MorphoGP: A Nonparametric Framework for Predicting Equilibrium Beach Profiles Under Tidal Influence

Xi Wu; Yanqing Wei; Hang Yin; Pengze Li; Hongshuai Qi; Xi Chen

cs.LGAug 19, 2026

MorphoGP: A Nonparametric Framework for Predicting Equilibrium Beach Profiles Under Tidal Influence

Authors: Xi Wu, Yanqing Wei, Hang Yin, Pengze Li, Hongshuai Qi, Xi Chen

Organizations: Artificial Intelligence Innovation and Incubation Institute, Fudan University, Shanghai 201203, China · Shanghai Academy of AI for Science (SAIS), Shanghai 200032, China · State Key Laboratory of Estuarine and Coastal Research, East China Normal University, Shanghai 200062, China · College of Ocean and Earth Sciences, Xiamen University, Xiamen 361104, China ·

Abstract

The prediction of equilibrium beach profiles under tidal influence is of fundamental importance for sustainable coastal development, informing shoreline protection strategies and managing coastal ecosystems under changing environmental conditions. However, it remains challenging due to the highly nonlinear interactions among wave, tide, and sedimentary processes. Traditional empirical and numerical models often exhibit limited adaptability across diverse coastal environments, with especially pronounced limitations in beach systems where tidal processes are important . To improve data-driven prediction under these conditions, this study proposes MorphoGP, a unified category-specific Gaussian process framework for predicting equilibrium beach profiles (EBPs) under tidal influence. The framework first introduces a ContourCluster model based on contrastive learning to classify tide-influenced beach morphologies automatically. Within each morphological category, a specialized Gaussian process expert learns statistical associations between environmental descriptors including waves, tides, and sediments and the beach profile's shape. A Gating Net then integrates the outputs of all experts through a probabilistic weighting mechanism to produce the final prediction. Evaluated on data from over 180 beach profiles from tide-influenced coasts along the Chinese coast, MorphoGP achieves improved predictive performance compared with conventional and deep learning models, reducing the test RMSE by about 59.3% compared with the best baseline and achieving a final RMSE of 0.297 m. The proposed framework provides a physically informed, data-driven tool for equilibrium beach-profile prediction under tidal influence and coastal management, while stronger process-level physical coupling remains an important direction for future development.

Figures & tables

Fig. 1: Overview of the proposed MorphoGP framework. Given environmental forcing descriptors

\mathbf{c}=[\text{wave},\text{tide},\text{sediment}]

and the cross-shore coordinate

s

, MorphoGP predicts the equilibrium beach profile

y(s)

. ContourCluster first groups observed profiles into

K

morphodynamic clusters. A set of cluster-specific Gaussian-process experts

\{G_{t}(\mathbf{c},s)\}_{t=1}^{K}

models profile variability within each cluster, while a gating network

g_{\Phi}(\mathbf{c})

outputs mixture weights

\mathbf{q}=\mathrm{softmax}(g_{\Phi}(\mathbf{c}))

. The final prediction is obtained by fusing the expert outputs via

\hat{y}(s)=\sum_{t=1}^{K}q_{t}\,G_{t}(\mathbf{c},s)

, yielding an uncertainty-aware profile estimate (inset).

Fig. 2: Illustration of shapelet-based matching on a representative Low Tide Terrace equilibrium beach profile [ 35 ] . Top: a discriminative subsequence (shapelet)

S

highlighted on the reference profile

r(x)

. Middle: z-normalized Euclidean distance profile

d_{u}

obtained by sliding a window

w_{u}

of length

L

along

r

. Bottom: normalized similarity weights

\chi_{u}

computed via the soft-min transform in ( 4 ), where larger

\chi_{u}

indicates higher local morphological similarity.

Fig. 3: Overall workflow for obtaining shapelets. The workflow integrates candidate extraction, a transformer-based embedding layer, and a selection strategy.

Fig. 4: Structure of the Transformer encoder. The encoder comprises three stacked Transformer layers, each equipped with three multi-head attention modules.

Fig. 5: Study area and locations of beach-profile surveys along the Chinese coast.

Model	$\bm{R^{2}}$ ( $\uparrow$ )	RMSE ( $\downarrow$ )	MAE ( $\downarrow$ )
KNN	$0.533\pm 0.041$	$0.848\pm 0.063$	$0.595\pm 0.048$
Linear Regression	$0.609\pm 0.038$	$0.776\pm 0.052$	$0.595\pm 0.045$
MLP	$0.479\pm 0.057$	$0.896\pm 0.071$	$0.601\pm 0.059$
Random Forest	$0.654\pm 0.035$	$0.730\pm 0.049$	$0.462\pm 0.037$
SVR	$0.569\pm 0.044$	$0.815\pm 0.058$	$0.550\pm 0.046$
Transformer	$0.600\pm 0.050$	$0.784\pm 0.061$	$0.527\pm 0.052$

TABLE I: Performance comparison of different models on the test set. Values are reported as mean

\pm

standard deviation across the five cross-validation folds. Arrows indicate desirable metric directions (

\uparrow

: higher is better;

\downarrow

: lower is better).

Fig. 6: Expert decomposition of the predicted cross-shore profile. (a) Predicted cross-shore profile with uncertainty. The solid curve shows the MorphoGP prediction, while the dashed curve denotes the observed elevation. The shaded band represents predictive uncertainty, providing an overall assessment of profile-scale accuracy. (b) Continuous expert mixture proportions along the profile. The stacked areas show the normalized mixture weights of individual experts, obtained by applying a softmax transformation to the pre-softmax gating logits, illustrating how expert responsibility varies smoothly in space. Local expert contributions at representative cross-shore locations. At selected positions along the transect, pie charts visualize the corresponding expert weight distributions, highlighting heterogeneous regions where different experts dominate the prediction.

Category	Variant	$\bm{R^{2}}$ ( $\uparrow$ )	RMSE ( $\downarrow$ )	MAE ( $\downarrow$ )
Reference	Full MorphoGP (ContourCluster + GP + RBF-ARD)	0.942	0.297	0.209
Classification	w/o ContourCluster (Single GP, RBF-ARD)	0.664	0.718	0.463
	Random grouping (GP experts + Gating Net)	0.612	0.772	0.507
Predictor	MLP (ContourCluster; replaces GP)	0.876	0.436	0.299
	Transformer (ContourCluster; replaces GP)	0.920	0.350	0.257
Kernel	Matérn-ARD ( $\nu=3/2$ ) (ContourCluster + GP)	0.935	0.316	0.224

TABLE II: Ablation study on key components of MorphoGP. The Reference is the full configuration (ContourCluster + GP predictor + RBF-ARD kernel). Each variant modifies only the component indicated by its category while keeping all other components identical to the reference. The random grouping control keeps the same number of groups, GP experts, and gating structure as MorphoGP, but replaces ContourCluster labels with random group assignments. Results for random grouping are reported as mean

\pm

standard deviation over five random seeds. Boldface indicates the best result across all rows.

Metric	Masselink–Short	ContourCluster
NMI with the other scheme	0.104	0.104
Silhouette score	-0.355	0.191

TABLE III: Comparison between the Masselink–Short

\Omega

–RTR classification and ContourCluster. NMI measures the label agreement between the two schemes; the silhouette score evaluates compactness and separation in the raw profile-geometry space.

Fig. 7: Representative learned shapelets projected back to observed cross-shore profile segments. S0 shows a low-gradient concave transition, S5 shows a convex slope-break-like transition, and S6 shows a steep beachface-like segment. These interpretations describe local profile-geometry patterns associated with learned subsequences and should not be regarded as deterministic geomorphological labels or direct process-level explanations.

Evaluation setting	RMSE ( $\downarrow$ )	$\bm{\overline{E}}_{\mathrm{unc}}$ ( $\uparrow$ )
Overall (random split / test folds)	0.297	0.071
Hainan held out	0.328	0.105
Fujian held out	0.351	0.113
Guangdong held out	0.314	0.097

TABLE IV: Region-held-out OOD evaluation for MorphoGP. In each run, one region is withheld for testing and the model is trained on the remaining regions. Reported RMSE and mean predictive variance

\overline{E}_{\mathrm{unc}}

are computed over all test points in the withheld region; “Overall” denotes the random-split reference.

Fig. 8: Pearson correlation heatmap of the 16 selected input variables. The analysis was conducted using all profile-point samples. Strong correlations are mainly observed within tidal-range, wave-height, and wave-period variable groups, indicating potential redundancy that should be considered when interpreting ARD-based feature relevance.

Fig. 9: Feature-wise mean ARD values with variance for all input parameters in the MorphoGP framework. Smaller ARD values indicate higher feature relevance. The most relevant variables are mainly associated with tidal range, directional wave climate, and sediment/morphological descriptors. Because several input variables are strongly correlated, the ARD results are interpreted at the level of physically meaningful variable groups rather than as strict causal evidence for isolated variables.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Model	$R^{2}$	RMSE (m)	MAE (m)
Bruun/Dean-type model	0.9855	0.1494	0.1040
Exponential model	0.9963	0.0750	0.0473

Appendix

TABLE V: Per-profile least-squares fitted analytical references. These results are reported as oracle-style curve-fitting references rather than predictive baselines.

Input parameter	Symbol	Unit	Definition / calculation
Cross-shore coordinate	$x$	m	Horizontal coordinate perpendicular to the shoreline.
Mean spring tidal range	MSR	m	Vertical difference between mean high water springs and mean low water springs.
Mean tidal range	MTR	m	Average vertical difference between adjacent high and low tides.
Relative tidal range	RTR	–	$\mathrm{RTR}=\mathrm{MSR}/H_{b}$ .
Breaker wave height	$H_{b}$	m	$H_{b}=0.39g^{1/5}(TH_{d}^{2})^{2/5}$ .
Deep-water wave height	$H_{d}$	m	Wave height in deep-water conditions.

Appendix

TABLE VI: Definitions of the input parameters used in this study.

Gaussian Process Regression

Gaussian process regression uses kernel-defined distributions over functions to make predictions and quantify uncertainty from observed data.

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Wave parameters in the nearshore are crucial for coastal engineering, shoreline protection, marine hazard assessment, and coastal management for climate resilience. Traditional monitoring systems like buoys and radar platforms offer accurate monitoring but can have high installation and maintenance expenses and limited spatial coverage. Passive ocean monitoring using video has been achieved by leveraging deep learning, however, many methods are not physically interpretable, feasible, and validated for oceanography. In thiswork, a Physics-Guided Deep Spatiotemporal Learning Framework for direct estimation of nearshore wave peak periods from passive coastal video stream is proposed. The framework combines automated temporal-variance based region-of-interest detection, multi-stage Sim-to-Real transfer learning, and physics-informed regularization to enhance the predictive accuracy and physical consistency. A variety of spatiotemporal architectures were assessed, such as transformer-based and recurrent-convolutional ones, alongside synthetic pretraining,silver-label adaptation, and expert fine-tuning. The results show that transformer-based architectures outperformed in terms of the accuracy of the instantaneous prediction, while lightweight recurrent-convolutional architectures achieved higher temporal stability and operational oceanographic skill. Ablation studies also demonstrated the benefits of physics-guided regularization in terms of trend-following consistency, and physically implausible predictions. Explainability auditing also helped to focus attention in hydrodynamically active surf-zone regions and showed good agreement with the physically derived wave propagation behavior. In general, the proposed framework shows the promise of physics-guided video-based deep learning systems for long-term coastal wave monitoring that are cost-efficient and operationally feasible.

Abstract

Gaussian Process Regression

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