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

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

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

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

arXiv:2608.18558v1Aug 19, 2026cs.LGcs.AIphysics.geo-ph

TL;DR

Predicting equilibrium beach profiles under tidal influence is difficult because tide-influenced beaches exhibit complex, variable morphologic responses and remain comparatively understudied. MorphoGP combines morphology-based classification, category-specific Gaussian-process experts, and probabilistic gating, consistently outperforming conventional machine-learning and deep-learning baselines across R2, RMSE, and MAE.

Problem
Tide-influenced beaches exhibit complex, variable morphologic responses, yet tidal effects remain comparatively understudied in equilibrium-profile research.
Method
MorphoGP combines morphology-based profile classification, category-specific Gaussian-process regression, and probabilistic gating to predict profiles from environmental descriptors.
Results
MorphoGP consistently outperforms conventional machine-learning and deep-learning baselines across R2, RMSE, and MAE on 183 Chinese coastal beaches.
Takeaways & Limitations
MorphoGP provides a probabilistic, morphology-aware framework for equilibrium beach-profile prediction with quantified uncertainty.
Takeaways & Limitations
MorphoGP predicts static morphodynamic indicators from aggregated environmental descriptors rather than resolving transient processes like process-based models.

Abstract

from arXiv · show

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.

I. INTRODUCTION … B. Process-Based Numerical Beach Morphology Models

Equilibrium beach profiles capture long-term morphological adjustment but are difficult to predict on tide-influenced coasts because tidal forcing creates heterogeneous, multimodal regimes. Existing empirical, process-based, and global data-driven models motivate MorphoGP’s morphology-conditioned probabilistic framework.

I. INTRODUCTION: Equilibrium beach profiles represent long-term adjustment to hydrodynamic forcing and sediment transport, while varying waves, sediment supply, and geological setting keep natural beaches continuously evolving.EBPs also contain information about sediment properties and geomorphic erosion and deposition.
I. INTRODUCTION: Macrotidal beaches exhibit large tidal ranges, strong tidal currents, and complex morphological responses that differ fundamentally from microtidal coasts.They occur across extensive coastlines and are socioeconomically important for coastal protection and ecosystems.
A. Progress in Equilibrium Beach Profile Research: Conventional equilibrium-profile models emphasize wave energy dissipation and sediment characteristics, assumptions that work best when tidal effects are weak or secondary.Classic formulations include the Dean profile and its extensions, while later models added exponential and segmented representations.
I. INTRODUCTION: Tidal currents and wave–tide interactions can dominate sediment transport and profile evolution, producing heterogeneous, multimodal profiles and multiple observed equilibrium-profile types.These types reflect distinct morphological regimes under different combinations of wave, tide, and sediment conditions.
I. INTRODUCTION: MorphoGP combines morphology-driven clustering, category-specific Gaussian process experts, and probabilistic model averaging to address tidal influence, regime diversity, uncertainty, and feature relevance.The study investigates over 180 Chinese tide-influenced beaches and analyzes the pronounced influence of tide-related factors on profile prediction.
A. Progress in Equilibrium Beach Profile Research: Existing tidal-beach classification schemes provide qualitative taxonomies but struggle to represent terrace-type and dissipative-type profiles shaped by macrotidal forcing.The Ω–RTR framework combines dimensionless fall velocity with relative tidal range to extend beach-state classification to tidal environments.
B. Process-Based Numerical Beach Morphology Models: Process-based numerical models resolve wave–current interactions, sediment transport, and energy dissipation, but require substantial computation, parameterization, and detailed initial and boundary conditions.Examples include DELFT3D and CSTM-ROMS.

C. Machine Learning and Deep Learning for Beach Profile Prediction … A. Overview

Prior work established machine-learning approaches for beach-profile prediction but often underrepresented tidal processes and lacked interpretability. MorphoGP addresses these gaps by combining morphology-aware unsupervised classification with category-specific Gaussian-process prediction and uncertainty-aware expert aggregation.

C. Machine Learning and Deep Learning for Beach Profile Prediction: Machine-learning studies progressed from empirical eigenfunction models and ANNs to Random Forests and CNNs for seasonal or long-term beach-profile prediction.Applications used beach geometry, wind, wave, and sea-level-rise information as inputs.
C. Machine Learning and Deep Learning for Beach Profile Prediction: Most existing studies inadequately considered tidal influence, often relied on synthetic datasets, and suffered from limited interpretability despite accurate predictions.These limitations are especially pronounced in macrotidal settings where tides strongly shape beach morphology.
C. Machine Learning and Deep Learning for Beach Profile Prediction: Gaussian processes provide MorphoGP with an interpretable, probabilistic approach that accounts for uncertainty and reveals the importance of environmental drivers.This is intended to model complex, nonlinear equilibrium beach-profile relationships under tidal influence.
D. Unsupervised Morphological Classification and Regime-Specific Ensemble Modelling: The Masselink–Short Ω–RTR framework classifies tide-influenced beach states using dimensionless fall velocity and relative tide range.These quantities represent wave forcing versus sediment settling and tidal range relative to breaking-wave height.
D. Unsupervised Morphological Classification and Regime-Specific Ensemble Modelling: Because parameter-based classes may miss profile geometry, MorphoGP develops ContourCluster to discover morphology-derived regimes for downstream Gaussian-process experts.The framework targets differences in profile curvature, intertidal width, bars, and terraces that similar Ω and RTR values may not capture.
D. Unsupervised Morphological Classification and Regime-Specific Ensemble Modelling: Regime-specific modeling replaces one global relationship with specialized predictors for heterogeneous beach states and combines their outputs through data-dependent weighting.Relevant states include reflective, dissipative, low-tide terrace, and tide-dominated morphologies.
A. Overview: MorphoGP predicts equilibrium profiles from wave, tide, and sediment descriptors using ContourCluster, category-specific Gaussian-process experts, and a gating network.For a new forcing vector, the gating network supplies expert weights, and predictions are assembled across cross-shore coordinates.

B. ContourCluster Model

ContourCluster creates objective morphology categories from beach-profile elevation sequences by learning discriminative shapelet representations with contrastive learning and clustering them with k-means. Its descriptor-guided training and post-hoc projection support interpretable geometric characterization without assigning definitive geomorphological or process-based meanings.

Module overview: ContourCluster embeds profile subsequences into a discriminative representation space using contrastive learning, then applies k-means to obtain K morphology categories.The module has two stages: shapelet extraction and clustering.
Shapelet extraction and selection: Shapelet candidates are interpolated, min–max normalized, encoded by a Transformer with positional encoding, and clustered by morphological similarity in the embedding space.The representative shapelet from each cluster is the candidate closest to its centroid.
Contrastive shapelet learning: Triplet construction uses local geometric descriptors to select positive and negative samples rather than relying on random sampling alone.Semi-hard negatives with high embedding similarity but descriptor distance exceeding τn are included to improve embedding robustness.
Geometric interpretation: Post-hoc projection locates representative shapelets on observed profiles and summarizes matched segments using position, elevation variation, slope, and curvature.These descriptions characterize geometric patterns, not definitive geomorphological labels or underlying sediment-transport mechanisms.

C. Category-Specific Experts

Category-Specific Experts use K independent Gaussian-process regressors, each dedicated to one inferred beach-morphology category. The experts map environmental drivers to profile elevations while quantifying predictive uncertainty, using ARD kernels whose hyperparameters are learned from category-specific data.

Expert Design: K independent regression models predict profile elevations for the corresponding morphology categories inferred earlier.Expert t ∈ {1, . . . , K} is responsible for the t-th category.
Expert Design: Gaussian processes with automatic relevance determination model mappings from environmental drivers to profile elevation and quantify predictive uncertainty.Each expert operates on an augmented input formed from a cross-shore coordinate and forcing vector.
Kernel Specification: Each expert uses an ARD squared-exponential covariance function with a zero mean function.The covariance function governs dependence between augmented inputs, with dimension-specific length scales.
Prediction: Category-specific training data produce Gaussian predictive distributions at new augmented inputs through the standard GP posterior.The predictive distribution is defined separately for each morphology category.
Training: Expert hyperparameters are learned by maximizing the log marginal likelihood using the category-specific training inputs and outputs.The category-specific covariance matrix is constructed from the ARD kernel.

D. Gating Net

The Gating Net combines category-specific Gaussian-process experts using cross-shore-dependent mixture weights conditioned on environmental forcing and position. Its evidence-aware contextual weighting and spatially varying expert combination avoid a single fixed mixture across the beach profile.

D. Gating Net: The gating network takes the environmental forcing vector c and cross-shore coordinate s to produce mixture weights for the K category-specific GP experts.Each expert supplies a Gaussian prediction p(y | s, c, M_t), and the gating weight is denoted q_ϕ(t | c, s).
D. Gating Net: Each expert’s logit combines normalized GP log-evidence with a learned contextual term, while β controls the trade-off between evidence and contextual weighting.The contextual term is produced by an MLP with input [c, s].
D. Gating Net: Including s allows mixture weights to vary across the upper beach, beachface, intertidal zone, and lower foreshore, reflecting spatially varying morphodynamic controls.This creates a spatially varying expert combination rather than one fixed mixture over the entire profile.
D. Gating Net: The augmented input z = [c, s] makes cross-shore position influence both each GP expert’s posterior and the gating weights.The predicted profile therefore reflects covariance-based variation within each expert and spatial variation in expert weighting.

IV. EXPERIMENTAL RESULTS AND DISCUSSION · A. Study area and Data · 1) Study Area:

The study analyzed 222 cross-shore profiles from 183 sandy beaches along the Chinese coast, spanning open-coast settings and diverse hydrodynamic and sedimentary conditions. Geographic and morphological information supported spatial analysis and links between local morphology, hydrodynamics, and sediment characteristics.

1) Study Area:: 222 cross-shore profiles were analyzed from 183 sandy beaches along the Chinese coast.The profiles formed the observational basis for the study’s experimental analysis.
1) Study Area:: The surveyed beaches were distributed along the Chinese coast.Their locations are presented in the study-area survey map.
1) Study Area:: The dataset covered open-coast settings across the surveyed beaches.This setting definition characterizes the spatial context of the profiles.
1) Study Area:: The study area encompassed a broad range of hydrodynamic conditions.This breadth enabled analysis across varying coastal forcing environments.
1) Study Area:: The study area also encompassed a broad range of sedimentary conditions.Sedimentary variability was included among the environmental characteristics represented in the data.
1) Study Area:: Basic geographic and morphological information was compiled for each beach.These data supported spatial analysis and links between local morphology, hydrodynamics, and sediment characteristics.

2) Beach profile and sediment data: … C. Performance

The study combines low-tide beach profiles, sediment measurements, modeled wave and tidal conditions, and derived descriptors within a controlled cross-validation framework. MorphoGP outperforms conventional baselines on profile prediction while representing cross-shore uncertainty, although its equilibrium-like inputs and reduced-order formulation limit direct process-model comparison.

2) Beach profile and sediment data:: Low-tide RTK-GPS surveys from April 2023 to September 2024 provide equilibrium-like intertidal profiles referenced to a common elevation datum.These profiles extend generally from the backshore to the low-water level and are practical approximations rather than complete long-term averaged equilibrium profiles.
2) Beach profile and sediment data:: Surface sediments were sampled synchronously, with at least one 500 g sample per beach collected at 5–20 cm depth and additional samples capturing spatial variability.Representative cross-shore profiles were selected according to coastal morphology and sediment distribution, with preferential sampling in the mid-tidal zone.
2) Beach profile and sediment data:: Grain-size distributions were measured using a vibrating sieve shaker across 0.063–4.00 mm meshes after drying samples at 60 ◦C for at least 24 h.Mean grain size, sorting, skewness, and kurtosis were calculated using classical sedimentological procedures.
3) Wave data:: Wave conditions were simulated with TOMAWAC at 183 nearshore observation points using unstructured meshes and combined GEBCO and CMAP bathymetry.Validation across nine typhoon events and 11 wave stations produced significant-wave-height mean bias no more than 0.14 m and wave-period mean bias no more than 0.57 s.
4) Tidal data:: Tidal conditions were simulated at the same 183 points with MIKE 21 and validated against long-term records from four national marine observation stations.Hourly water levels covered 00:00 on 1 January 2023 to 00:00 on 1 January 2024, supporting calculation of mean spring and mean tidal ranges.
5) Derived parameters:: Derived inputs combine beach slope, cross-shore width, dry-beach presence, sediment composition and grain-size descriptors with wave and tidal parameters.The dry beach is defined as the supratidal zone between mean spring high water and the foredune toe or vegetation.
B. Implementation Details: Five-fold beach-level cross-validation was executed on one NVIDIA A100 GPU, with identical optimization and stopping protocols across models.A fixed random seed, Adam learning rate of 1×10−3, up to 200 epochs with early stopping, and batch size 16 were used.
C. Performance: 0.942 is MorphoGP’s highest R2, while RMSE = 0.297 and MAE = 0.209 are its lowest reported errors, compared with RF values of 0.654, 0.730, and 0.462.The comparison includes data-driven regressors and analytical profiles, while process-based models are excluded because their transient, site-specific requirements do not match the dataset and task.

D. Morphological Validity of ContourCluster · E. Ablation and Sensitivity Analysis

ContourCluster forms geometry-based morphology regimes that differ from Masselink–Short classes while retaining recognizable local profile patterns, though these patterns are not direct process explanations. Ablation results show that morphology-aware partitioning, GP prediction, and the RBF kernel materially support MorphoGP performance, with five clusters favored in sensitivity analysis.

D. Morphological Validity of ContourCluster: ContourCluster is compared with Masselink–Short Ω–RTR classification, which remains process-oriented but is not designed to optimize profile-geometry compactness.The comparison evaluates whether learned categories represent meaningful profile-geometry regimes rather than arbitrary numerical partitions.
D. Morphological Validity of ContourCluster: NMI is 0.104, while silhouette scores are 0.191 for ContourCluster versus -0.355 for Masselink–Short, indicating distinct and more compact geometry-based groups.ContourCluster therefore does not simply reproduce traditional Ω–RTR classes in raw profile-geometry space.
D. Morphological Validity of ContourCluster: Representative shapelets correspond to a low-gradient concave transition, a convex slope-break transition, and a steep beachface-like segment with relatively stable slope.These patterns are represented by S0, S5, and S6, respectively.
D. Morphological Validity of ContourCluster: The shapelets are statistical subsequences associated with local profile geometry, not deterministic geomorphological units or direct process-level representations.Their correspondence with observed local patterns nevertheless helps reduce ContourCluster’s black-box nature and supports its use as a morphology-aware clustering module.
E. Ablation and Sensitivity Analysis: Removing ContourCluster reduces R2 from 0.942 to 0.664 and increases RMSE from 0.297 to 0.718, showing that a single global GP cannot capture heterogeneous morphology-related relationships.The ablation experiments use the same data split and training protocol for fair comparison.
E. Ablation and Sensitivity Analysis: Random grouping yields R2 = 0.612, RMSE = 0.772, and MAE = 0.507, indicating that learned contour partitions outperform arbitrary grouping.The comparison is conducted across five random seeds.
E. Ablation and Sensitivity Analysis: The GP-based MorphoGP outperforms MorphoGP-MLP and MorphoGP-Trans across all metrics, achieving the highest R2 and lowest RMSE and MAE.The input features and ContourCluster module remain unchanged across prediction architectures.
E. Ablation and Sensitivity Analysis: Among RBF, Matérn ν = 3/2, and Rational Quadratic kernels, RBF provides the best overall trade-off among R2, RMSE, and MAE; silhouette score peaks at k = 5.The cluster-count sweep covers k ∈{2, 3, . . . , 8} under identical beach-level cross-validation splits.

F. Region-Held-Out Transfer and OOD Uncertainty Assessment · G. Relevance Analysis

Region-held-out tests show that MorphoGP maintains relatively low transfer error while increasing predictive uncertainty on unseen regions, although the protocol represents only a moderate domain shift. ARD and correlation analyses identify tidal-range and morphological descriptors as strong, relatively independent, and comparatively stable sources of predictive information, while correlated wave and sediment variables require cautious interpretation.

F. Region-Held-Out Transfer and OOD Uncertainty Assessment: Predictive variance Var[ŷ(x, c)] serves as the uncertainty score, with larger values indicating lower confidence, and Eunc summarizes mean predictive variance across test points.Forcing descriptors include waves, tides, sediments, and associated variables at cross-shore coordinate x.
F. Region-Held-Out Transfer and OOD Uncertainty Assessment: Leave-one-region-out testing withholds Hainan, Fujian, or Guangdong and trains on the remaining regions, while Zhejiang and Shandong are not separate held-out groups because of limited samples.The evaluation is designed as external validation within the South China coastal dataset.
F. Region-Held-Out Transfer and OOD Uncertainty Assessment: 0.314–0.351 RMSE in region-held-out tests versus 0.297 in the random-split setting, while higher Eunc indicates reduced confidence on unseen geographic regions.Despite increased uncertainty, transfer errors remain relatively low across the three held-out regions.
F. Region-Held-Out Transfer and OOD Uncertainty Assessment: The stable transfer performance is attributed to overlapping environmental descriptor ranges across South China and to morphology-based category-specific Gaussian process experts rather than geographic labels.The protocol cannot establish universal OOD generalization under stronger shifts, including unfamiliar tidal ranges, sediment supply, coastlines, or beach types.
G. Relevance Analysis: Within each Gaussian process expert, smaller ARD length scales indicate higher output sensitivity, while Pearson correlations assess redundancy among the 16 input variables.ARD values were summarized across 50 independent training runs.
G. Relevance Analysis: Tidal-range and morphological descriptors provide strong, stable predictive information, while wave-related ARD values vary more across runs and correlated Hd and Hb require cautious interpretation.ARD relevance indicates predictive sensitivity, not definitive causal attribution.
G. Relevance Analysis: Annual Mean Spring Tidal Range has the smallest ARD value, followed by Annual Mean Tidal Range and Frequency of Dominant Wave Direction, indicating highest model sensitivity.A second tier includes Dimensionless Fall Velocity, Dominant Wave Direction, and Sorting Coefficient, whereas x, RTR, Mean Grain Size, Hd, and Hb show weaker direct sensitivity.

V. CONCLUSION

MorphoGP is a morphology-aware Gaussian process framework that predicts equilibrium-like intertidal beach profiles from wave, tide, and sediment descriptors while quantifying uncertainty. Experiments on Chinese coastal beaches show improved performance and interpretable morphological representations, while future work targets stronger physical coupling and broader generalization.

Framework: MorphoGP combines unsupervised contour classification, probabilistic Gaussian process regression, and gating-based expert aggregation for morphology-conditioned profile prediction.The framework organizes profiles into data-driven morphological regimes and learns statistical relationships between environmental forcing and observed profile geometry rather than explicitly simulating complete coastal morphodynamics.
Performance: 183 beaches along the Chinese coast show that MorphoGP consistently outperforms conventional machine learning and deep learning baselines on R2, RMSE, and MAE.ContourCluster produces compact, well-separated morphological groups, while ablation studies support the effectiveness of the clustering and Gaussian process expert components.
Uncertainty and interpretation: Predictive variance increases under region-held-out evaluation, indicating reduced confidence under moderate domain shift.Feature relevance analysis suggests tidal-range and morphological descriptors dominate profile-shape variability, with wave and sediment variables providing complementary information.
Uncertainty and interpretation: Learned shapelets and expert representations capture recurring local geometric patterns and provide an interpretable view of sub-profile morphological variability.These representations are statistical descriptors of observed geometry, not direct process-based physical units.
Contributions: MorphoGP’s main contributions are morphology-driven regime decomposition, GP-based predictive modeling, quantified uncertainty, and spatially varying expert aggregation.The framework provides probabilistic, morphology-aware equilibrium beach-profile prediction.
Future work: Future work will extend MorphoGP to continuous spatio-temporal profile evolution, stronger physical constraints, and broader cross-regional datasets for improved generalization under domain shifts.These directions address the framework’s current limits in process-level physical coupling and cross-regional applicability.

APPENDIX A PER-PROFILE FITTED ANALYTICAL REFERENCES · APPENDIX B DEFINITION OF INPUT PARAMETERS

Appendix A distinguishes per-profile least-squares curve fitting from predictive beach-profile modeling, while Appendix B defines the study’s input parameters and notes measurement limitations. The fitted analytical references use observed target profiles during parameter estimation and therefore serve as oracle-style references rather than predictive baselines.

APPENDIX A PER-PROFILE FITTED ANALYTICAL REFERENCES: Table V reports per-profile least-squares fitted analytical references as oracle-style curve-fitting references, not predictive baselines.These references concern the classical analytical formulations evaluated in the appendix.
APPENDIX A PER-PROFILE FITTED ANALYTICAL REFERENCES: Unlike MorphoGP, the fitted analytical models use each observed target profile to estimate parameters during inference.MorphoGP predicts unseen equilibrium beach profiles from environmental descriptors without using target profile elevations.
APPENDIX A PER-PROFILE FITTED ANALYTICAL REFERENCES: For each observed test profile, Bruun/Dean-type parameter A and exponential parameters B and k were optimized independently by least squares.The resulting fitted profiles were evaluated with the same metrics as the main experiments.
APPENDIX A PER-PROFILE FITTED ANALYTICAL REFERENCES: Per-profile fitting substantially improves the analytical formulations when the target profile is available.The appendix states that this demonstrates strong shape-fitting capacity but differs from the predictive task.
APPENDIX B DEFINITION OF INPUT PARAMETERS: Table VI summarizes the input parameters used in the study.The appendix places these definitions in the context of the study’s predictive framework.
APPENDIX B DEFINITION OF INPUT PARAMETERS: Φx denotes the grain-size value at which the cumulative percentage reaches x%.This notation clarifies the grain-size input definition.
APPENDIX B DEFINITION OF INPUT PARAMETERS: The lowest surveyed elevation may not always reach the mean low water spring level because of field-survey limitations.This note qualifies interpretation of the surveyed elevation data.
APPENDIX B DEFINITION OF INPUT PARAMETERS: Detailed derivations of standard morphodynamic parameters, including Ω and RTR, follow conventional definitions.The appendix directs readers to standard definitions for these parameters.