AquaCast: Urban Water Forecasting

• AquaCast is a transformer model designed for short- to medium-term forecasting of urban water height and discharge using both sensor history and rain forecasts.
• It distinguishes endogenous variables from exogenous inputs by embedding precipitation data with Conv1D patching and attention to capture temporal and inter-variable dependencies.
• The model shows significant performance improvements in hydrological forecasting, proving valuable for operational urban drainage management.

Searching arXiv for AquaCast and closely related hydrology/forecasting papers to ground the article in current literature. AquaCast is a precipitation-informed, multi-input transformer for short- to medium-term forecasting of urban water dynamics—specifically water height and discharge—across distributed drainage sensors. It was developed to address the steep, fast-response hydrology of Lausanne, Switzerland, where rainfall surges can induce abrupt hydraulic changes with minimal lag. Unlike conventional multi-to-multi forecasters, AquaCast explicitly distinguishes endogenous variables, which are the hydrological targets to be forecast, from exogenous drivers, here precipitation history and forecast reports, embedding and fusing them while forecasting targets only. The model captures inter-variable and temporal dependencies via attention, and incorporates future exogenous information as an input token, avoiding the need to predict weather, which is already available from operational forecasting services (Abdollahinejad et al., 11 Sep 2025).

1. Definition and scope

AquaCast addresses the task of multi-horizon forecasting of urban water dynamics using both endogenous history and exogenous drivers. The endogenous targets are water height and/or discharge at sensors, while the exogenous variables are precipitation history and forecast. Rainfall is treated as an exogenous time series whose short-term influence on hydrodynamics is critical in steep urban terrains, and the rationale for exogenous factors is explicit: using only endogenous histories often misses imminent rainfall-induced peaks, whereas incorporating precipitation history and especially forecast reports improves anticipatory capability (Abdollahinejad et al., 11 Sep 2025).

The model’s stated contributions are fourfold. It introduces a multi-input, multi-output transformer that fuses endogenous and exogenous sequences but forecasts endogenous targets only; it explicitly separates endogenous versus exogenous variables, so precipitation is embedded and consumed as context rather than predicted; it uses a precipitation-informed embedding in which history is encoded with Conv1D patching while future precipitation is projected to an extra token per exogenous variable; and it captures inter-variable and temporal dependencies without a graph router, via joint multivariate embeddings and attention. Empirical evaluation is reported on a real four-sensor LausanneCity dataset and on three large-scale synthesized datasets spanning increasing temporal complexity across 100 nodes (Abdollahinejad et al., 11 Sep 2025).

This framing places AquaCast in a design space distinct from approaches that either forecast all variables jointly or require homogeneous history and forecast lengths. A plausible implication is that the separation between forecast targets and forecast covariates is intended to align the model with operational settings in which weather reports are externally available and should be ingested rather than re-predicted.

2. Formal problem formulation

At time $t$, AquaCast operates on three tensors. Endogenous history is denoted

$$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$

exogenous history is

$$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$

and exogenous forecast reports for future steps are

$$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$

The goal is to forecast the endogenous targets only: $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$ The mapping is written as

$$\widehat{\mathbf{Y}}_{t+1:t+H} = f_\theta\big(\mathbf{E}_{t-L+1:t}, \mathbf{X}_{t-L+1:t}, \mathbf{F}_{t+1:t+H}\big),$$

where $f_\theta$ is AquaCast’s transformer-based model (Abdollahinejad et al., 11 Sep 2025).

The distinction between endogenous and exogenous variables is structurally central rather than merely semantic. Future precipitation enters as an input token, not as an output head. This design eliminates the need to forecast exogenous variables and enables the model to attend to their short-term influences more effectively (Abdollahinejad et al., 11 Sep 2025). In the reported experiments, the look-back window is fixed at $L = 96$ time steps, corresponding to one day at 15-minute resolution, while forecast horizons are $H \in \{96, 192, 480, 720\}$, corresponding to 1, 2, 5, and 7 days (Abdollahinejad et al., 11 Sep 2025).

AquaCast is evaluated in both multi-to-single and multi-to-multi regimes. The data block states that the model is multi-input and multi-output, yet a substantial part of the analysis emphasizes that forecast is focused solely on endogenous variables. This is not contradictory: the outputs can still be multi-target endogenous forecasts across sensors and variables, while exogenous inputs remain contextual.

3. Architecture and embedding strategy

AquaCast uses a precipitation-informed multi-input transformer built around a unified embedding for history, forecast exogenous tokenization, a transformer encoder, and an MLP prediction head (Abdollahinejad et al., 11 Sep 2025).

For historical inputs, AquaCast applies two stacked Conv1D layers jointly across all input series, meaning endogenous and exogenous history are concatenated and projected into a compact token space. The stack maps from $\mathbb{R}^{\text{History\_Length}}$ to $$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$0, with a ReLU between layers. The first Conv1D layer performs patching inspired by PatchTST, but it is not channel-independent; instead, each temporal token integrates multivariate context, so inter-variable information is already embedded in the tokens. The second Conv1D layer refines higher-level abstractions, allowing downstream attention to focus on temporally and cross-variable relevant components without requiring a separate cross-variate module or router (Abdollahinejad et al., 11 Sep 2025).

Future precipitation is handled differently. The exogenous forecast tensor $$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$1 is passed through a trainable linear projector from $$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$2 to $$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$3, producing one embedded token per forecasted exogenous variable. These forecast token or tokens are appended to the embedded history sequence. This allows different history and forecast lengths and resolutions while maintaining a common attention space (Abdollahinejad et al., 11 Sep 2025).

The transformer encoder consists of standard self-attention and feedforward blocks with residual connections and LayerNorm. Attention is applied over the full token set, comprising both joint temporal patches and exogenous forecast tokens. The paper reports the following single-head notation: $$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$4

$$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$5

$$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$6

Positional encodings are not specifically reported; instead, Conv1D patching over time provides implicit temporal locality (Abdollahinejad et al., 11 Sep 2025).

The decoder is a single-layer perceptron that converts encoder outputs into multi-step forecasts of endogenous variables only. Because exogenous variables are not forecasted, their role is entirely contextual. This suggests a deliberate asymmetry between environmental forcing and hydrological response, with the former serving as conditioning information for the latter.

4. Training protocol, preprocessing, and datasets

Each time series is standardized to zero mean and unit variance using training-set statistics, and predictions are de-standardized to the original scale. Periods with constant water height for one hour, defined as four 15-minute steps or longer, are flagged as missing; third-order spline interpolation fills gaps. Two height sensors originally sampled at 1-minute resolution are downsampled to 15-minute to match discharge resolution and unify the dataset. Optimization uses Adam with MSE loss on an RTX 4090 GPU for up to 100 epochs, with early stopping patience of 10. Depth, heads, $$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$7, feedforward dimension, dropout, batch size, and weight decay are tuned but not explicitly disclosed (Abdollahinejad et al., 11 Sep 2025).

The training objective is

$$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$8

The principal real-world benchmark is LausanneCity. It comprises four sensors across the Ouchy area: two ultrasonic level sensors at 1-minute resolution and two Doppler discharge sensors at 15-minute resolution. After preprocessing, all series are unified to 15-minute frequency. The dataset contains 49,673 steps per sensor over 517 days from July 2023 to December 2024, with train, validation, and test splits of 362, 52, and 103 days, approximately 70/10/20%. The data block reports ADF $$\mathbf{E}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_e},$$9-values below 0.05 for all four series. Hourly MeteoSwiss precipitation records for Lausanne are up-sampled to 15-minute by zero insertion and used as exogenous input; in the RainFull condition, future rain reports are also injected as forecast information (Abdollahinejad et al., 11 Sep 2025).

Three synthesized datasets are used to assess scalability and generalization. They are generated via a continuous differential model over Lausanne terrain and pipe network, involving approximately 3,500 nodes and linking a precipitation field, terrain watershed segmentation, and pipe-network delays proportional to pipe geometry. Each synthesized benchmark has 100 nodes and is driven by one of three precipitation sources: MeteoSwiss records for SynthLow, Lorenz Attractor for SynthMid, and Random Fields for SynthHigh. Temporal complexity is quantified through permutation entropy $$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$0 and Jensen–Shannon statistical complexity $$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$1: $$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$2

$$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$3

with reported median complexity values of approximately 0.207, 0.236, and 0.265 for SynthLow, SynthMid, and SynthHigh, respectively (Abdollahinejad et al., 11 Sep 2025).

5. Empirical performance and scaling behavior

The baseline used for comparison is PatchTST, described as a channel-independent transformer with patching. It does not support heterogeneous history and forecast lengths and therefore cannot train on RainFull cases; these entries are marked NA. Evaluation uses point-wise metrics MSE, MAE, RMSE, and variance-explained $$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$4, as well as a sequence-level DTW-based sample accuracy summarized by normalized AUC across thresholds $$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$5. The reported formulas include

$$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$6

(Abdollahinejad et al., 11 Sep 2025)

On LausanneCity, AquaCast is reported to exhibit state-of-the-art performance using only endogenous variables, with further gains from exogenous inputs and forecast tokens. The following summary values are reported for mean performance across four sensors.

Setting	H=96	H=720
AquaCast multi-to-single, NoRain MSE	0.4829	0.6012
AquaCast multi-to-single, RainHist MSE	0.4413	0.5852
AquaCast multi-to-single, RainFull MSE	0.1811	0.2215
AquaCast multi-to-single, NoRain $$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$7	0.2133	0.0698
AquaCast multi-to-single, RainFull $$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$8	0.6889	0.6415
PatchTST MSE	$$\mathbf{X}_{t-L+1:t} \in \mathbb{R}^{L \times N \times d_x},$$9 to 0.4886	0.6035 to 0.6433

For AquaCast multi-to-multi, the reported mean MSE progresses from 0.5012 to 0.4538 to 0.1925 at $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$0, with $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$1 from 0.1863 to 0.2602 to 0.6669 as the configuration changes from NoRain to RainHist to RainFull. At $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$2, mean MSE progresses from 0.5985 to 0.5793 to 0.2294, while $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$3 rises from 0.0512 to 0.0834 to 0.6231 (Abdollahinejad et al., 11 Sep 2025).

Sensor-level results underscore the magnitude of the precipitation-conditioning effect. For Sensor 1.A at $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$4, AquaCast in the multi-to-single setting reduces MSE from 0.4901 in NoRain to 0.1596 in RainFull; $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$5 rises from 0.3259 to 0.7811, and AUC from 0.8573 to 0.9350. For Sensor 3.C at $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$6, AquaCast multi-to-single reduces MSE from 0.6124 to 0.1811 with RainFull, a 70.42% reduction. For Sensor 4.D at $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$7, MSE decreases from 0.3878 to 0.2184, a 43.68% reduction (Abdollahinejad et al., 11 Sep 2025).

The synthesis also quantifies the effect of moving from single-target to multi-target prediction. Under RainFull, mean MSE degradation is reported as 6.77% at $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$8 and 3.54% at $$\mathbf{F}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_f}.$$9, indicating model scalability to multi-target forecasting with minimal loss (Abdollahinejad et al., 11 Sep 2025).

On the synthesized 100-node datasets, the gains remain substantial. For SynthLow at $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$0, AquaCast RainFull attains MSE 0.0489 versus PatchTST 0.6297, with $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$1 and AUC 0.9004. At $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$2, the corresponding MSEs are 0.1964 and 0.8876, with $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$3. For SynthMid at $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$4, AquaCast RainFull yields MSE 0.0348 versus 0.6638, with $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$5 and AUC 0.9498; at $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$6, MSE is 0.0987 versus 1.3443, with $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$7. For SynthHigh at $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$8, AquaCast RainFull yields MSE 0.2727 versus 0.7190 and $$\widehat{\mathbf{Y}}_{t+1:t+H} \in \mathbb{R}^{H \times N \times d_e}.$$9; at $$\widehat{\mathbf{Y}}_{t+1:t+H} = f_\theta\big(\mathbf{E}_{t-L+1:t}, \mathbf{X}_{t-L+1:t}, \mathbf{F}_{t+1:t+H}\big),$$0, the reported values are 0.2969 versus 0.8168 and $$\widehat{\mathbf{Y}}_{t+1:t+H} = f_\theta\big(\mathbf{E}_{t-L+1:t}, \mathbf{X}_{t-L+1:t}, \mathbf{F}_{t+1:t+H}\big),$$1 (Abdollahinejad et al., 11 Sep 2025).

These results are paired with two explicit trends. First, adding rain history alone produces clear gains at short horizons but smaller gains at longer horizons, because relevant rain impacts may fall outside the look-back window. Second, including rain forecast gives the largest boost across all horizons, while PatchTST shows no benefit from rain history and cannot exploit heterogeneous future inputs (Abdollahinejad et al., 11 Sep 2025).

6. Interpretation, related systems, and operational significance

AquaCast’s reported qualitative case studies indicate that rainfall events induce immediate deviations in water height and discharge, and that forecasts align closely with the timing and magnitude of these peaks, especially in RainFull configurations. The improvement in DTW-based AUC is presented as evidence of better shape alignment beyond point-wise error minimization (Abdollahinejad et al., 11 Sep 2025).

From a systems perspective, AquaCast’s architecture stands apart from two other 2025 forecasting systems appearing in adjacent application domains. “SimCast: Enhancing Precipitation Nowcasting with Short-to-Long Term Knowledge Distillation” (Yin et al., 9 Oct 2025) addresses precipitation nowcasting from radar sequences using a SimVP backbone, short-to-long term knowledge distillation, weighted MSE, and a diffusion-based refinement named CasCast. “ORCAst: Operational High-Resolution Current Forecasts” (Garcia et al., 21 Jan 2025) is a multi-stage, multi-arm network for global ocean surface current nowcasts and forecasts over one week, trained on satellite data and drifters. Both systems are distinct from AquaCast in target variable, data modality, and operational horizon, but together they clarify that the “Cast” nomenclature spans multiple hydro-environmental forecasting tasks rather than a single framework (Yin et al., 9 Oct 2025); (Garcia et al., 21 Jan 2025).

A plausible implication is that AquaCast occupies the urban hydrodynamics niche within a broader landscape of data-driven environmental forecasting systems. In that landscape, SimCast emphasizes precipitation field evolution, ORCAst emphasizes ocean surface current vectors, and AquaCast emphasizes water height and discharge at distributed urban drainage sensors. The data block also includes a 2019 study on sloping coastal aquifers, which synthesizes governing equations, optimization, contamination transport, and seawater intrusion modeling for a different class of water-management problem (Mondal et al., 2019). That work is not an AquaCast model, but it illustrates that the term “AquaCast” can also be used in design guidance for water forecasting and optimization systems beyond urban drainage.

Operationally, AquaCast is presented as useful for fast-response catchments where short lag times make forecast precipitation particularly informative. The practical motivation stated in the source is urban drainage operations, including pre-emptive storage allocation, separation of stormwater and sewage streams, and mitigation of overflow risks (Abdollahinejad et al., 11 Sep 2025). This suggests that the main operational value lies not only in lower average error, but also in earlier anticipation of rainfall-induced peaks.

7. Limitations, misconceptions, and future directions

A common misconception would be to treat AquaCast as a generic graph-based spatiotemporal model for sensor networks. The available description does not support that characterization. The model learns inter-variable dependencies implicitly through joint embedding and self-attention without requiring an explicit graph or cross-variate router (Abdollahinejad et al., 11 Sep 2025). Another possible misconception is that AquaCast predicts precipitation; the source explicitly states the opposite. Precipitation history and forecast reports are embedded and consumed as context, while only endogenous variables are forecast (Abdollahinejad et al., 11 Sep 2025).

The main limitations reported are practical rather than conceptual. Gains depend on the quality and timeliness of precipitation forecasts, so forecast errors in rain will propagate to water forecasts. Steep urban settings with short lags benefit most; flatter cities with longer hydrological response times may require longer look-back windows or different fusion strategies. Extreme rainfall may induce dynamics beyond the historical training distribution. Sensor gaps, constant readings, and Doppler uncertainty of approximately 15% can affect training and validation. Additionally, specific architectural hyperparameters such as heads, $$\widehat{\mathbf{Y}}_{t+1:t+H} = f_\theta\big(\mathbf{E}_{t-L+1:t}, \mathbf{X}_{t-L+1:t}, \mathbf{F}_{t+1:t+H}\big),$$2, depth, dropout, and batch size were tuned but not disclosed, and code and synthetic generator links are referenced but not concretely provided (Abdollahinejad et al., 11 Sep 2025).

The paper’s stated future directions include spatial graph modeling to encode network topology explicitly, probabilistic forecasting and uncertainty quantification for decision support, integrating weather model uncertainty, attention interpretability analyses, and efficient attention for longer horizons and larger node counts (Abdollahinejad et al., 11 Sep 2025). These directions indicate that the present formulation prioritizes deterministic accuracy and flexible exogenous conditioning over explicit topology modeling or uncertainty-aware outputs.

In summary, AquaCast’s novelty lies in a precipitation-informed, multi-input transformer that fuses endogenous and exogenous histories and future exogenous reports while forecasting endogenous targets only. Its embedding combines joint multivariate Conv1D patching with a single token per exogenous forecast variable, enabling heterogeneous history and forecast lengths. Across both LausanneCity and 100-node synthesized datasets, the reported results indicate consistent gains over PatchTST, especially when future rain reports are provided, and these gains persist from one-day to seven-day horizons (Abdollahinejad et al., 11 Sep 2025).