The Urban Uncertainty Hypothesis: A Probabilistic Deep Learning Framework for Morphological Flux in Complex Cities

Cities are increasingly recognised not as static artefacts but as complex adaptive systems characterised by non-linear dynamics, feedback loops, and emergent behaviours [1]. Yet, traditional urban planning often remains tied to deterministic representations that prioritise fixed spatial boundaries and stable land-use categories. In rapidly changing urban contexts, this creates a methodological tension: the more planning analysis concentrates on precise morphological delineation, the more difficult it becomes to anticipate unstable socio-functional flux. This paper frames that tension as the Urban Uncertainty Hypothesis, a heuristic proposition intended to guide empirical modelling rather than a universal law. In such contexts, the master plan may lose relevance when organic morphological change proceeds faster than institutional planning cycles [2].

While recent advancements in Deep Learning (DL)—particularly Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs)—have demonstrated unprecedented capabilities in modelling complex urban transformations, they introduce a new epistemological challenge: the “black box” problem [3]. Current predictive models act as opaque oracles; they generate high-accuracy forecasts of urban sprawl or land-use change but fail to explain why these changes occur [4]. For urban planners and policymakers, a prediction without explanation is of limited utility [5]. The inability to trace the causal pathways within these algorithms creates a barrier to their adoption in strategic decision-making, where accountability and interpretability are paramount [6].

This research addresses this critical gap by proposing a novel, interpretable predictive framework. We introduce a hybrid architecture that couples Convolutional Long Short-Term Memory (ConvLSTM) networks—designed to capture spatiotemporal dependencies—with Explainable AI (XAI) techniques. By integrating interpretability layers, specifically shapley additive explanations (SHAP), the proposed model moves beyond mere forecasting. It decodes the decision boundaries of the neural network, allowing the hidden drivers of morphological change—infrastructural, social, or economic—to be visualised. This shift from black-box to glass-box modelling is essential for moving towards what we term probabilistic urbanism.

This framework is evaluated using Al-Mansour District, Baghdad, as a primary case study because of its significant geopolitical disruptions and rapid urban changes in recent decades. By training the model on morphological data from 2000 to 2025, this study seeks to: (1) measure the balance between urban form stability and functional change; (2) produce scenario-based projections of future morphological change and explicitly document their uncertainty; and (3) offer clear, interpretable explanations for observed transformations.

This section presents the computational framework used to operationalise the Urban Uncertainty Hypothesis. The methodology adopts a hybrid deep-learning approach, combining ConvLSTM networks for spatiotemporal prediction with XAI techniques for interpretability. The workflow consists of four stages: data collection and preprocessing, model architecture, interpretability integration, and performance evaluation, as summarised in Figure 1.

Table 1 summarises the dataset characteristics and model input configuration.

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Table 1. Summary of dataset characteristics and model input configurations

Source: Authors.

3.1 Study area and data preparation

Al-Mansour District, Baghdad, represents a complex case for modelling urban morphological change because of its organic spatial structure, rapid urbanisation, and exposure to socio-political instability. A multi-temporal database was constructed for the period 2000–2025 at five-year intervals (T = 6: 2000, 2005, 2010, 2015, 2020, and 2025). The database consists of three main layers: (1) Land Use/Land Cover (LULC) series extracted from the classification of satellite images and historical maps within major urban categories (urban fabric, vegetation cover, water, bare land); (2) the topology of the street network, represented by a structural rigidity index from which ΔS is derived using geometric-topological grid metrics; and (3) socio-functional factors based on density proxies that reflect the intensity of use and attraction. All inputs were standardised spatially and radiometrically and transformed into a unified 512 × 512 patch representation, organised as a four-dimensional tensor (T, C, H, W) to preserve fine morphological detail. Figure 2 illustrates the observed morphological evolution, while Table 2 details the data sources, preprocessing decisions, and analytical roles of the input layers.

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Table 2. Data sources, preprocessing decisions, and analytical role of each input layer

3.2 Spatiotemporal predictive model

To address the limitations of traditional CA in capturing non-linear dependencies, this study employs a ConvLSTM architecture informed by recent urban-growth forecasting work [19]. Unlike standard LSTMs, which manage temporal sequences but do not explicitly preserve spatial topology, ConvLSTM replaces matrix multiplications with convolutional operations within recurrent transitions, enabling the model to learn temporal evolution (flow) and spatial structure (form) simultaneously [20]. The core ConvLSTM cell is governed by Eqs. (1)-(5), determining the flow of information through the input (it), forget (ft), and output (ot) gates:

$i_t=\sigma\left(W_{x i} * X_t+W_{h i} * H_{t-1}+W_{c i} \circ C_{t-1}+b_i\right)$    (1)

$f_t=\sigma\left(W_{x f} * X_t+W_{h f} * H_{t-1}+W_{c f} \circ C_{t-1}+b_f\right)$    (2)    

$C_t=f_t \circ C_{t-1}+i_t \circ \tanh \left(W_{x c} * X_t+W_{h c} * H_{t-1}+b_c\right)$   (3)

$o_t=\sigma\left(W_{x o} * X_t+W_{h o} * H_{t-1}+W_{c o} \circ C_t+b_o\right)$    (4)

$H_t=o_t \circ \tanh \left(C_t\right)$    (5)

where, ∗ denotes the convolution operator, ∘ denotes the Hadamard (element-wise) product, $\mathrm{X}_{\mathrm{t}}$ is the input tensor (urban map at time t ), $H_{\mathrm{t}}$ is the hidden state (morphological memory), and $\mathrm{C}_{\mathrm{t}}$ is the cell state (accumulated urban knowledge). The Mansour-Net model is a specialised architecture developed to implement this framework, using skip connections to link encoder and decoder layers and preserve fine spatial details such as street edges that are often lost during down-sampling.

3.3 Interpretability framework: From black box to glass box

A critical contribution of this research is the decoding of the predictive model using XAI. The framework applies SHAP, a game-theoretic approach to feature attribution, to calculate the marginal contribution of each urban feature (e.g., proximity to main roads, neighbourhood density) to the model prediction. For a prediction f(x) at location (i, j), the SHAP value ϕk assigns an importance score to feature k. This process generates uncertainty heatmaps in which high SHAP values indicate strong drivers of change (deterministic zones), while conflicting or low-confidence attribution values indicate areas of high entropy, interpreted as uncertainty zones that support the operationalisation of the proposed ΔS–ΔF framework.

3.4 Operational definitions of ΔS and ΔF

To avoid treating ΔS and ΔF as purely metaphorical terms, this study operationalises them as bounded, reproducible indices computed from the calibrated probabilistic output of the ConvLSTM model. Let i denote a raster cell, N the total number of cells, c one of K land-use classes, $\mathrm{y}_{\mathrm{i}, \mathrm{c}, \mathrm{t}}$ the one-hot encoded observed class, and $\mathrm{p}_{\mathrm{i}, \mathrm{c}, \mathrm{t}+1}$ the calibrated softmax probability predicted for the next time step. A small $\varepsilon=10^{-8}$ is added inside logarithms to avoid undefined values.

Observed morphological stability, $\Delta S^{\text {obs}}$. For historical periods where two observed maps are available, stability is estimated from the actual transition rate. For cell $\mathrm{i}, \mathrm{T}_{\mathrm{i}, \mathrm{t}}=1$ if the observed class changes between t and $\mathrm{t}+1$, and 0 otherwise:

$\Delta S^{o b s}=1-(1 / N) \Sigma_i T_{i, t}$    (6)

Values close to 1 indicate a stable urban fabric; values close to 0 indicate extensive change. This metric is used for the historical periods 2000–2025.

Forecast functional flux, ΔF. For the 2030 forecast, where observed ground truth is unavailable, functional flux is represented as a predictive-entropy proxy:

$\Delta F=-[1 /(N \log K)] \Sigma_i \Sigma_c p_{i, c} \log \left(p_{i, c}+\varepsilon\right)$    (7)

The index is bounded in [0, 1]. Values near 0 indicate high confidence in a single outcome; values near 1 indicate several plausible outcomes. High ΔF is interpreted as a proxy for socio-functional instability, not as directly measured flow.

Forecast spatial precision, ΔS(pred). For future years, spatial precision is the complement of predictive entropy:

$\Delta S^{\text {pred}}=1-\Delta F$    (8)

Consequently, $\Delta S^{\text {pred}}$ and $\Delta \mathrm{F}$ are complementary uncertainty indicators in the forecast period, while $\Delta \mathrm{S}$ (obs) remains the preferred empirical stability measure for historical validation. Because Eq. (8) defines $\Delta \mathrm{S}$ (pred) as the algebraic complement of $\Delta \mathrm{F}$, their inverse movement in the forecast period is a definitional identity and is not interpreted as an empirical finding; the hypothesis is instead tested against observed data, as described below.

Testing the Urban Uncertainty Hypothesis. The hypothesis is not assumed to be proven by the complementary formulas alone. It is evaluated by examining whether high-ΔF cells overlap with observed transition zones, low-confidence prediction areas, and morphologically unstable edges. For historical periods, the testable expectation is that cells with lower ΔS(obs) show higher predictive entropy and higher classification error. The hypothesis is therefore treated as a diagnostic framework requiring empirical validation, not as a universal law.

3.5 Experimental design and evaluation metrics

3.5.1 Calibration and temporal split

To assess forecast performance under non-stationary dynamics, the study uses a strictly chronological validation design rather than random pixel shuffling. Each pair of consecutive maps is treated as a transition sample. The training set contains 2000→2005, 2005→2010, and 2010→2015; the validation/calibration set uses 2015→2020; and the held-out test set uses 2020→2025. The 2025→2030 stage is treated only as a future scenario protocol and is not used to compute validation accuracy. The accompanying code currently performs held-out validation and provides threshold-sensitivity routines; it does not, by itself, constitute a confirmed 2030 projection unless calibrated model outputs are generated from the actual latest satellite inputs. This split reduces temporal leakage and forces the model to predict an unseen later period. Because ΔF relies on probability outputs, model confidence is calibrated on the validation transition only and then evaluated unchanged on the held-out test transition. Table 3 summarises this chronological split.

Table 3. Chronological split used for training, validation, testing, and forecasting

3.5.2 Baselines and ablation study

The proposed framework is benchmarked against three predictive baselines: CA-Markov (rule-based), U-Net (pure spatial deep baseline), and vanilla ConvLSTM (spatiotemporal baseline without additional feature design). To isolate the contribution of each information layer, an ablation protocol is specified: Model A (LULC only); Model B (LULC + street topology, ΔS-related inputs); Model C (LULC + socio-functional features, ΔF-related inputs); and Model D, the proposed model (LULC + street topology + socio-functional features). Model D-Cal adds probability calibration (temperature scaling), and Model D-UQ optionally adds MC Dropout/ensemble for uncertainty decomposition. Explainability (SHAP/Grad-CAM) is applied post-hoc to the final predictive model and evaluated separately for faithfulness and stability; it is not treated as a predictive ablation component.

3.5.3 Validation metrics and reporting protocol

All evaluations are reported on the held-out 2020→2025 transition at the common raster resolution, across three levels. Classification metrics: overall accuracy, macro-F1, per-class precision/recall, Cohen’s Kappa, and mIoU. Spatial-structure metrics: Structural Similarity Index (SSIM), built-up-area error, and boundary preservation. Probabilistic metrics: Negative Log-Likelihood (NLL), Brier Score, and Expected Calibration Error (ECE) for calibration; and Spearman correlation between predictive entropy and pixel-level error, with AUROC for error detection, for uncertainty quality. Until the held-out metrics are available, the 2030 output is described as a scenario projection, not a verified forecast. Table 4 summarises the evaluation protocol.

Table 4. Validation metrics required for defensible model assessment

Note: LULC = land use/land cover.

This study presented the Urban Uncertainty Hypothesis as a theoretical and computational framework for examining non-linear morphological change in complex urban environments. The proposed hybrid ConvLSTM-XAI architecture addresses two limitations in current planning analytics: the rigidity of conventional master-planning assumptions and the opacity of many deep-learning models. The held-out validation and baseline comparison show that the proposed model provides an interpretable probabilistic workflow, although a pure spatial U-Net achieved the strongest accuracy on several conventional metrics. The contribution of this work therefore lies in interpretability and uncertainty quantification rather than in maximising classification accuracy.

The empirical application in Al-Mansour District demonstrates the value of combining held-out validation, temporal scenario modelling, uncertainty diagnostics and explainability for analysing morphological flux. The updated 2030 outputs are now computed from the temporal ConvLSTM routine rather than retained as illustrative placeholders. The argmax-based scenario estimates 61.68% urban fabric, while the threshold analysis shows a wide sensitivity band. The generation of uncertainty heatmaps through SHAP, Grad-CAM and predictive entropy remains the main methodological contribution, translating model outputs into spatially interpretable evidence that planners can critically review.

The study has limitations. The raw spatial datasets are context-sensitive and cannot be shared publicly; the held-out accuracy is moderate; the historical sequence is short and temporally uneven; and the 2030 projection has not yet been independently validated against future observations, planning records or ensemble forecasts. Future research should incorporate alternative forecasting scenarios, expand the framework with real-time economic, mobility and demographic data, and test the Urban Uncertainty Hypothesis across multiple districts and cities.