

The loss function combines target errors and a weak continuity regularization term:

$$\mathcal{L}\left(\mathrm{\Theta }\right)={\displaystyle \frac{{\lambda }_c}{N{Q}_c}}{\displaystyle \sum _{i=1}^N}{∥{\widehat{\tilde{\mathbf{c}}}}_i-{\tilde{\mathbf{c}}}_i∥}_2^2+{\displaystyle \frac{{\lambda }_s}{N{Q}_s}}{\displaystyle \sum _{i=1}^N}{∥{\widehat{\tilde{\mathbf{s}}}}_i-{\tilde{\mathbf{s}}}_i∥}_2^2+{\displaystyle \frac{{\lambda }_{\Delta }}{N(Q_c+Q_s)}}{\displaystyle \sum _{i=1}^N}{∥{\widehat{\tilde{\mathbf{r}}}}_i-{\tilde{\mathbf{r}}}_{i,0}∥}_2^2+{\lambda }_{\mathrm{\Theta }}{\parallel \mathrm{\Theta }\parallel }_2^2.$$

(15)

Here, N is the number of training sequences, ${\tilde{\mathbf{r}}}_{i,0}$ is the last observed standardized stacked response in the input window, ${\lambda }_c$ and ${\lambda }_s$ balance the two task groups, ${\lambda }_{\Delta }$ controls response-continuity regularization, and ${\lambda }_{\mathrm{\Theta }}$ denotes weight decay.

Figure 5 illustrates how the shared temporal encoder and the two task-specific residual heads are connected.

5. Experimental Design

5.1. Benchmark Algorithms

The proposed model is compared with six baselines. Persistence uses the last observed target vector as the forecast. Ridge regression represents a strong linear autoregressive benchmark. Random forest and Extra Trees represent bagged tree ensembles. XGBoost represents boosted tree learning. The GRU network represents a recurrent deep learning baseline. These models cover naïve persistence, linear regression, ensemble learning, boosted nonlinear regression, recurrent neural forecasting, and the proposed attention-based multi-task temporal fusion. In addition, MTTF-Net-S is evaluated as a spatial-prior ablation. It uses the same neural architecture and training settings as MTTF-Net but appends the ten distance-weighted contextual variables defined in Equations (8) and (9).

All methods used the same input window, target horizon, feature matrix, and target vector. The chronological split contained 2978 training sequences, 638 validation sequences, and 639 test sequences. Standardization was fitted only on the training split. Tree and Ridge models were trained on the combined training and validation data after hyperparameters were fixed. Neural models used the validation split for early stopping.

The input window length was fixed at $L=21$ samples because the response interval was approximately 8 h; thus, the model received about 7 days of history and covered repeated short-term thermal and operational cycles. To verify that this choice was not arbitrary, the target autocorrelation was inspected at representative lags. The mean target autocorrelation remained high at lag 3 (0.975), lag 6 (0.961), lag 12 (0.946), and lag 21 (0.928), indicating that a one-week window preserves strong response memory while avoiding an unnecessarily long input sequence. The forecast horizon was set to $H=6$ samples, corresponding to approximately 48 h, because this horizon is long enough for operational warning yet short enough to remain relevant to routine inspection and maintenance decisions.

5.2. Metrics

For a target group with K scalar observations in the test set, the mean absolute error, root mean squared error, and coefficient of determination are

$$MAE={\displaystyle \frac{1}{K}}{\displaystyle \sum _{k=1}^K}\left|y_k-{\widehat{y}}_k\right|,$$

(16)

$$RMSE=\sqrt{{\displaystyle \frac{1}{K}}\sum _{k=1}^K{\left(y_k-{\widehat{y}}_k\right)}^2},$$

(17)

$$R^2=1-{\displaystyle \frac{{\sum }_{k=1}^K{\left(y_k-{\widehat{y}}_k\right)}^2}{{\sum }_{k=1}^K{\left(y_k-\overline{y}\right)}^2}}, \overline{y}={\displaystyle \frac{1}{K}}{\displaystyle \sum _{k=1}^K}{y}_k.$$

(18)

Metrics are reported for all targets together and separately for crack and rebar stress groups. For the “all-target” rows, MAE, RMSE, and the flattened $R^2$ were computed after flattening the complete multi-output prediction vector. Because rebar stress channels have larger numerical ranges and variances than crack width channels,

Table 5. Test set performance for 48 h ahead joint prediction. MAE and overall RMSE are computed after flattening all target values; macro $R^2$ is the arithmetic mean of the per-target $R^2$ values.

Model	MAE	Overall RMSE	Flattened ${\mathit{R}}^2$	Macro ${\mathit{R}}^2$	Crack RMSE (mm)	Stress RMSE (MPa)
Persistence	1.1393	2.6590	0.9796	0.9114	0.0164	3.7604
Ridge	1.0683	2.2968	0.9848	0.9360	0.0149	3.2481
Random Forest	1.5439	3.3355	0.9680	0.7307	0.0269	4.7170
Extra Trees	1.4552	3.0957	0.9724	0.7265	0.0239	4.3780
XGBoost	1.1011	2.3957	0.9835	0.9150	0.0144	3.3880
GRU	1.3949	2.9686	0.9746	0.8531	0.0196	4.1982
MTTF-Net	1.0035	2.3813	0.9837	0.9345	0.0141	3.3676

additionally reports a macro-averaged $R^2$, defined as the arithmetic mean of the per-target $R^2$ values. This macro metric gives each response channel equal weight and helps avoid over-interpreting the flattened score.

6. Results

6.1. Overall Prediction Accuracy

Table 5 summarizes the overall test set performance and separates crack RMSE from rebar stress RMSE so that the different physical units are not mixed in interpretation.

The benchmark results show that the 48 h ahead task remains strongly autoregressive. Ridge regression has the lowest overall RMSE (2.2968) and stress RMSE (3.2481). However, MTTF-Net has the lowest overall MAE (1.0035) and the lowest crack RMSE (0.0141), indicating that the nonlinear shared temporal encoder is particularly useful for the crack response group. Compared with persistence, MTTF-Net reduces the overall RMSE from 2.6590 to 2.3813, corresponding to a 10.44% reduction, and improves the flattened overall $R^2$ from 0.9796 to 0.9837. The macro-averaged $R^2$ values are 0.9360 for Ridge and 0.9345 for MTTF-Net, confirming that the two models are very close when each target is weighted equally. Random forest, Extra Trees, and GRU are less competitive on this dataset, which suggests that simple nonlinear model capacity is not sufficient when the response is dominated by linear temporal inertia.

The RMSE comparison in Figure 6 separates crack targets, rebar stress targets, and the mixed all-target group so that the different physical units are not hidden.

6.2. Distance-Informed Spatial Ablation

To evaluate the practical value of the plan-distance information,

Table 6. Ablation test of the plan-based distance-informed input. MTTF-Net-S uses the same architecture, chronological split, and training protocol as MTTF-Net but appends ten inverse-distance spatial context features derived from Table 3.

Model	Features	MAE	All RMSE	Flat. ${\mathit{R}}^2$	Macro ${\mathit{R}}^2$	Crack RMSE (mm)	Stress RMSE (MPa)
MTTF-Net	29	1.0035	2.3813	0.9837	0.9345	0.0141	3.3676
MTTF-Net-S	39	1.0712	2.5052	0.9819	0.9309	0.0157	3.5428

compares the original MTTF-Net with MTTF-Net-S under the same data split, forecast horizon, optimizer settings, and early-stopping rule.

MTTF-Net-S increases the input dimension from 29 to 39 by appending five distance-weighted rebar context variables for the crack gauges and five distance-weighted crack context variables for the rebar stress gauges. The spatial prior gives a controlled sensitivity test rather than a new source of target information: all added variables are computed from measured historical responses inside the look-back window. In the independent test period, MTTF-Net-S remains close to the original model in macro-averaged $R^2$ (0.9309 versus 0.9345) and slightly reduces the RMSE of the R03-01 stress channel from 5.1404 MPa to 5.0925 MPa. However, it is not superior in the overall metrics, with the overall RMSE increasing from 2.3813 to 2.5052. This mixed outcome is informative because it shows that the plan distances capture useful engineering context for some local channels, while horizontal distance alone is too limited to improve all responses when the input already contains target histories and local temperature channels. A more complete spatial model would require three-dimensional gauge coordinates, embedment depth, structural zoning, and preferably finite-element- or inspection-based connectivity information.

6.3. Per-Target Performance of MTTF-Net

Table 7. Per-target test performance of MTTF-Net for 48 h ahead prediction task. Crack errors are in millimeters, and rebar stress errors are in MPa.

Target	Group	Unit	MAE	RMSE	${\mathit{R}}^2$
crack_J01-01	crack	mm	0.0211	0.0296	0.9860
crack_J02-05	crack	mm	0.0061	0.0083	0.9935
crack_J10-01	crack	mm	0.0025	0.0031	0.9703
crack_J13-01	crack	mm	0.0035	0.0045	0.9979
crack_J15-07	crack	mm	0.0029	0.0037	0.9081
rebar_R02-02	rebar_stress	MPa	0.6450	0.8378	0.9902
rebar_R02-03	rebar_stress	MPa	3.1063	4.2088	0.9906
rebar_R03-01	rebar_stress	MPa	3.2830	5.1404	0.8950
rebar_R03-02	rebar_stress	MPa	0.5122	0.6609	0.9946
rebar_R15-01	rebar_stress	MPa	2.4528	3.3806	0.6192

reports the per-target MTTF-Net errors and $R^2$ values, with units listed explicitly for each response group.

The per-target results in Table 7 show that MTTF-Net maintains high $R^2$ values for most response channels, while the R15-01 stress channel is more difficult during the test period. The model is especially accurate for crack channels because their amplitudes are smaller and their temporal evolution is smoother in the test period. Stress channels have larger numerical ranges and more abrupt local variations, which explains their higher absolute RMSE.

Figure 7 shows representative test period time series, and Figure 8 compares observed and predicted values for the two response groups using their respective physical units.

6.4. Feature Relevance

Random forest feature importance was used only as an auxiliary exploratory tool. It was not used to select the final model inputs. Figure 9 shows that historical response channels dominate the benchmark feature ranking, while water level, rainfall accumulation, and temperature channels provide contextual information. This is consistent with the strong performance of persistence and Ridge regression and confirms that any nonlinear model must first respect the response continuity structure of the monitoring data. The large importance assigned to the local temperature channel T-R02-03 should be interpreted cautiously. This channel records a local thermal condition near a stress monitoring zone and covaries with nearby structural responses in the aligned monitoring record. In impurity-based random forest importance, a set of correlated predictors often concentrates importance on one representative variable, even when several physical drivers act together. Therefore, the high score is interpreted as an exploratory indication of local thermo-mechanical coupling and correlated feature competition, not as a claim that this temperature channel is the only or dominant physical driver.

6.5. Computational Cost

Table 8. Wall-clock training and inference time measured on local workstation. Time is reported in seconds.

Model	Time (s)	Epochs
Persistence	0.00	–
Ridge	0.02	–
Random Forest	76.21	–
Extra Trees	34.92	–
XGBoost	28.76	–
GRU	57.85	61
MTTF-Net	46.21	27

reports the measured wall-clock time for each model on a local workstation.

The experiments were executed on a local workstation CPU. MTTF-Net required early-stopped neural training but remained tractable for the dataset size. The computational burden was acceptable for offline model development and periodic retraining; for real-time warning deployment, inference cost was negligible compared with data acquisition and database access.

7. Discussion

7.1. Why Ridge Is a Strong Baseline

The results demonstrate that the dataset has a strong linear autoregressive component. This is reasonable for dam monitoring: crack and stress responses usually vary continuously under slowly changing thermal and hydraulic actions. Therefore, a high-capacity nonlinear model is not automatically superior. This finding is important because some recent SHM studies emphasize deep learning without comparing sufficiently strong persistence and linear baselines. In the present study, Ridge regression was not treated as a weak reference but as a serious benchmark.

7.2. Computational Cost Versus Marginal Gain

The comparison with Ridge regression is important for practical deployment. Ridge training required approximately 0.02 s, whereas MTTF-Net required 46.21 s in the reported CPU experiment. Therefore, if the operational task is limited to rapid routine screening dominated by slowly varying stress responses, Ridge regression is a more economical baseline. The proposed MTTF-Net is justified in scenarios where operators need a unified crack–stress forecasting interface, where crack response deviations are of primary concern, where nonlinear thermal or rainfall-related interactions may appear, or where the model will be periodically retrained offline and then used for low-cost inference. In such settings, the additional training cost is modest relative to the monitoring cycle and may be acceptable because MTTF-Net provides the best crack RMSE and the best overall MAE while remaining competitive in overall RMSE.

7.3. Value of Joint Multi-Task Fusion

Although Ridge has the lowest overall RMSE, MTTF-Net provides the best crack RMSE and the best overall MAE. This suggests that joint nonlinear temporal fusion can better track some local crack-response deviations, while stress prediction remains more sensitive to the large-amplitude and abrupt components of rebar measurements. From an engineering perspective, the proposed framework is useful because it provides a unified crack–stress prediction interface and can be extended with more monitoring channels, physical constraints, or probabilistic warning intervals.

7.4. Role of the Plan-Based Spatial Prior

The added spatial ablation helps clarify the role of sensor location information. The plan layout confirms that several crack and rebar stress gauges are physically close, especially J15-07–R15-01 and the J02-05 group near R02/R03 gauges. MTTF-Net-S gives a comparable but not overall superior result, with a small improvement for the R03-01 stress channel and slightly lower global accuracy. This should not be interpreted as evidence that spatial mechanics are irrelevant. Rather, it indicates that the available horizontal distances are a useful but limited proxy for structural coupling. Rebar stress and crack opening are affected by local reinforcement arrangement, embedment depth, concrete joint condition, temperature gradients, water pressure, and restraint conditions, none of which can be fully represented by a single plan-distance value. Therefore, the plan-based prior is useful for transparency and engineering interpretation, while stronger spatial modeling should rely on verified three-dimensional coordinates and structural connectivity.

7.5. Limitations

Several limitations should be acknowledged. First, the added plan layout distances improve the engineering transparency of the shared crack–rebar representation, but they remain two-dimensional proxies and do not include embedment depth, reinforcement arrangement, joint conditions, or finite element connectivity. Second, the experiments are based on one monitored dam and one chronological test period, so transferability should be verified on additional structures or operating stages. Third, the present model provides deterministic point forecasts; operational warning would benefit from calibrated uncertainty intervals around the predicted crack and stress responses.

8. Conclusions

This study constructed a reproducible manuscript package for joint 48 h ahead prediction of dam crack responses and rebar stress from measured multi-source monitoring data. The aligned dataset contains 4281 common response timestamps and 4255 supervised sequences from five crack targets and five rebar-stress targets. No synthetic target responses were created.

The proposed MTTF-Net uses a shared Transformer encoder, attention pooling, response-residual decoding, and multi-task loss. In the chronological test period, Ridge regression achieved the lowest overall RMSE of 2.2968, while MTTF-Net achieved the lowest crack RMSE of 0.0141, the lowest overall MAE of 1.0035, and the second-best overall RMSE of 2.3813. Compared with persistence, MTTF-Net reduced the overall RMSE by 10.44%. The findings indicate that the monitored structure exhibits strong linear temporal inertia, but nonlinear multi-task temporal fusion improves crack response forecasting and offers a flexible framework for integrated crack–stress warning. The plan-based distance-informed ablation further shows that explicit spatial metadata can be incorporated without fabricating target data but that horizontal distance alone is not sufficient to deliver a consistent accuracy gain. This distinction is important for future graph-based extensions, where richer structural connectivity should be introduced before strong spatial mechanics claims are made.

Future work will focus on three directions: introducing richer structural metadata such as three-dimensional sensor coordinates, embedment depth, reinforcement layout, structural zoning, and finite-element connectivity; adding calibrated uncertainty intervals through conformal, quantile, or Bayesian forecasting; and validating the framework on additional dams, construction stages, and hydrological conditions.