Floods are one of the most catastrophic natural disasters causing hugely negative socioeconomic, urban and agricultural impacts and taking thousands of humans lives every year all around the globe (Borah and Buragohain, 2025; Nandi and Reddy, 2022; Grimaldi et al., 2019). Particularly in semi-arid regions, where the occurrence of seasonal thunderstorms and the expansion of agricultural land along riverbanks, the accurate prediction of flood hydrograph and flood inundation extent presents significant challenges (Bougherra and Mansour, 2023).    
       
The floods caused by river overflows have demonstrated their harmful effects in several regions, damaging crops and reducing sugar-cane yield (Berning et al., 2000). The flood risk management projects are usually require accurate hydraulic models to evaluate hydraulic consequences in terms of the depth, duration and velocity of the water and, on the other hand, accurate damage models able to quantify the vulnerability of the different land uses affected in order to estimate the socio-economic consequences of flooding (Brémond et al., 2013).
       
The flood-routing studies aim to predict downstream discharge and channel storage based on known upstream flow conditions, by other words is to optimize the outflow hydrograph using inflow hydrograph (Yoon and Padmanabhan, 1993; Perumala et al., 2017; Reggiani and Todini, 2018).
       
Many effective methods have been used for flood routing in the world, we cite, Viviroli et al., (2009); Liang (2010); Seyoum et al., (2012); Fenton (2019); Alhumoud (2022) and Khalifeh et al., (2020). These studies employed either the analytical approach, which relies on hydrodynamic equations primarily the Saint-Venant where we cannot include all the details and the full set of parameters governing the problem and requires the use of complicated numerical techniques such as the finite difference method, finite element method or the finite volume method (Dottori and Todini, 2011; Fenton, 2019). Or a simplified approach that considers the temporal evolution of storage within the studied reach. The latter is based on a volume balance within the system, with the Muskingum model serving as a well-known example of this method. including the flow rate, water depth and location, the drawback of this method is the neglect of the river’s morphology and physical characteristics which leads to uncertain parameters.
       
In the last years, the new algorithms and numerical techniques, have shown considerable effectiveness in solving hydrological problems. Mohan (1997) and Kim et al., (2001) implemented the genetic algorithm (GA) and the harmony search (HS) algorithm. In order to estimate the parameters of their Muskingum model, Niazkar and Afzali (2015) adopted the honey-bee mating optimization (HBMO) algorithm and compared the outputs to the results of 17 other algorithms. Noteworthy in this algorithm is the fast convergence to the optimum value within a broad range of values. Benefiting from the Grey Wolf Optimizer (GWO) algorithm (Akbari and Hessami-Kermani, 2021) optimized the parameters of the Muskingum model. Cai et al., (2014) use the dynamic simulation property of a cellular automaton was used to make further progress in flood routing for define submerged area.
       
The main idea of this work is to develop a numerical model suitable for simulating flood routing by optimising the temporal variation of flood discharge downstream or outflow hydrograph in a river reach. The connectionist models, such as artificial neural networks and fuzzy logic, have demonstrated their performance and efficiency in addressing hydrological and agricultural problems (Anandakumar et al., 2019; Hadj-Miloud and Djili, 2022; Han et al., 2010; Sidi, 2024). Among these models is the hybrid model, adaptive neuro-fuzzy inference system ANFIS employed in this study. The ANFIS approach is based on a stochastic method that utilises only the impact of the system’s main apparent parameters, along with comparing the results obtained to those of another method applied in flood routing. Here, we incorporated the time derivative of discharge because it reflects the rate of change and trend of the discharge function. The outflow discharge can differ for the same discharge value depending on whether the flow is increasing or decreasing. The aforementioned model has thus been employed in two case studies, in each of which the outflow hydrograph was developed after the calibration of the ANFIS model. It has been indicated that this model provides relatively more accurate results. It should be noted that although it has significative result the ANFIS system in other works (Jianhua, 2015; Han, 2010; Talei, 2010).

Model perspective
 
In hydrodynamics field, flood propagation is governed by the Saint-Venant equations, which are represented by the continuity equation and the momentum conservation equation simplified and formulated in one dimension (1D) as;

 
Where,
Q= Flow in time t.
A= Cross section of river.
h= Flow hight.
S_f and S₀= Slops of the free water surface and river bed respectivly.
q= The lateral flow.
       
The model is simplified through the Muskingum linear simplified model, as shown below:

 
It should be noted that a nonlinear model also exists.
Where,
Q^t_int and Q^t_out  = A discharge of inflow and outflow at time (t).
       
The coefficients C₀, C₁ and C₂ are functions of the parameters K, X and Δt.
       
While, K represents the travel time of water through a river reach and know as storage time constant and X is a dimensionless parameter that controls the relative influence of inflow and outflow on the storage within the river reach known as the weighing factor (Alhumoud, 2022; Akbari and Hessami-Kermani, 2021).
       
The majority of researchers in this context work either on the discretization of the hydrodynamic model or on the determination of the parameters of the Muskingum model.
       
At present, stochastic models demonstrate strong performance and efficiency in the field of hydrology, particularly connectionist models associated with artificial intelligence. These models generally rely on learning from the historical data of the studied system (Sidi et al., 2024). Their effectiveness mainly depends on two factors: the choice of the applied method characteristics and the selection of the system’s explanatory variables (inputs).            
       
In this study, we present an alternative perspective inspired by the previously presented models, seeing that the variation of the outflow, Q^t+Δt_out , at the section’s outlet at time t+Δt depends on the inflow and outflow at time t noted Q^t_int  Q^t_out as well as the variation in the inflow expressed by the derivative (∂Q/∂t)^t_int , this parameter has a significant influence on the rate of decrease or increase of the outflow at the downstream section under study (Fig 1).


 
Innovation and modelling tool
 
Given the complexity of the problem and in order to define and characterize the model, we will rely on a neuro-fuzzy system known as the Adaptive Neuro-Fuzzy Inference System (ANFIS).
        
Which takes Q^t_int , Q^t_out and (∂Q/∂t)^t_int as explanatory variables of the model (Inputs) and Q^t+Δt_out as the endogenous variable (Output).
        
In the initial phase, we focused a network, noted ANFIS (3Var) with three input variables, Q^t_int , Q^t_out and  (Fig 1).
Where,

 
In a second step, we made use of the two states of the derivative at t and t+Δt denoted by (∂Q/∂t)^t_int and (∂Q/∂t)^t+Δt_int (Fig 1). In this case, the network used consists of four input variables, ANFIS (4Var).
Where,

 
Adaptive neuro-fuzzy inference system (ANFIS)
 
The Adaptive Neuro-fuzzy Inference System (ANFIS) utilizes a multilayer neural network consisting of five (5) layers (Fig 2). Each layer of the network corresponds to the execution of a specific step in a Takagi-Sugeno type fuzzy inference system (Jang, 1993).


       
The first layer of an ANFIS architecture comprises a number of neurons equal to the number of input variables n multiplied by the number of fuzzy subsets p present in the inference system. The number (n) of subsets typically ranges from 2 to 5. To avoid model complexity and the proliferation of fuzzy rules, we selected n=3, which facilitates the learning process of the ANFIS system.
       
In our cases we have n.p=3.3=9 for three input variables and n.p=4.3.=12 for four input variables.  
       
This configuration enables the system to transform crisp input values into degrees of membership across different fuzzy sets, facilitating the fuzzification process essential for subsequent inference operations.
       
We incorporated the derivative of discharge in time because it reflects the rate of change and trend of the discharge function. The outflow discharge can differ for the same discharge value depending on whether the flow is increasing or decreasing. This derivative parameter is incorporated in two possible forms, either in form (∂Q/∂t)^t_int or form (∂Q/∂t)^t+Δt_int.
       
The second hidden layer is designed to compute the activation level of the premises. Each of its neurons represents the premise of a fuzzy inference rule. The activation functions of these neurons depend on the operators present in the rules (And or Or).
 

 
In our cases we obtain 3³ as 27 rules for the network of three input variables and 3⁴ as 81 for the network of four input variables.
       
The third hidden layer aims to normalize the degree of rule activation by computing the ratio between the i^th rule weight and the total sum of all rule weights. This operation is known as normalization step.
       
The fourth hidden layer is used to determine the parameters of the consequent part of the rules a, b, c, d, e. The function of each neuron in this layer is as follows:

- For our ANFIS model with three input variables we have;

 
-  For our ANFIS model with four input variables we have;

f⁴_k   is the output of k^th neuron in the fourth hidden layer. 
Where, 
 = The output of the third layer.
a, b, c,e= The consequent parameters.
       
The fifth output layer contains a single neuron in this layer, which computes the overall output as the sum of all incoming signals, that is:

 The study was conducted at Ahmed Zabana University of Relizane in Algeria, during the year 2025.

The models were applied to two real-world cases. The first case involved a non-smooth hydrograph from the River Wye in the United Kingdom. The second case concerned a single-peak hydrograph from the Karun River in Iran. The optimized outflows are computed by both model variants as ANFIS(3var) and ANFIS(4Var) for the characteristics presented in Table 1, in order to compare them with the results obtained by the GWO algorithm.


 
Case study 1
 
This is a non-smooth flood hydrograph belonging to the River Wye in the United Kingdom, which has been studied in many researchers works (Akbari and Hessami-Kermani, 2021).
       
In this case, the system data are presented in the form of inflows and outflows. The total number of data points is 33 observations, with a time step of 6 hours (Table 2).


       
It should be noted that, the performance of the GWO method is evaluated by comparing its results with those obtained in various relevant research studies  (Akbari and Hessami-Kermani, 2021).
       
The ANFIS(3Var) model with three input parameters Q^t_int , Q^t_out and (∂Q/∂t)_int showed discharge errors of up to 10.4%, with an average error of 3.1%. This error was reduced by adding at system a fourth explanatory parameter (∂Q/∂t)^t+Δt_int resulting in a maximum error of 7.7% with an average error  of 1.6% (Table 1), which is lower than that obtained by the GWO model as 9.7%. Moreover, a clear opposite trend in the error variation is observed in Fig 3. This observation has led us to conclude that the ANFIS(4Var) model, demonstrates considerable performance in this case.


       
The ANFIS models were implemented in MATLAB on Windows 10 64-bit. The datasets, obtained from previous studies (Akbari and Hessami-Kermani, 2021) , were fully pre-processed and serve as a reliable basis for model validation.
       
The effectiveness of the ANFIS (4 Var) model is observed in the output hydrograph as show in Fig 4, where the calculated discharges are almost superimposed on the observed discharges with Nash–Sutcliffe Efficiency (NSE) of 0.95.


 
Case study 2
 
After successfully demonstrating its effectiveness in the previous example, the ANFIS (4Var) model was applied to another real-world case on a river in Iran. The Karun River, the longest in southwestern Iran, receives inflow from the Dez River-another key component of the region’s ecosystem. Target discharges were measured at two gauging stations, Godar and Gotvand, with inflows ranging from 380 to 1.300 m³/s. In this hydrograph, the time step used was two hours (Akbari and Hessami-Kermani, 2021; Moghaddam et al., 2016). The optimized outflows presented in Table 3 were computed by the ANFIS (4Var) model in order to compare them with the results obtained by the GWO algorithm appliqued also in this case by (Akbari and Hessami-Kermani, 2021).


       
In this stage, the calculation of outflows by the ANFIS(4Var) model recorded a Root Mean Square Error (RMSE) as 25.14m³/s a maximum error of 19.7% with an average error MAPE of 3%, compared to the maximum and average errors obtained by the GWO model of 18.9% and 3.4%, respectively. This observation allows us to conclude that the ANFIS(4Var) model demonstrates a high level of performance and reliability in this case study.
       
Fig 5 reveals a distinctly opposite trend in the variation of errors, which provides further insight into the model’s behavior. This contrasting pattern underscores the consistency and robustness of the ANFIS(4Var) model when applied to this case study. Based on these findings, we can confidently conclude that the ANFIS(4Var) model demonstrates not only considerable performance but also strong predictive capability and reliability, making it a suitable tool for accurately simulating and forecasting.


       
The ANFIS(4var) effectiveness is also clearly reflected by the output hydrograph in Fig 6, where the simulated discharges are nearly indistinguishable from the observed values superimposed, with an average deviation equal 1.8% and a NSE of 0.98. This close alignment between calculated and measured flows highlights the model’s ability to accurately reproduce the hydrological behavior of the system, thereby reinforcing its suitability for forecasting and decision-making in similar rivers.

This work aimed to develop and evaluate a numerical model capable of simulating flood routing by optimizing the temporal variability of downstream discharge. By employing the Adaptive Neuro-Fuzzy Inference System (ANFIS), The study demonstrated that a stochastic, data-driven methodology is capable of representing the flow dynamic in river reach with only a minimum of explanatory parameters. Two configurations were explored: a three-input version and an extended four-input version that incorporates an additional derivative term to better represent temporal flow changes. Application to the River Wye showed that the four-input ANFIS significantly lowered average errors at 1.6% compared with the three-input model with 3.1% and even outperformed the benchmark GWO algorithm with 9.7%. The performance of the GWO method results is evaluated and compared with  those obtained using other methods applied in various  relevant research studies. The convergence between simulated and observed hydrographs confirmed the accuracy of the proposed approach. A second application to the Karun River under different flow conditions further validated these findings, with the four-input ANFIS again producing lower average errors as 3 % than the GWO model with 3,4% where a simulated and observed hydrographs were nearly superimposed with an average deviation varies between 0.8% and 1.8%.
       
Across both case studies, the ANFIS model demonstrated not only improved predictive performance but also robustness and consistency. Ultimately, it depends on the appropriate selection of input parameters, with particular emphasis on the derivative parameter (“Q/”t). These results highlight its suitability for operational flood forecasting and decision-making, especially where detailed hydraulic data are limited. Overall, the research confirms that the ANFIS (4Var) model is a powerful and reliable tool for simulating and forecasting flood routing in diverse hydrological settings.

The authors declare that there are no conflicts of interest regarding the publication of this article.