A Novel Bayesian Probabilistic Approach for Debris Flow Accumulation Volume Prediction Using Bayesian Neural Networks with Synthetic and Real-World Data Integration

Abstract

Debris flow events are complex natural phenomena that are challenging to predict, especially when data are limited or uncertain. This study presents a novel probabilistic approach using Bayesian Neural Networks (BNN) to predict possible volumes of debris flow accumulation by using synthetic and real-world data. Synthetic datasets are created based on statistical distributions informed by geomorphological and hydrological knowledge, allowing the model to learn typical behaviors when real data is scarce. BNN provide uncertainty quantification by modeling neural weights as probability distributions. The model resulting from validation on synthetic data and two real datasets from China and South Korea show strong predictive performance (R² > 0.98) and close alignment between predicted and observed volumes, even in the presence of outliers. The key strength of this integrated approach lies in its integration of synthetic data generation, real data augmentation based on Bootstrapping, expert knowledge and Bayesian deep learning to overcome limitations of traditional statistical models, improving debris flow forecasting and enabling more informed and resilient risk management strategies.

1. Introduction

The research starts from the primary consideration of the phenomena of Debris-flow and their potential accumulation volumes capable of obstructing riverbeds, thereby causing upstream and downstream impacts due to damming. The aim of this work is to propose a probabilistic approach for predicting accumulation volumes by employing Bayesian Neural Networks (BNN) with both synthetic and real-world data.

In recent years, Artificial Intelligence (AI) has increasingly been applied in geosciences to model complex natural phenomena, thanks to its ability to process large datasets and uncover both linear and non-linear relationships among variables [1]. Among AI techniques, Artificial Neural Networks (ANN) have gained particular attention due to their biological inspiration and strong performance in pattern recognition and predictive tasks. Their use has been proven effective in hydrology [2], sediment transport [3], debris flow susceptibility [4], and flood prediction [5]. However, the application of AI to debris flow is still challenging, primarily due to the intrinsic complexity of these events characterized by high spatial and temporal variability, nonlinearity, and uncertainty in triggering conditions [6].

A debris flow is a type of mass wasting process that acts on natural and engineered slopes. It is the movement of a mass of rock, dry or wet debris, under the influence of gravity [7]. Predicting the magnitude and accumulation volume of such events is essential for risk assessment and civil protection planning but is hindered by data scarcity, especially in regions where events are infrequent or monitoring infrastructures are lacking [8]. In addition, any accumulation of material along riverbeds affected by neighboring, earth flow, wet or dry debris flow, could obstruct the path of the river causing its overflow. Traditional numerical, empirical or physically based models often struggle to generalize across different geographic contexts due to their deterministic nature and rigid parameterizations [9].

Mathematical and numerical modeling is extensively employed to study debris flows, enabling analysis of their evolution, forecasting, and impact assessment. These models are grounded in differential equations derived from conservation laws (mass, momentum, and energy), often supplemented by empirical or theoretical closure relations. Among the most advanced approaches is Computational Fluid Dynamics (CFD) [10], also used for simulating sediment transport in both laminar and turbulent regimes. However, due to the computational time demands of full 3D models, simplified frameworks like the Shallow Water equations [11] or Reduced Complexity Models (RCMs), such as Cellular Automata (CA) [12], are frequently adopted. Discretization of the system domain, typically via mesh-based methods using simple geometric elements, is essential for calculating physical quantities such as stress, velocity, and flow height. Common discretization techniques include the Finite Element Method (FEM) [13], Finite Difference Method (FDM), Finite Volume Method (FVM), and Boundary Element Method (BEM) [14]. Mesh-free approaches, like Smoothed Particle Hydrodynamics (SPH) [15,16], and hybrid techniques such as the Particle Finite Element Method (PFEM) [17,18,19], offer alternatives that bypass the complexities of mesh generation. Given the inherent variability and uncertainty in physical and geo-mechanical parameters, probabilistic approaches, especially the Monte Carlo Method [20,21], are widely used for stochastic simulations. Additionally, statistical tools support the analysis of sediment grain size distributions [22] in debris flows due to erosion as well [23].

As an alternative we propose to apply a novel probabilistic approach based on BNN. In the Bayesian paradigm, probability expresses the degree of belief in an event and is updated through Bayes’ theorem [24]. BNN extends traditional ANN by treating weights as probability distributions rather than fixed values, allowing not only accurate predictions but also explicit uncertainty quantification, an essential capability in risk-sensitive decision-making [25,26].

Given the scarcity of empirical data, we can address this issue by generating synthetic data and applying domain-based assumptions to create plausible input scenarios, or by leveraging bootstrapping methods [27] on real datasets. These synthetic datasets are designed to reproduce the statistical and physical properties of real-world variables influencing debris flow dynamics (e.g., slope, rainfall intensity, soil type, lithology), allowing the model to learn typical behavior patterns even when empirical data are insufficient.

Our two-step framework includes the following: (1) generation and validation of synthetic datasets based on geomorphological and hydrological distributions, and (2) implementation of a BNN to predict debris flow accumulation volumes, first tested on synthetic data and subsequently validated on real-world datasets from China [28] and South Korea [29].

The proposed framework introduces an innovative approach to debris flow volume prediction by combining strong generalization with robust uncertainty quantification and demonstrates promising performance across different geomorphological contexts.

A key element of our methodology is synthetic data generation, particularly in contexts where real data are scarce. By calibrating parameters that statistically characterize debris flow, we can build tailored datasets for specific geographical areas and develop local predictive models even without direct observations. Integrating these synthetic datasets into a Bayesian framework yields highly specific probabilistic estimates of volume accumulation. This adaptability is a core strength of our approach, providing a flexible and scalable solution for hazard assessment.

Although regions with comparable characteristics may benefit from the application of generalized models, the most reliable and contextually robust predictions are achieved through the generation of region-specific synthetic datasets coupled with the development of specialized models.

2. Materials and Methods

This section details the Python-based methodology adopted for synthetic data generation and the application of advanced deep learning models, specifically BNN, for debris flow volume prediction.

The primary objective was to develop a robust framework for debris flow volume prediction that not only provides accurate predictions but also quantifies the associated uncertainty, a crucial aspect in geoscientific applications where the intrinsic variability of natural phenomena is significant. The methodology is organized into several key phases, briefly outlined below and discussed in greater detail in the subsequent sections.

Bayesian Neural Network Modeling (Section 2.1): This delves into the theory and implementation of BNN, emphasizing their capability to quantify predictive uncertainty, a distinct advantage over traditional neural networks.

Datasets (Section 2.2): This section describes the real datasets and the creation of a synthetic dataset designed to replicate the statistical properties and inter-variable relationships of real debris flow data. The synthetic dataset is used to validate the BNN in debris flow scenarios, while the real datasets employed for training the BNN provide realistic patterns and variability, enabling the models to learn from actual debris flow events and ensuring their applicability to real-world conditions.

Application of Bayesian Neural Networks (Section 2.3): This section presents the application of BNN to synthetic and real debris flow data, detailing data preparation, model design, training, and uncertainty quantification, with a focus on reproducibility and methodological transparency.

2.1. General Introduction to Bayesian Neural Network Architecture

In this section, we outline the fundamental aspects of the BNN architecture, focusing on both the underlying theory and its implementation in Python.

2.1.1. “Bayes by Backprop”: Weight Uncertainty in Neural Networks

Traditional ANN, unlike their Bayesian counterparts, provides point estimates for their weights and biases. This means that each weight or bias parameter is represented by a single, deterministic value, leading to single-point predictions without an inherent measure of uncertainty. This limitation is particularly critical in safety-sensitive applications like natural hazard prediction, where understanding confidence in a prediction is as important as the prediction itself. The BNN meets this need by placing probability distributions over the network’s weights and biases, enabling the quantification of predictive uncertainty (Figure 1) [30].

In a BNN, instead of learning fixed weight values, the primary goal is to infer the posterior distribution over the weights, $P(w|D)$, given the observed data, where $w$ represents the network weights and $D$ is the dataset. Directly computing this true posterior is analytically intractable due to the high dimensionality of the weight space and the non-linearity of neural networks. Consequently, Variational Inference (VI) is commonly employed to approximate the true posterior with a simpler, tractable distribution, $q(w|\theta )$ parameterized by $\theta$ [30]. Here, $q(w|\theta )$ is referred to as the variational distribution or approximate posterior distribution. The objective of VI is to minimize the KL divergence between the approximate posterior $q(w|\theta )$ and the true posterior $P(w|D)$:

$${\theta }^{*}= arg min KL[q(w|\theta )\left|\right|P(w|D)]$$

(1)

The KL divergence, KL $[q(w|\theta )\left|\right|P(w|D)]$, measures the “distance” or information gain between the two distributions. The resulting cost function is variously known as the variational free energy [31,32,33] or the expected lower bound [34,35]. For simplicity (Equation (1)) could be denoted as follows:

$$F(D,\theta ) = \mathrm{KL}[q(w|\theta )\left|\right|P\left(w\right)]-E_{q(W|\theta )}[logP(D|w)]$$

(2)

where KL $[q(w|\theta )\left|\right|P\left(w\right)]$ is the complexity cost which acts as a regulator, encouraging the approximate posterior to remain close to the chosen prior distribution over the weights, $P(w$), helping to prevent overfitting and promoting more robust models [30]. The complexity cost reflects the “complexity” of the model in terms of how much it deviates from our prior knowledge about the weights; $E_{q(W|\theta )}[logP(D|w)]$ is the likelihood cost, often referred to as the expected negative log-likelihood, encourages the model to fit the training data well, representing how well the model, with its sampled weights, explains the observed data. The total cost function (Equation (2)) embodies a trade-off between satisfying the complexity of the data $D$ (by minimizing the likelihood cost) and satisfying the simplicity prior $P\left(w\right)$ (by minimizing the complexity cost). Exactly minimizing this cost naively is computationally prohibitive. Instead, gradient descent and various approximations are used [30]. During training, samples of weights are drawn from the approximate posterior distribution $q(w|\theta )$. Gradients are then backpropagated through the network to update the variational parameters $\theta$ of this approximate posterior. This specific training procedure, where the variational parameters are updated via backpropagation, is known as “Bayes by Backprop” [30]. This method allows for efficient training of BNN by leveraging standard gradient-based optimization techniques.

Once the BNN is trained, predictive uncertainty can be estimated by performing multiple forward passes (stochastic inferences) through the network. This process, often referred to as Monte Carlo Sampling from the Approximate Posterior distribution [30], provides a robust measure of the model’s uncertainty. For a given input, multiple sets of outputs are sampled from the learned approximate posterior, leading to an ensemble of predictions. The mean of these predictions provides the most probable output, while the standard deviation quantifies the predictive uncertainty [30].

2.1.2. Implementation of “Bayes by Backprop” in TensorFlow

Implementing “Bayes by Backprop” in TensorFlow (and Keras) is significantly facilitated by libraries like TensorFlow Probability (TFP) [36,37]. TFP provides the necessary building blocks for constructing probabilistic models, including neural network layers with distributed weights. Here the key steps are briefly outlined below for conceptual implementation

Defining Bayesian Layers: Instead of traditional Keras layers like Dense or Conv2D which have deterministic weights, one uses TFP’s specialized layers such as tfp.layers.DenseVariational. These layers do not learn a single weight value but rather the parameters of a distribution (e.g., mean and standard deviation for a Gaussian distribution) from which the weights will be sampled during inference and training [30].

Defining the Approximate Posterior Distribution$q(w|\theta )$: For each weight in the network, we must define how its posterior distribution will be approximated. TensorFlow Probability makes this straightforward by allowing “kernel_posterior_fn” and “bias_posterior_fn” to be specified within the variational layers. These functions define the distribution from which weights will be sampled and how its parameters will be learned [30].

Defining the Prior Distribution ($P\left(w\right)$): Similarly, “kernel_prior_fn” and “bias_prior_fn” define the prior distribution for the weights and biases [30].

Sampling and Propagation: During the model’s forward pass, the variational layers sample weights from their approximate posterior distributions. These sampled weights are then used for computations just as in standard ANN. This sampling process introduces the necessary stochasticity to estimate uncertainty.

Defining the Loss Function: The loss function is the sum of the likelihood cost and the complexity cost. TensorFlow Probability automatically handles the calculation of the KL $[q(w|\theta )\left|\right|P\left(w\right)]$ term (the complexity cost) when using its variational layers. The likelihood cost $-E_{q(W|\theta )}[logP(D|w)]$ is typically the negative log-likelihood of the model on the data (e.g., tfp.losses.NegativeLogLikelihood for distributions). These two terms are combined to form the cost function [30].

Optimization: A standard optimizer (such as Adam or SGD) is used to minimize the loss function. Gradients are computed with respect to the variational parameters (θ, $\theta$ i.e., the means and standard deviations of the weight distributions) and update these parameters to bring $q(w|\theta )$ closer to $P(D|w)$ [30].

Predictive Uncertainty: After training, to obtain uncertainty estimates, a code with multiple forward passes is executed using the same input. Each pass will sample a new set of weights from the learned distribution, leading to slightly different predictions. The mean of these predictions provides the point estimate, while the variance or standard deviation of the predictions provides the measure of uncertainty [30]. In practice, TensorFlow Probability greatly simplifies this process, abstracting away much of the complexity of sampling and KL divergence calculation, allowing scientists, engineers and developers to focus on defining the model and its architecture.

2.2. Datasets

The datasets from China [28] and Korea [29] used in this study rappresent real collected data, selected because they document well-characterized debris flow events and provide detailed information on accumulation volumes and triggering conditions. In addition, they were publicly available and widely cited in the international literature, making them reliable benchmarks for testing new modeling approaches. Their features, such as geomorphological setting, variability in triggering conditions, and availability of measured volumes, made them particularly suitable for validating the proposed methodology.

In contrast, a synthetic dataset will be designed to reproduce a general geographical context, relying on general distributions reported in the literature. This dataset will primarily serve as a flexible foundation: it will be adaptable and calibratable in a general way, and it will provide a first step for validating the BNN architecture.

Once the BNN framework will be successfully validated with the synthetic dataset, its subsequent application to the Chinese and Korean datasets will demonstrate how the method can be effectively transferred to real-world cases when actual measurements become available.

2.2.1. Real Datasets

The dataset sourced from [28] comprises observed debris flow events in the Sichuan region of China, specifically focusing on 60 typical debris-flow catchments that experienced events between 2008 and 2018. This dataset is characterized by variables such as basin area (A), topographic relief (H), channel length (L), distance from nearest seismic fault (D), and average channel slope (J). These morphological factors were measured using ArcGIS from 10 to 30 m Digital Elevation Models (DEMs). Additionally, the dataset includes the total volume of co-seismic debris flow debris (V) generated in each catchment, accounting for the initiation of many debris flows in the Wenchuan area from such deposits. The observed debris-flow volume ($V_0$), defined as the total volume of debris discharged during a single event, was compiled from technical journals, reports, and unpublished documents from local authorities. The values of $V_o$ range from a few tens to several hundreds of cubic meters, including both wet and solid debris. The study area’s debris flow events mostly occurred in catchments smaller than 10 ${\mathrm{km}}^2$, with topographic relief between 500 and 2000 m, channel lengths less than 7 km, and at less than 6 km from the seismic fault. The authors of the paper reported in [28] conducted correlation analyses, including Pearson correlation coefficient (R) and maximal information coefficient (MIC), between the morphological features (A, H, L, D, J), total volume of co-seismic landslide debris (V), and debris-flow volume ($V_0$). They found positive linear correlations for V, A, L and H (with values of R 0.89, 0.71, 0.58 and 0.39, respectively), a null correlation for D (R = 0.023) and a negative correlation for J (R = −0.354).

The dataset from [29] includes 63 historical debris flow events that occurred in South Korea between 2011 and 2013. The related paper discussed events that were caused by continuous rainfall over a 24 h period. These events are characterized by 15 predictive variables categorized into morphological, meteorological, and geological factors. Morphological factors, derived from high-resolution 5 m DEMs, include watershed area (Aw), channel length (Lc), watershed relief (Rw), mean slope of stream (Ss), mean slope of watershed (Sw), Melton ratio (Rm is a dimensionless index calculated as the relief (elevation difference) of a drainage basin divided by the square root of its area, used to assess basin shape and potential for flash flooding), relief ratio (Rr), form factor (Ff), and elongation ratio (Re). Meteorological factors are derived from pluviometric data obtained from the Korea Meteorological Agency (KMA) and include continuous rainfall (CR), rainfall duration (D), average rainfall intensity (Iavg), and peak rainfall intensity (Ipeak), and antecedent rainfall (AR). Geological factors primarily consist of Lithology, which is converted into a numerical Geological Index (GI) based on its erodibility. The Authors [29] performed correlation analysis using Pearson’s correlation coefficient ($R$) between these 15 predisposing factors and the debris-flow volume. The volume values range from hundreds to thousands of ${\mathrm{m}}^3$. They identified six significant factors: Aw (R = 0.828), Rw (R = 0.816), Lc (R = 0.769), CR (R = 0.500), GI (R = 0.295), and D (R = 0.251).

However, due to weak correlations, GI and D were excluded from their ANN modeling. Slope parameters (Ss, Sw), other rainfall factors (AR, Iavg, Ipeak), and watershed shape indices (Rm, Rr, Ff, Re) were also excluded due to low or no significance. Consequently, four remaining predisposing factors (Aw, Lc, Rw, and CR) were selected by [29] for the development of their ANN model.

2.2.2. General Description of Synthetic Dataset Generation

The generation of the synthetic dataset was carried out following a structured framework designed to ensure both the physical realism of each independent variable and the statistical coherence among them. The process involved two main steps: modeling the marginal distributions of the eight independent variables and imposing realistic inter-variable correlations through mathematical transformations. The following subsections describe these procedures in detail.

Modeling of Independent Variables and Distributions

Eight independent variables, crucial for debris flow and debris-flow susceptibility and dynamics, were chosen and synthesized. Each variable’s distribution was carefully parameterized to mimic observed geophysical properties, referring to established literature.

Grain Size: The diameter of sediment grains is represented by a log-transformed and truncated Generalized Hyperbolic (GH) [38]. With truncation between 0.1 and 100 mm to guarantee physically plausible particle sizes is reported as an example of the kind of distribution we have used:

$$\begin{array}{cc}\hfill f\left(x,\lambda ,\alpha ,\beta ,\delta ,\mu \right)& ={\displaystyle \frac{{\left({\alpha }^2-{\beta }^2\right)}^{\lambda /2}}{\sqrt{2\pi }{\alpha }^{\lambda -1/2}{\delta }^{\lambda }{K}_{\lambda }\left(\delta \sqrt{{\alpha }^2-{\beta }^2}\right)}}\hfill \\ \hfill & ·\left[K_{\lambda -1/2}·\alpha ·{\left(\sqrt{{\delta }^2+{\left(x-\mu \right)}^2}\right)}^{\lambda -1/2}\right]e^{\beta ·\left(x-\mu \right)}\hfill \end{array}$$

(3)

where $x=logD$ is the logarithmic value of the sediment grain diameter; $\lambda =p=0.5$ shape index; $\alpha =1.5$ scale parameter; $\beta =0.3$ asymmetry parameter; $\delta >0 \left(=1\right), \mu =0.7$ position and scale position; $K_{\lambda -1/2}\equiv K_{\lambda -1/2}$ modified Bessel function of the second kind of zero order. Then, (Equation (3)) was normalized. Figure 2 shows the related plot.

Rainfall: extreme rainfall events, recognized as primary triggers for debris flows, were modeled using a truncated power-law distribution [39]. For this dataset, proper coefficients were utilized, ensuring a realistic decay for high intensity events. The rainfall values, representing 24 h accumulations, were constrained between 50 mm (xmin) and 600 mm (xmax), fully consistent with the five severity classes, following the classification used by [40], reflecting physically plausible rainfall intensities capable of triggering debris flows.

Soil Composition: This variable, representing overall composition of fine content, was simulated using a Beta distribution scaled between 0% and 100%. This broad range allows for the representation of diverse soil types, from cohesionless to cohesive, as supported by studies on soil parameter variability [41].

Friction Angle (ϕ): This is a crucial parameter in debris flow analysis, is modeled using a truncated normal distribution [9]. Specifically, the distribution is centered around 30° with a standard deviation of 5°. The values are restricted to a physically meaningful and realistic interval between 20° and 45°, corresponding to typical friction angles found in natural soils.

Friction (μ): a critical parameter for granular flow, is modeled using a scaled Beta distribution [11]. This choice is particularly appropriate for variables naturally bounded between two values, allowing for accurate representation of its skewed nature and defined domain. The generated values are then rescaled to the interval [0.05, 0.4], which corresponds to typical friction coefficients observed in both granular and cohesive soils.

Slope Angle (θ): a primary control parameter on debris flow initiation and evolution, is modeled using a truncated normal distribution [42]. Specifically, the slope angle is drawn from a distribution centered at 25° with a standard deviation of 7°. The values are constrained to a physically realistic range, truncated between 10° and 45°. These limits reflect the natural range of hillslope gradients typically associated with debris flows.

Density ($\rho$): a key factor in estimating mass movement dynamics, is simulated using a truncated normal distribution. This approach incorporates typical values observed for various geotechnical materials, from loose mud to denser rocky debris. Specifically, the distribution has a mean of 1800 kg/m³ and a standard deviation of 150 kg/m³. The values are bounded between 1300 kg/m³ and 2300 kg/m³ to reflect the realistic range of densities found in granular and cohesive soils [11].

Visco-Turbulent Parameter (ξ): The visco-turbulent parameter (ξ), which distinguishes between wet and dry debris flows, is generated by sampling from a mixture of triangular distributions. This approach effectively captures the bimodal behavior observed in landslide dynamics [11]. The simulation randomly selects between “granular” and “earth flow” regimes with equal probability ($p$ = 0.5). For granular flows, ξ values range from 100 to 200, with a peak at 150. Conversely, for earth flows, ξ ranges from 200 to 1000, peaking at 500.

A comprehensive overview of the synthetic data generation framework, obtained through Python language, is provided in Figure 3.

Imposition of Inter-Variable Correlation Structure

To assign realistic interdependencies among the variables in our synthetic dataset, we predefined a Pearson cross-correlation matrix-based R on assumed variable relationships. Since R must be positive-definite for decomposition (all eigenvalues positive), we applied the “Nearest Positive Definite Adjustment” (NPDA) algorithm [43] to enable decomposition. The application of the NPDA algorithm, mathematically formalized as finding the corrected matrix ${\mathit{R}}^{*}$ that minimizes the Frobenius distance can be expressed as follows:

$${\mathit{R}}^{*}= arg\underset{\left\{\mathit{M} \succ 0\right\}}{min} {\left|\left|\mathit{R} - \mathit{M}\right|\right|}_F$$

(4)

We seek the positive definite matrix M that minimizes the distance from R. The adjusted R was then factorized using Cholesky decomposition [44]:

$$\mathit{R}=\mathit{L}{\mathit{L}}^{\mathit{T}}$$

(5)

where L is a lower triangular matrix. The process involved generating a set of eight normally distributed, uncorrelated variables. These were then transformed to achieve the desired correlations using the Cholesky decomposition [44] of the adjusted matrix. Cholesky decomposition breaks a symmetric, positive-definite matrix (no negative eigenvalue) into a lower triangular matrix and its transpose for efficient computation. For distribution imposition, the eight uncorrelated standard normal samples Z, were transformed into correlated samples:

$$\mathit{Y}=\mathit{Z}{\mathit{L}}^{\mathit{T}}$$

(6)

Each correlated normal variable was mapped into the uniform space using the cumulative distribution function (CDF) [45]:

$${\mathit{u}}_{\mathit{i}}=\mathit{\Phi }{\mathit{Y}}_{\mathit{i}}$$

(7)

where $\mathit{\Phi }$ denotes the standard normal CDF, each uniform variable ${\mathit{Y}}_{\mathit{i}}$ was transformed into the target distribution (already defined in subsection Modeling of Independent Variables and Distributions) using the inverse CDF ${\mathit{F}}_{\mathit{i}}^{\mathbf{-}\mathbf{1}}$ [45]:

$${\mathit{x}}_{\mathit{i}} = {\mathit{F}}_{\mathit{i}}^{-1}\left({\mathit{Y}}_{\mathit{i}}\right)$$

(8)

This procedure guarantees that the synthetic dataset simultaneously preserves the marginal distributions of each variable (physical realism), and the cross-variable dependencies defined by the correlation matrix (statistical realism).

Dependent Variable Volume Generation

The volume of the debris flow accumulation was assumed as the dependent variable, generated as the predicted output of a specialized neural network model, rather than derived from a direct, explicit experimental, numerical or analytical formula. We adopted this approach to simulate volumes exhibiting non-linear characteristics consistent with real-world debris-flow phenomena, specifically their observed heavy-tailed (power-law) distributions behavior [46]. In particular, the target distribution was defined as:

$$p\left(V\right)\propto V^{-\alpha }, V \ge V_{min}$$

(9)

where α is the scaling exponent and $V\_min$ the minimum threshold volume considered. The total volume available for the debris flow is implicitly accounted for within the variability range used in the simulations. This range reflects both the mass generated by instability and the balance between erosion and deposition. Different numerical values can be used to represent other scales. The input is directly provided by the user, who is expected to have geological, engineering, or related knowledge of the system.

The neural network’s architecture included a custom layer with specific weight constraints, followed by standard dense layers with ReLU (Rectified Linear Unit) activation function and a final output layer with softplus activation to ensure positive volume predictions. The custom input layer enforced feature-specific sign constraints on the weights to inject prior physical knowledge; for kernels associated with grain_size, rainfall, slope_angle, and density we enforced non-negativity, while for soil_composition, friction_angle, dry_friction, and xi we enforced non-positivity:

$$w_1, w_2, w_5, w_6 \ge 0; w_3, w_4, w_7, w_8 \le 0.$$

(10)

Each dense layer transformation can be expressed as follows:

$${\mathit{h}}^{\left(l\right)} = \sigma \left( {\mathit{W}}^{\left(l\right)} {\mathit{h}}^{\left(l-1\right)}+ {\mathit{b}}^{\left(l\right)} \right)$$

(11)

where ${\mathit{h}}^{\left(l\right)}$ is the hidden representation at layer l, ${\mathit{W}}^{\left(l\right)}$ and ${\mathit{b}}^{\left(l\right)}$ are the trainable weights and biases, and $\sigma \left(·\right)$ is the activation function. Specifically:

$${\sigma }_{ReLU\left(x\right)}=\mathit{max}\left(0, x\right), {\sigma }_{softplus\left(z\right)}=\mathit{log}\left(1+ e^z\right)$$

(12)

ensuring non-linearity and positivity of the predicted volumes. The network was constrained to maintain correlations by attempting to predict the power-law distribution, minimizing the Kullback–Leibler (KL) divergence [47] between the predicted volume distribution $p_{\theta }\left(V\right)$ and a target power-law distribution $q\left(V\right) \propto V^{-\alpha }$. This implicitly calibrated the relationship between the independent variables and the resulting synthetic volumes.

$$L_{KL}=\int p_{\theta }\left(V\right)·\mathit{log}{\displaystyle \frac{p_{\theta }\left(V\right) }{ q\left(V\right)}} dV$$

(13)

This implicitly calibrated the relationship between the independent variables and the resulting synthetic volumes. The final prediction of the network can thus be written as follows:

$$\widehat{\mathit{V}} = f_{\theta }\left(x\right)$$

(14)

where x is the input feature vector and $\mathit{\theta } = \left\{{\mathit{W}}^{\left(l\right)}, {\mathit{b}}^{\left(l\right)}\right\}$ the set of trainable parameters.

2.3. Application of Bayesian Neural Networks

This section describes the application of BNN for predicting debris flow volumes using both the synthetically generated dataset and real-world debris flow datasets (Flow-chart reported in Figure 4). The entire prediction framework was developed in Python, primarily utilizing the TensorFlow and TensorFlow Probability libraries, ensuring a consistent approach across different data types. In particular, it is worth noting that an overview of Section 2.1 is illustrated in the blue flowchart on the left-hand side of Figure 4.

2.3.1. Data Preparation and Preprocessing

For both synthetic and real-world datasets, a meticulous preprocessing pipeline was applied. All independent variables were scaled using z-score normalization (standardization). The dependent variable, debris flow volume, exhibiting a heavy-tailed distribution, was first subjected to a logarithmic transformation to reduce skewness and stabilize variance, and subsequently also scaled with z-score normalization. The pre-processed data was then split into training set (80%) and testing set (20%) to evaluate generalization performance.

2.3.2. Selected Bayesian Neural Network Architecture

A consistent BNN model was constructed using Dense Variational layers, which inherently introduced a probability distribution over the network’s weights and biases. The model architecture comprised multiple hidden Dense Variational layers with appropriate activation functions. Specifically, for the Prior Distribution $P\left(w\right)$ “kernel_prior_fn” and “bias_prior_fn” were defined. For the Approximate Posterior Distribution $q(w|\theta )$ “kernel_posterior_fn” and “bias_posterior_fn” were defined. The “kl weight” for each Dense Variational layer was set in relation to the number of samples in the training set (by scaling with the number of training samples), following best practices to balance the likelihood and KL divergence terms in the cost function. The model used a final Dense output layer with a single unit to predict the mean of the transformed volume. Compiled with the Adam optimizer and the cost function loss, the model also incorporated the Huber loss function. Huber loss was selected for its superior robustness to outliers, a valuable characteristic when handling potentially noisy or heavy-tailed target variables.

2.3.3. Training, Evaluation, and Uncertainty Analysis

The training protocol involved an initial hyperparameter search over a limited grid (e.g., varying learning rates, activation functions, and number of layers). The model yielding the lowest validation loss was selected as the “best model” based on its performance on unseen validation data. Subsequently, this best model was retrained on the full training dataset for an extended number of to ensure thorough convergence and optimize its parameters.

Predictive uncertainty was quantified by performing multiple stochastic forward passes through the trained model. For each test sample, 100 forward passes were executed. During each pass, weights were sampled from their learned posterior distributions. The mean and standard deviation of these predictions were then calculated for each test sample, providing both the most probable prediction and a measure of its uncertainty. The predicted mean and standard deviation values were then inverse-transformed back to the original volume scale, taking into account the prior logarithmic and z-score transformations. This allows for direct interpretation of predictions and uncertainty in their original units (e.g., cubic meters).

The performance of the BNN model was evaluated using several established metrics and comprehensive visualizations. These included the calculation of Mean Squared Error Percentage (MSE%) and Coefficient of Determination ($R^2$) [48,49] to assess prediction accuracy.

Visualizations included generation of observed vs. predicted plots to visually inspect agreement and histograms with Kernel Density Estimation (KDE) of the predictions. All results, trained model parameters, and learned weight distributions were saved for full reproducibility and detailed post-analysis.

2.3.4. Application to Synthetic Data

The framework described above was first applied to the synthetic dataset generated in Section 2.2.2. This allowed for a controlled environment to rigorously evaluate the predictive capabilities and uncertainty quantification of the BNN model under ideal conditions, where the true underlying data characteristics were known. For training purposes, 10,000 samples were synthetically generated, with 8000 for training and 2000 for testing.

2.3.5. Application to Real-World Debris Flow Data

Subsequently, the BNN framework was applied to real-world debris flow datasets from the Sichuan region in China and from South Korea.

Regarding the dataset from [28], in our article we selected $V_0$ as the dependent variable, while the others were treated as independent variables. $V_0$ values range from tens to hundreds of ${\mathrm{m}}^3$, including both wet and solid debris. For the dataset in [29], in our work we will use all the available variables without discarding any; in this case, the selected dependent variable is V.

Given the typically limited number of original samples in both real-world datasets, a data augmentation technique based on bootstrapping [27] with added noise was employed to generate a larger and more robust dataset. This approach was crucial for model generalization, reducing the risk of overfitting and ensuring a more robust evaluation of model performance, especially when dealing with scarce or inherently dependent geoscientific data. Specifically, many synthetic data points were generated by resampling existing data with replacement and then adding a small amount of Gaussian error to each resampled point. This focused on maintaining a balanced representation of outliers and creating diversified synthetic samples while preserving the underlying statistical properties of the original data. For small and heterogeneous geoscientific datasets, data augmentation must be applied cautiously to avoid artifacts. Instead of advanced generative models like Generative Adversarial Networks (GANs) or autoencoders, we adopted a conservative approach using bootstrapping with Gaussian noise. This preserves the statistical properties of the original data while reducing the risk of unrealistic patterns.

For these real-world applications, the model configuration, training protocols, and evaluation metrics remained consistent with the general framework, providing comparable results and insights into the BNN’s performance on synthetic data. Also here, as with the synthetic data, for training purposes, 10,000 samples were synthetically generated, with 8000 for training and 2000 for testing.

3. Results

This section systematically presents the performance evaluation of the synthetic data generation methodology and the three distinct BNN models developed for predicting debris flow volume: the “Model from Sichuan region (China)”, the “Model from Korea”, and the “Synthetic Model”. The analysis focuses on validating the synthetic data’s statistical fidelity and assessing the predictive accuracy and uncertainty quantification capabilities of each BNN model.

3.1. Generation of Synthetic Dataset

3.1.1. Independent Variable Correlation Matrix

An initial Pearson correlation matrix $R$ for the eight independent variables was predefined by us. To ensure the mathematical validity required for synthetic data generation, this matrix underwent nearest positive definite adjustment and subsequent Cholesky decomposition. This procedure guaranteed that the matrix could be used to transform a set of uncorrelated variables into a set exhibiting the desired correlations, while maintaining positive semi-definiteness. The successful implementation of this process, coupled with sampling from the predefined probability distributions of each independent variable, is visually confirmed by the resulting correlation matrix presented in Figure 5.

Figure 5 is a heatmap visualizing the Pearson correlation matrix among all pairs of independent variables in the synthetically generated dataset. The application of Cholesky decomposition and the “nearest positive definite” adjustment were critical to generating a synthetic dataset where independent variables exhibit the desired interdependencies. For instance, friction angle and dry friction exhibit a significant positive correlation (0.59), which is geotechnically expected given their shared physical basis in material resistance. Conversely, rainfall demonstrates negative correlations with friction angle (−0.48) and xi (−0.45), while showing a positive correlation with slope angle (0.34) and a weak negative correlation with grain size (−0.11). These correlations align with geophysical hypotheses concerning the detrimental influence of water on soil stability and the role of slope geometry. density also shows notable negative correlations with xi (−0.50) and soil composition (−0.39), and a weak positive correlation with friction angle (0.31).

3.1.2. Correlation of Independent Variables with the Dependent Variable

Figure 6 is a bar chart displaying the Pearson correlation coefficient between each independent variable and the dependent synthetic variable, ‘Volume’.

Variables such as rainfall, grain size, and slope angle exhibit relatively strong positive correlations with volume. Friction angle and dry friction consistently show negative correlations with volume. Soil composition and density demonstrate very weak, near-zero correlations with volume. Xi (ξ) shows a weak positive correlation, indicating a minor influence compared to dominant factors like rainfall.

3.1.3. Predicted Volume Distribution

Figure 7 is a histogram with KDE illustrating the distribution of the generated synthetic debris flow volumes. This result provides critical validation for the neural network-based volume generation process described in subsection ‘Dependent Variable Volume Generation’. The primary objective of simulating volumes with characteristics consistent with observed real-world distributions, particularly heavy-tailed behavior (approximating a power-law distribution), was successfully achieved. In the synthetic dataset developed in our work, we assumed a variability range for the total accumulated volume between 0 and 1000 cubic meters.

The distribution is markedly asymmetric and highly right-skewed, with the vast majority of synthetic volumes concentrated at the lower end

3.2. BNN Performance

3.2.1. Model Performance Evaluation

This section evaluates the performance of the three BNN models, each trained on different datasets: the “Synthetic Model” (trained on the synthetically generated data), the “Model from Sichuan region (China)”, and the “Model from Korea”. The latter two models were specifically developed using real-world debris flow data from their respective regions, as detailed in Section 2.2.1. While the synthetic dataset was designed to be statistically pristine and internally consistent, offering an idealized setting for model learning, the real-world datasets introduce the complexities, noise, and uncertainties inherent to observational data. These differences are clearly reflected in the model’s prediction metrics.

Table 1. Predictive performance for all three BNN models.

Model Name	$\mathbf{Validation} \mathbf{Value} [{\mathbf{m}}^3]$	$\mathbf{Predicted} \mathbf{Value} [{\mathbf{m}}^3]$	$\mathbf{Predicted} \mathit{\sigma } \left[{\mathbf{m}}^3\right]$	$95\% \mathbf{Confidence} \mathbf{Interval} (\pm 2\mathit{\sigma })$
Synthetic Model	32.7	32.4	0.26	(31.87, 32.92)
Model from Sichuan region (China)	45.57	44.63	3.62	(37.39, 51.87)
Model from Korea	5445.99	5953.52	233.94	(5485.64, 6421.41)

demonstrates excellent predictive performance for all three BNN in estimating volumes:

The Synthetic Model exhibits outstanding predictive accuracy, with a predicted value of 32.4 m³ closely aligning with the reference value of 32.7 m³; its 95% confidence interval of (31.87; 32.92), though slightly broader than the point estimate, fully contains the true value, demonstrating strong robustness. Similarly, the model developed for the Sichuan region in China shows solid performance, predicting 44.63 m³ versus an actual value of 45.57 m³, with a wider confidence interval of (37.39, 51.87) that still encompasses the validation value, confirming its reliability. The Korean model, which operates on significantly larger volumes, predicts 5953.52 m³ against a validation value of 5445.99 m³; although this model features the widest confidence interval among the three, ranging from 5485.64 to 6421.41 m³, it still captures the validation value, underscoring the model’s precision even at higher magnitudes. Figure 8a–c present a scatter plot comparing the predicted versus debris flow volume validation values for all three BNN models. The dashed red line represents the “Ideal Prediction” (y = x), where predicted values perfectly match validation values. Points closely clustering around this line indicate superior model performance.

The coefficient of determination $R^2$ confirmed the robustness of all the models under investigation. The Synthetic Model exhibited the highest explanatory power, with an $R^2$ value of 0.999, indicating an almost perfect agreement between predicted and observed data. The model developed for the Sichuan region (China) also demonstrated excellent performance, with an $R^2$ of 0.995, while the model from Korea, although slightly lower, still achieved a remarkably high value of 0.983, confirming its strong predictive capability.

Both the “Synthetic Model” and the “Model from Sichuan region (China)” demonstrate predictions remarkably close to the ideal line across a broad range of volumes, signifying exceptionally high accuracy and generalization capabilities. The “Model from Korea” also exhibits a good fit, although with slightly more scatter, which is quantitatively reflected in its comparatively lower $R^{2 }$ value.

3.2.2. Prediction Uncertainty Analysis

Figure 9a–c shows the predictive uncertainty for each BNN model, a key advantage of the Bayesian approach. Blue dots represent the actual, observed debris flow volume values. Red ‘x’ marks represent the model’s predicted mean for each sample, derived from the ensemble of stochastic forward passes. The yellow shaded area represents the range of one standard deviation (±1 ${\sigma }^2$) around the predicted mean, quantifying the inherent uncertainty or spread of the predictions. A narrower yellow band indicates higher model confidence and lower predictive uncertainty.

Figure 9a visualizes the prediction uncertainty for the “Synthetic Model”. The exceptionally narrow yellow band, particularly prominent for smaller volumes, indicates very high confidence and remarkably low uncertainty in its predictions. This directly aligns with its excellent error metrics and high $R^2$ value. The predicted means (red ‘x’) consistently and precisely track the true values (blue dots), demonstrating the model’s capacity to learn the precise, deterministic relationships embedded in the synthetic data, while still providing a measure of epistemic uncertainty inherent to the BNN framework.

Figure 9b visualizes the prediction uncertainty for the “Model from Sichuan region (China)”. This model exhibits a wider yellow band compared to the Synthetic Model, particularly as the predicted volume increases, suggesting a greater degree of uncertainty for larger debris flow events. However, the predicted means generally follow validation values well, indicating strong overall performance despite the increased uncertainty for higher-magnitude events. This wider band is typical for models trained on real-world data, reflecting both aleatoric (inherent data variability) and epistemic (model uncertainty due to limited data) uncertainties. The dataset from [28] for the Sichuan region, while providing valuable insights into co-seismic debris flow debris, naturally contains more inherent variability than a perfectly controlled synthetic dataset.

Figure 9c visualizes the prediction uncertainty for the “Model from Korea”. This model exhibits the widest yellow band among the three, especially for higher volumes, indicating the greatest degree of uncertainty in its predictions. The predicted means also show greater divergence from the values compared to the other two models, which is entirely consistent with its higher MRE and lower $R^2$. The pronounced uncertainty for larger volumes suggests that the model struggles to confidently generalize to the tails of the distribution or that the inherent variability in the “Korea” dataset is higher. The diverse and potentially less correlated variables in the [29] dataset, some of which were eliminated due to weak correlations during their original ANN modeling, but which we have retained, could contribute to this increased uncertainty and decreased predictive confidence for the Korean model.

Figure 10a–c presents histograms and KDE for the predicted Volume. The plot also provides key statistical moments: skewness and kurtosis values, which quantitatively describe the shape and tail of the data distribution.

The skewness and kurtosis values reveal significant differences among the analyzed models. The model from the Sichuan region (China) shows a skewness of 1.735 and a kurtosis of 2.347, indicating a moderately asymmetric distribution with slightly heavier tails compared to a normal distribution. The Korean model presents a higher skewness (2.1) and a kurtosis of 4.272, suggesting stronger asymmetry and a more pronounced tendency toward heavy-tailed behavior. Finally, the synthetic model exhibits the most extreme values, with a skewness of 3.898 and a kurtosis of 16.884, reflecting a highly skewed distribution characterized by extremely heavy tails.

3.2.3. Error Metric Comparison

Figure 11 presents a bar chart comparing the values of the Mean Relative Error (MRE) (%) metric for each BNN model, where lower values indicate better predictive performance. MRE expresses the average absolute difference between predicted and actual values relative to the actual values.

The Synthetic Model achieves the lowest errors (MRE: 1.67%), demonstrating exceptional predictive accuracy due to the high-quality, well-structured synthetic dataset. The Model from Korea shows moderate error levels (MRE: 9.74%), indicating good performance despite the increased complexity of real-world data. The Model from the Sichuan region (China) records the highest error values (MRE: 22.11%), yet still delivers reasonable accuracy given the variability and challenges of real debris flow data. Overall, the synthetic model outperforms the others, validating the effectiveness of synthetic data in training BNN, while the Korea and Sichuan models remain capable within the context of more complex, real-world conditions.

The slightly higher error percentage for the China model can be better understood by examining its residual plot (Figure 12a) and recognizing that the dataset follows a power-law distribution, there are far more small-volume events than large-volume ones.

As a result, aggregate error metrics like MRE are disproportionately influenced by small-volume predictions, where even minor absolute errors can cause large relative errors. This statistical imbalance skews overall error figures, despite the model performing reliably across the full range of values. For instance, in very low validation values (0–150 m³), a small absolute error, such as 10 m³ on a validation value of 10 m³, produces a 100% relative error. However, it is essential to assess whether such errors fall within the pre-established confidence intervals. For example, if the confidence interval for a 20 m³ prediction is ±10 m³, a prediction of 30 m³ still lies within the acceptable range, even though it results in a 50% MRE. At intermediate and high volumes (e.g., 1000–2000 m³), residuals can be as large as 100–200 m³. While these may seem significant in absolute terms, they have minimal impact on relative error and overall model evaluation, provided they fall within the expected uncertainty range.

In essence, a thorough evaluation of prediction errors always requires analyzing residual plots in conjunction with intervals to determine their acceptability.

4. Discussion

The performance evaluation of the three BNN models (Table 1, Figure 8 and Figure 11) provides crucial insights into the efficacy of BNN for debris flow volume prediction and the impact of data characteristics on model performance.

The “Synthetic Model” consistently exhibited outstanding predictive accuracy across all metrics, with an $R^2$ of 0.998 and the lowest MRE (1.67%). Its 95% confidence interval consistently contained validation values, and the prediction uncertainty visualization (Figure 9) showed an exceptionally narrow yellow band.

A significant advantage of the Bayesian approach is its ability to quantify predictive uncertainty. The visualization of prediction uncertainty (Figure 9) vividly illustrates how confidence levels vary across the models and volume ranges. The “Synthetic Model’s” remarkably narrow uncertainty band highlights its high confidence, while the wider bands for the “Sichuan” and “Korea” models accurately reflect the increased uncertainties associated with real-world data. These uncertainties are not merely noise but represent the inherent variability within the data (aleatoric uncertainty) and the model’s confidence due to limited data (epistemic uncertainty).

This superior performance is a direct consequence of the pristine nature of the synthetic dataset, free from complexities inherent in observational data. It validates the foundational capacity of BNN to learn precise, deterministic relationships when trained on high-quality, statistically consistent data, while still providing a measure of epistemic uncertainty.

The power-law highly skewed (3.898) and kurtotic (16.884) distribution of the synthetic volumes further demonstrates the model’s ability to accurately capture extreme event characteristics (Figure 9). This characteristic was intentionally engineered in the synthetic data generation process to mimic extreme event distributions and test the models’ ability to handle such non-Gaussian targets. This information is crucial for understanding the intrinsic statistical properties of the data each model was trained on and how these characteristics might influence model performance, error distribution, and the resulting predictive uncertainty. The differences in skewness and kurtosis between the real-world datasets (Sichuan and Korea) and the synthetic data reflect the nuanced statistical properties of observed natural events versus a precisely engineered distribution.

It is crucial to consider that all the datasets follow a power-law distribution, meaning there are far more small-volume events than large-volume ones. Consequently, aggregate error metrics like MRE are influenced by small-volume predictions, where even minor absolute errors can significantly inflate relative error percentages. This statistical imbalance skews the overall error figures, even though the model performs reliably across the full range of values.

The varying performance of models trained on real-world data, such as those from the Sichuan region in China, underscores the inherent challenges of working with natural datasets. The “Model from Sichuan region (China)” achieved a good $R^2$ of 0.995, despite higher MRE (22.11%) compared to synthetic models. Its wider confidence intervals and broader prediction uncertainty band (Figure 9) are characteristic of models based on real-world data, reflecting both aleatoric uncertainty (inherent data variability) and epistemic uncertainty (model uncertainty due to limited data).

Despite this, the model still accurately captures the general trend of validation values, confirming its reliability in a more complex environment. The dataset from [28], while invaluable for co-seismic debris flow studies, naturally contains more inherent variability than a perfectly controlled synthetic dataset.

The “Model from Korea,” while still exhibiting good performance with an $R^2$ of 0.983, recorded the widest confidence intervals despite lower error metrics (MRE: 9.74%). Its prediction uncertainty (Figure 9) showed the widest yellow band, especially for higher volumes, indicating a greater degree of uncertainty. This could be attributed to the inherent variability and potential complexities of the [29]. The model’s struggle to confidently generalize to the tails of the distribution suggests that the statistical properties, including potentially less correlated variables, might contribute to increased uncertainty and decreased predictive confidence for larger magnitude events. Nevertheless, we retained the variables, and we have demonstrated how the model still captures the validation value within its confidence interval, underscoring its utility even with more challenging data.

Beyond the evaluation of model performance, it is important to highlight how our approach provides a concrete alternative to the limitations of traditional debris-flow modeling techniques. Physically based and numerical models (e.g., FEM, FVM, SPH, PFEM) [13,14,15,16,17,18,19] are grounded in conservation laws and have been widely applied to debris-flow dynamics, but their applicability is often hindered by high computational costs, the need for extensive parameter calibration, and difficulties in generalizing across diverse geomorphological contexts. Similarly, empirical and statistical methods, including Monte Carlo simulations [20,21], can incorporate uncertainty but typically require large observational datasets that are rarely available in debris-flow prone areas [8]. Our probabilistic framework based on Bayesian Neural Networks addresses these challenges by combining the predictive capabilities of AI with the explicit treatment of uncertainty. Unlike deterministic AI models applied to debris flows [4,5], which provide point predictions without quantifying confidence levels, BNNs allow us to capture both aleatoric and epistemic uncertainty [24,25,26], which is crucial for risk-sensitive applications in hazard assessment and civil protection.

Looking ahead, a promising research direction lies in integrating physics-based models with probabilistic AI approaches such as BNNs. While numerical and physically based frameworks (e.g., CFD, shallow-water approximations, SPH, PFEM) [10,11,12,13,14,15,16,17,18,19] provide mechanistic insight into debris-flow processes and allow explicit modeling of governing dynamics, they often struggle to incorporate data-driven adaptability and uncertainty quantification. Conversely, BNNs excel in generalization, computational efficiency, and probabilistic forecasting but rely on the quality and representativeness of training datasets. A fusion of these two paradigms, embedding physically informed constraints into AI models, or using AI to emulate and accelerate high-fidelity simulations, could yield hybrid tools that are both physically interpretable and probabilistically robust. Such an integrated approach would combine the strengths of deterministic process-based modeling with the flexibility and uncertainty-awareness of probabilistic AI, providing more reliable and scalable solutions for debris-flow hazard assessment and civil protection planning. Among others approaches, this AI technique could be enhanced by integrating other methods, such as particle swarm optimization and genetic algorithms, particularly for identifying sets of numerical parameters that characterize complex constitutive laws [50].

The successful generation of synthetic data, with its pre-defined and intentionally implemented correlations, underscores the robustness of our methodology. Applying Cholesky decomposition and the “nearest positive definite” adjustment proved critical in ensuring mathematical validity and realistic interdependencies among independent variables as dictated by our synthetic assumptions. For instance, the strong positive correlation between friction angle and dry friction (0.59) and the negative correlation between rainfall and friction angle (−0.48) were specifically designed to be geotechnically correct and consistent with established geophysical principles. These findings confirm that the synthetic dataset not only adheres to predefined statistical properties but also precisely emulates the complex relationships intended for our idealized debris flow systems.

Variables such as rainfall, grain size, and slope angle exhibit relatively strong positive correlations with volume. This aligns directly with geomorphological understanding: increased precipitation often acts as a trigger for larger events, coarser grain sizes can contribute to greater mobilized volumes, and steeper slopes inherently present higher instability leading to larger debris flows. Friction angle and dry friction consistently show negative correlations with volume. This is physically correct, as higher internal friction angles and effective friction coefficients imply greater resistance to shear failure, thus correlating with smaller debris flow volumes or preventing large-scale movement. Soil composition and density demonstrate very weak, near-zero correlations with volume within the synthetic model. This suggests a less pronounced direct linear influence on volume, which may reflect the more complex, non-linear ways these parameters interact within a real debris flow system or the specific parameterization choices during synthetic data generation. xi (visco-turbulent parameter) shows a weak positive correlation, indicating a minor influence compared to dominant factors like rainfall.

It is crucial to emphasize that while our synthetic data generation methodology successfully replicated these complex, pre-defined relationships, creating a synthetic dataset that perfectly mirrors reality, with all its inherent nuances and unpredictability, is an intrinsically more difficult and complex endeavor in practice. The simplifying assumptions that enabled the generation of this idealized dataset do not always apply with the same ease to real-world scenarios, where data is often noisy, incomplete, and influenced by a wide array of factors that are not always easily modellable or fully understood. The ability to generate targeted synthetic datasets for specific contexts is particularly valuable, as it allows the data to be easily adapted to different geographical areas. This adaptation is achieved by calibrating the parameters that statistically describe the debris-flow phenomenon for each region, thereby capturing the intrinsic characteristics of the local territory. Since the dataset was synthetic and based on assumptions, the primary objective was to ensure a good capacity for generalization, where “generalization” refers to the ability to reproduce global characteristics of the phenomenon. As highlighted by [51], when generating a synthetic dataset tailored to the Italian context, it would be preferable to adopt passbands with thresholds ranging from 4 mm/h to 400 mm/h. Nonetheless, these thresholds must be adjusted according to the specific conditions of each region.

Furthermore, the intentional correlations between independent variables and the dependent variable, ‘Volume’ (Figure 7), demonstrate our deliberate design choice to embed intuitive relationships. The positive correlations of rainfall, grain size, and slope angle with volume align perfectly with how these factors are known to contribute to larger debris flow events. Conversely, the negative correlations with friction angle and dry friction are physically consistent, as higher resistance leads to smaller volumes. The weak, near-zero correlations for soil composition and density, while seemingly counterintuitive, could reflect the complex, non-linear interactions of these parameters within real debris flow systems, or specific parameterization choices during synthetic data generation. This highlights our approach’s capacity to mimic complex real-world behaviors through deliberate design, while acknowledging the simplifications necessary for such emulation.

A key achievement of the synthetic data generation was the successful emulation of a heavy-tailed, power-law-like distribution for debris flow volumes (Figure 7). This was a deliberate design objective, achieved through the minimization of Kullback–Leibler (KL) divergence during the neural network training. The resulting highly right-skewed distribution, with a distinct long tail, accurately represents the natural phenomenon where small events are frequent, and large, catastrophic events are rare but contribute significantly to overall mobilized volume. This statistical fidelity to real-world debris flow characteristics is paramount, as it provides an ideal environment for training and evaluating predictive models, particularly those designed to capture extreme events.

A distinct, long, and thin “tail” extends towards significantly higher volume values. This characteristic accurately represents the natural phenomenon where small-volume debris flows are prevalent, while rare, catastrophic large-volume events contribute disproportionately to total mobilized volume. The visually observed distribution strongly aligns with a power-law or heavy-tailed distribution, which was explicitly targeted and calibrated for in the theoretical framework and the neural network’s loss function during synthetic volume generation. This confirms the successful emulation of a crucial statistical property of natural debris flow events.

These approaches directly address the challenges outlined in the introduction, particularly the scarcity and heterogeneity of empirical debris-flow data [8]. Traditional physically based and numerical models [9,10,11,12,13,14,15,16,17,18,19] rely heavily on dense observational datasets and calibrated parameters, which are often unavailable in regions with limited monitoring infrastructures. Similarly, probabilistic approaches such as Monte Carlo simulations [20,21] require extensive empirical input to achieve reliable results. Our framework offers an alternative by generating synthetic datasets that preserve key geomorphological and hydrological correlations while maintaining statistical fidelity to natural heavy-tailed distributions. This not only alleviates the limitations posed by incomplete or noisy field measurements but also enables the development of localized predictive models tailored to specific regions, even in the absence of comprehensive observational records.

Ultimately, the integration of synthetic data generation with BNN [24,25,26] provides a flexible and scalable solution that mitigates the fundamental problem of data scarcity, while remaining adaptable to diverse geomorphological contexts.

The performance differences among the models trained on real-world data highlight the importance of data quality, completeness, and inherent variability. Future research could focus on further refining synthetic data generation techniques to incorporate more nuanced complexities found in diverse real-world debris flow datasets, acknowledging that replicating true complexity is an ongoing and challenging objective. Additionally, exploring advanced BNN architectures or hybrid modeling approaches could potentially mitigate some of the uncertainties observed in models trained on highly variable field data.

5. Conclusions

This study highlights the capabilities of BNN in quantifying debris flow volumes and their associated uncertainties. It also demonstrates the remarkable potential of synthetic data for training and validating BNN models applied to complex geological phenomena, such as debris flow prediction. The ability to generate statistically consistent and controlled datasets provides an idealized environment for model development, enabling precise calibration and rigorous evaluation of predictive performance. Although real-world data inevitably pose significant challenges, the superior performance of the ‘Synthetic Model’ sets a benchmark for what can be achieved with high-quality, well-structured datasets.

The Synthetic Model, trained on this idealized dataset, consistently demonstrated exceptional predictive accuracy, achieving an $R^2$ of 0.999 and remarkably low error rates (MRE of 1.67%). This outstanding performance highlights the foundational capacity of BNN to learn precise relationships when provided with high-quality, statistically consistent data. Crucially, the Synthetic Model also provided a clear measure of epistemic uncertainty through its exceptionally narrow prediction uncertainty band. In contrast, BNN trained on real-world data from the Sichuan region (China) and Korea, while still exhibiting excellent performance ($R^2$ of 0.995 and 0.983, respectively), showed wider confidence intervals and higher uncertainty, particularly for larger volumes. These differences underscore the inherent challenges of real-world data, including its variability, incompleteness, and noise. Despite these challenges, the real-world models effectively captured the general trends and statistical properties of debris flow volumes, demonstrating the utility of BNN even in less-than-ideal conditions. The ability of BNN to quantify both aleatoric and epistemic uncertainties emerged as a significant advantage, providing crucial insights into model confidence and data variability.

Our synthetic data generation method stands out as an inherently cutting-edge framework. The ability to create a targeted synthetic dataset for specific contexts is particularly valuable, as it allows us to easily adapt the data to different geographical areas. This is achieved by calibrating the parameters that statistically describe the debris flow phenomenon for each region. This synthetic data can then be used in combination with the Bayesian framework we have implemented, enabling the development of highly specific local models for Bayesian prediction of volume accumulation. This represents the core strength of our approach: even in the absence of real data, we are able to generate synthetic ones to obtain a probabilistic characterization of potential accumulation. Interestingly, areas with similar features could benefit from the application of the same predictive model. However, the ideal solution remains the generation of a specific synthetic dataset and, consequently, a dedicated model.

This research not only confirms the significant potential of BNN for debris flow hazard assessment but also validates a robust methodology for synthetic data generation. The findings provide a strong benchmark for what is achievable with high-quality data and offer a promising direction for future advancements in debris flow forecasting, especially in data-scarce or heterogeneous regions. Future work should focus on further refining synthetic data generation to incorporate more nuanced real-world complexities and exploring advanced BNN architectures or hybrid modeling approaches to mitigate uncertainties observed in highly variable field data.