## Abstract

ANN was used to create a storage-based concurrent flow forecasting model. River flow parameters in an unsteady flow must be modeled using a model formulation based on learning storage change variable and instantaneous storage rate change. Multiple input-multiple output (MIMO) and multiple input-single output (MISO models in three variants were used to anticipate flow rates in the Tar River Basin in the United States. Gamma memory neural networks, as well as MLP and TDNNs models, are used in this study. When issuing a forecast, storage variables for river flow must be considered, which is why this study includes them. While considering mass balance flow, the proposed model can provide real-time flow forecasting. Results obtained are validated using various statistical criteria such as RMS error and coefficient of correlation. For the models, a coefficient of correlation value of more than 0.96 indicates good results. While considering the mass balance flow, the results show flow fluctuations corresponding to expressly and implicitly provided storage variations.

## HIGHLIGHTS

Gamma memory usage along with storage in river flow prediction.

Comparison and finding the best model to work on in real-time scenario using various ANN models.

Incorporating storage rate change along with flow values such as discharge and gauge height.

Use of mass balance flow and continuity equation satisfying river flow studies.

Practical applicability of ANN-based models in real-world situations.

## INTRODUCTION

Channel flooding is a complex dynamic process characterized by spatial and temporal variations inflow parameters. River flow and flooding are highly complicated processes that are space and time dependent, necessitating the use of space and time-dependent functions to represent them. Researchers have investigated the efficacy of techniques used in river flow modeling that rely on the application of artificial neural networks (ANNs). Giles *et al.* (1997) utilized one such model, namely multilayer perceptron (MLP), to predict flow variable(s), which is a feedforward network. Many investigators have used MLP, but MLP being a static network does not account for memory elements, hence cannot identify time-related variations in input sequences. Memory in ANN can be accounted for using two parameters, namely feedback and feedforward delays. Memory by feedback delays is accounted for by the self-recurrent network, given by Elman & Zipser (1988), while memory in feedforward delays by Time-Delay Neural network as provided by Singh *et al.* (2021b).

In 2015, Choudhury *et al.* used gamma memory neural network to update time-variant patterns in input sequences, since it has an adjustable memory parameter that assimilates both feed-forward and feedback delays. In situations where time-bound input data set patterns are unknown, ANNs with adaptable and updatable memory characteristics are significantly better and more efficient than static ANNs. Choudhury & Ullah (2015) in their work have used focused ANN, namely multiple gamma memory neural network (MGMNN) where it can spontaneously select best memory parameters, such as memory depth, hence utilizing the updating characteristics of time-varying river flow pattern in input flow sequences (Singh *et al.* 2021a).

The flow rate and depth of an unstable flow are constantly changing throughout time. The flow rate and storage that are interconnected over time are governed by river reach and upstream drainage parameters based on the geomorphologic structure of a river. In river flow modeling, the principles of continuity and mass balancing flow are always important. In most of the literature (Than *et al.* 2021; Zakaria *et al.* 2021) routing type ANN models used do not account for storage variation, hence do not comply with continuity law. Sil & Choudhury (2016) used fractional storage to formulate the upstream and downstream flow forecasting models, whereas in the year 2015, Choudhury & Roy (2015) used flow rate and flow depth based on learning characteristics of actual and fractional-storage variation that comply with mass balance flow in forecasting concurrent flow in river reach. They have implicitly incorporated storage rate change, which is contemporaneous with the flow rate change, into their study. In the case of river flow and flooding, the utilization of storage variables is critical. The goal of this study is to explicitly include storage change factors as well as flow rates at a specific time interval. This study is a continuation of Choudhury & Roy's work where the utilization of storage is considered both explicitly and implicitly in the flow forecasting technique of river channels, storage, which is the most significant factor when it comes to river flow studies, cannot be overlooked. In the forecasting river flow study, the use of instantaneous storage rate change and storage factors, as well as flow rate and flow depth, has been utilized. Model forms such as multiple inputs multiple outputs (MIMO-1 & MIMO-2) that Choudhury & Roy 2015 utilized in their paper based on static and dynamic ANN (MLP, TDNN, and GMNN) give this paper an extension of employing in conjunction with storage model form.

## METHODOLOGY

*et al.*(2021), storage variable and flow rates are interlinked and governed by the following equation:where

=storage parameter calculated explicitly at time

**t**=discharge at the upstream section calculated at time

**t**=flow rate/discharge obtained at the downstream section at time

**t**=river basin characteristics

Equations (2) and (3), giving discharge at time for upstream and downstream bounding sections in a river system, are obtained as per the work in Choudhury & Roy in 2015. When forecasting river flow, they do not explicitly account for the consideration of storage rate change variables. Storage, which is an important element for making a prediction in a basin channel, must be taken into account while modeling a river system.

*et al.*(2017)

Here, *c***1**, *c***3** = Muskingum Model parameters that represent river flow properties at a common section in an equivalent flow while , = upstream hydrograph evolution parameters and define the initial flow condition at the upstream and the downstream stations that produce no downstream flow after a time interval, Δt (Barbetta *et al.* 2017; Hadiyan *et al.* 2020; Zakaria *et al.* 2021).

*et al.*2021; DeVries & Principe 1992) which is defined as section reach properties on which , depends.

Storage rate change splits into two complementary parts as characteristics flow variation as:

for equivalent inflow

And for downstream flow.

Equation (4) can be split into *N* different parts. While Equation (5) depicts no flow at all upstream gauging stations at time , having initial flow state given by [(1− ) upstream flow shift factors], while (1−) at downstream depicts fractional storage in the river system.

The fractional-storage change is complementary and sum to actual storage change. Models used in forecasting are MIMO ANNs models by Choudhury (2007) and Choudhury & Roy (2015) that predict upstream and downstream flow and storage rate change. For predicting flow at upstream and downstream stations, ANN having similar number of input and output nodes may be taken as , as inputs and , 0 as desired outputs data set. For prediction in the downstream flow section of a river channel , as inputs and 0, as outputs can be used. MIMO-1 ANNs can be used to describe these prediction models. However, in addition to river flow, gauge height and storage rate change characteristics can be analyzed at the same time. Storage refers to the average or mean of all gauge heights from both inflow and outflow stations. The storage rate change parameter is represented by the average mean depth of all gauging stations. Combining two MIMO-1 ANNs like , and, , for learning, the actual storage variation will be termed as the MIMO-2 ANN model given in Choudhury & Roy's (2015) work. MISO ANNs, on the other hand, forecast for one single station and learn arbitrary storage change where training networks can be , as inputs and as one single output.

## CHARACTERISTICS OF MEMORY ELEMENTS IN DYNAMIC ANN

Three major components in dynamic ANN govern the memory element part in the neural network, namely depth, order, and resolution denoted by *D*, *P*, and *μ*. Memory depth at an instance, as the name implies, remembers how far the input parameter can be stored in the system. It refers to the size of a window into the past. The number of delay sections with transfer function *G* (*z*) is represented by memory order. The first tap is always initialized as current input and assigned as tap-zero. Current input, i.e., *I*(*t*) is having memory of *P*+1 number of taps. Memory depth initializes the number of taps and the delay between each tap (Tap Delay) in the delay line input. The length of the memory window in samples is the product of these two quantities. Order (*P*) of the memory remains tap minus one always. On the other hand, memory resolution refers to the fineness with which information is stored in individual taps. According to Choudhury & Roy (2015), it can be expressed in terms of memory taps and interpreted as taps per unit time step. It means that the resolution will be one-fourth if the data are stored in four time steps per unit time step. The solution in TDNN is always unity and cannot be changed, indicating that memory order and depth are always equal. However, in the case of Gamma Memory, where resolution fluctuates during training and is updated over a set period of time resolution=*P*/*D*.

To optimize the best memory depth based upon input, the gamma memory can update its memory depth parameter. Being recursive in nature, the gamma parameter adapts the memory depth during learning. The gamma memory adapts to choose an appropriate ratio of memory depth/resolution, which is very important in dynamic modeling with neural networks. Initially, the gamma parameter assigns its value to one. During adaptation, the gamma parameter decreases and then searches for further deepening memory depth. Memory depth is the ratio of the number of taps and gamma parameters. Gamma memory structure becomes stable when its value reaches 0.5.

### Time-delay neural network

Here, the transfer function *g*(*t*)=*Z*^{−1} is generating a kernel. It is a unit delay operator operating on discrete variable *I*(*t*) that gives a delayed version of *I*(*t*−1).

*I*(

*t*) as input processing from input to hidden layer neuron having memory order

*P*(Choudhury & Ullah 2015; Singh

*et al.*2021a) is the weight vector and is a vector of input having

*P*delayed lines. represents synaptic weights, while is biased. Here, the output of neuron

*i*can be a function of

*f*[

*u*

_{i}(

*t*)], which may activate neuron

*i*.

*et al.*(2020) gave the equation for calculating the output of neuron

*j*, where

*j*=1,2,. . .,

*N*+1, having

*N*+1 input nodes and a single hidden layer of

*m*nodes with memory order

*P*, given bywhere

Here is input to the hidden node for multiple input variables at time (*t*). And *r*=1, 2, 3, . ., *N*+1. *f* is the activation function of hidden node ‘*i*’ while *F*(.) is the activation function of output neuron *j.* is bias and is a weight connection connecting *p*th tap of *r*th node to *i*th neuron.

The fact that TDNN employs the backpropagation technique makes it ideal for river flow investigations. The inability to update memory elements while training to learn actual storage properties is also a drawback, resulting in poor outcomes and sub-optimal solutions.

## GAMMA MEMORY NEURAL NETWORK

*P*with one single input and multiple outputs that are based on linear structure. They also described impulse response in continuous time of the

*p*th tap as given by

*p*th tap can be computed recursively from one time lag (

*p*−1)th tap. Filter weights () can be updated by feed-forward adaptation. Impulse response of

*p*th tap () having order

*P*in a focused gamma memory neural network can be given bywhere first tap response is input.

*x*remain zero for

*p*=2, 3…

*P*, which is the same as the convolution memory model suggested by DeVries & Principe (1992), Arslan (2021), and Singh

*et al.*(2021b). Changing the derivative with a first-order forward difference, the following equation can be obtained

*et al.*2021)

=weight of the connection that joins neuron

*i*to the*p*th tap in the memory filter.

Figure 2(a) depicts focused gamma memory networks.

The input layers in a focussed MGMNN are recursive where memory parameter is a special back propagation, through time.

## NEURAL NETWORK TRAINING ALGORITHM

*d*(

*t*) implies the desired output while latter

*y*(

*t*) signifies corresponding network output, respectively. In the case of MLP and TDNN, which are feed-forward networks, since mapping is instantaneous and error gradient is not depending on time, the weights in the network get updated by applying the back-propagation technique (Cartuyvels

*et al.*2021). The simple partial derivative is used to update the network weights while training, as given by Werbos (1990) in Equation (20).here, is the summation of the product of and (

*t*) from

*j*=1 to

*N*

_{1}, i.e.

Here *N*_{1} is the number of nodes in the previous layer.

is negative of the product of learning rate to simple partial derivative, i.e. – .

Here is learning rate while the latter is a simple partial derivative.

Rate of increment in the weight can be computed using ordered derivative as

= ordered derivative of the error function concerning weight.

Here = *E*_*w*_{i,j} which is the product of *E*_net_{i}(*t*) and *x _{j}*(

*t*).

_{i}(

*t*) is the function of current activation only in nodes

*j*, and for the recurrent network such as gamma memory net

_{i}(

*t*) can be given as (Than

*et al.*2021):

Here *T* is the trajectory length, while [net_{i}(*t*)] is the derivative of the transfer function concerning net_{i}(*t*).

### Model application and results

The use of ANNs to predict concurrent flows in the Tar-Pamlico river in the United States has been investigated. Figure 2(c) depicts the research area. In this diagram, all of the gauging stations are portrayed as dots, with Enfield, Hilliardstone, and Rockymount as upstream flow stations and Tarboro as a downstream flow station. Data has been collected from the USGS streamflow archive (https://waterdata.usgs.gov/nc/nwis/current/?type=flow&group_key=basin_cd) where concurrent flow records for the aforesaid gauging stations from 29 July 2004 to 1 October 2004, are utilized. This study used 786 consecutive data sets with stream flow and gauge height spread at 2-h intervals. Peak flow rates of 3,400, 1,420, and 6,280 for Enfield, Hilliardstone, and Rockymount stations, and 14,133 for Tarboro outflow station, have been compared using Peak flow criterion, and it has been determined that flood episodes are moderate to low for the assigned time period. The model architecture is chosen through trial and error, whereas the network design is determined by the training, which is detailed in Table 1(b). For training purposes, the first 65 percent of data sets are randomly selected or sequentially selected. Fifteen per cent are utilized for cross-validation, while the remaining 20% are used for network performance testing. The river network with four flow measuring sections is separated into a river system with three upstream flow sections located at Enfield, Hilliardstone, and Rockymount, while Tarboro works as a common downstream outflow section in forecasting concurrent flow of the Tar basin. MIMO-1 networks are trained to learn fractional-storage rate change utilizing flow rates as input for all four sections at time *t* + *t*, with zero flow rate assigned to all but the forecasting section as the intended output. As inputs and outputs, the MIMO-2 ANNs are trained to learn actual storage characteristics using concurrent flow data sets separated at 2-h time intervals for all sections. MISO ANN models, on the other hand, which are based on learning arbitrary storage fluctuations, are trained using only the section used in predicting as input and flow rate for t + *Δ*t as the desired output. Equation (22) is used for calculating the rate of change of error with respect to weights where is observed while is forecasted flow rates at section *r* at time *t*. Performance result and model performances in forecasting the testing data sets with a lead time of 2 h are given in Table 1(b) for MIMO-2 models. The greatest RMSE values estimated from all gauging sites, namely Enfield, Hilliardstone, Greenville, and Rockymount, are less than 10% of the measured mean flow rate. Forecasted flow rates for various sections, including explicitly calculated storage rate change having a lead time of 2-h using various models like MGMNN, are shown in Figures 1(c), 3(c), 4(c) and 5(c). The values obtained in the graph clearly show that the highest deviation of the predicted peaks from the observed data sets is computed to be less than 280 cfs, which is less than that of deviation of estimated peaks by using computer software mentioned earlier. The *R* value obtained is mostly more than 0.90, which shows satisfactory results given that *R* = 1 is a perfect fit model.

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 4017.92 | 4367.16 | 47,145.18 | 15,5231 | 44,829.61 | 43,854.74 |

NMSE | 0.0646 | 0.63503 | 0.2833 | 0.2066 | 0.07879 | 0.07535 |

MAE | 52.8300 | 53.8488 | 172.95 | 341.690 | 171.165 | 169.291 |

Min Abs Error | 0.0673 | 1.53316 | 1.5096 | 0.66911 | 1.54003 | 0.79339 |

Max Abs Error | 167.181 | 146.563 | 572.23 | 983.924 | 543.570 | 503.691 |

R | 0.98268 | 0.96226 | 0.9203 | 0.94103 | 0.9866 | 0.98802 |

(b) | ||||||

MISO | MLP | P = 0 | 6–9–1 | 35 | 2,000 | Static Back-propagation |

TDNN | P = 2 | 12–6–1 | 65 | 2,000 | ||

MGMNN | P = 2 | 12–6–1 | 92 | 2,000 | Back-propagation through time, momentum = 0.7, Learning rate = 0.01 | |

MIMO | MLP | P = 0 | 6–9–6 | 42 | 5,000 | Static Back-propagation |

TDNN | P = 2 | 12–4–6 | 98 | 10,000 | ||

MGMNN | P = 2 | 12–4–6 | 98 | 10,000 | Back-propagation through time, momentum = 0.7, Learning rate = 0.01 |

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 4017.92 | 4367.16 | 47,145.18 | 15,5231 | 44,829.61 | 43,854.74 |

NMSE | 0.0646 | 0.63503 | 0.2833 | 0.2066 | 0.07879 | 0.07535 |

MAE | 52.8300 | 53.8488 | 172.95 | 341.690 | 171.165 | 169.291 |

Min Abs Error | 0.0673 | 1.53316 | 1.5096 | 0.66911 | 1.54003 | 0.79339 |

Max Abs Error | 167.181 | 146.563 | 572.23 | 983.924 | 543.570 | 503.691 |

R | 0.98268 | 0.96226 | 0.9203 | 0.94103 | 0.9866 | 0.98802 |

(b) | ||||||

MISO | MLP | P = 0 | 6–9–1 | 35 | 2,000 | Static Back-propagation |

TDNN | P = 2 | 12–6–1 | 65 | 2,000 | ||

MGMNN | P = 2 | 12–6–1 | 92 | 2,000 | Back-propagation through time, momentum = 0.7, Learning rate = 0.01 | |

MIMO | MLP | P = 0 | 6–9–6 | 42 | 5,000 | Static Back-propagation |

TDNN | P = 2 | 12–4–6 | 98 | 10,000 | ||

MGMNN | P = 2 | 12–4–6 | 98 | 10,000 | Back-propagation through time, momentum = 0.7, Learning rate = 0.01 |

*a* = input nodes; *b* = hidden node; *c* = output node.

The network architectures employed and other essential aspects of training the model form are presented in Table 1(b), whereas the model architecture is based on a trial and error approach in the research reported in this work. The memory order *P* indicates whether the data are in the present observation or in the lagged form. *P* = 0 always depicts current observation, while value 1 means memory order in current observation and one lagged input. Figures 1(a), 3(a) and 4(a) depict the flow forecast of observed and predicted for one particular inflow gauging station using the MIMO-1 model form having MLP memory, including the storage rate change variable. Figures 1(b), 3(b) and 4(b) presented the graph for TDNN memory using the MIMO-1 model form for flow forecast in outflow station. Values obtained for the same are presented in tables having MLP memory in Tables 2(a), 3(a), and 4(a) for inflow station, and Table 5(a) for outflow station. Tables 2(b), 3(b), and 4(b) show the TDNN memory parameter values for inflow gauging stations, while Table 5(b) shows those for outflow gauging stations.

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 2,206.416 | 2.79 × 10^{−7} | 0.00012 | 0.00027 | 2,173.492 | 4,750.32 |

NMSE | 0.0350 | – | – | – | 0.034 | 0.0189 |

MAE | 32.9729 | 0.0004 | 0.00887 | 0.0137 | 33.049 | 42.527 |

Min Abs Error | 0.3006 | 1.233 × 10^{−5} | 2.66 × 10^{−5} | 0.0001 | 0.0574 | 0.6218 |

Max Abs Error | 136.8921 | 0.0008 | 0.0208 | 0.0292 | 135.143 | 199.804 |

R | 0.9952 | – | – | – | 0.995 | 0.994 |

(b) | ||||||

MSE | 4,481.87 | 0.0001 | 9.21 × 10^{−5} | 4.65 × 10^{−5} | 4,074.788 | 62,873.20 |

NMSE | 0.0710 | – | – | – | 0.0646 | 0.2503 |

MAE | 59.013 | 0.0118 | 0.0090 | 0.0058 | 56.5728 | 189.57 |

Min Abs Error | 0.0329 | 1.621 × 10^{−5} | 3.74155 × 10^{−5} | 4.2 × 10^{−5} | 3.3458 | 5.8211 |

Max Abs Error | 167.66 | 0.0167 | 0.0136 | 0.0157 | 162.5547 | 700.67 |

R | 0.9784 | – | – | – | 0.9784 | 0.974 |

(c) | ||||||

MSE | 4,907.743 | 0.0006 | 0.0001 | 9.6 × 10^{−5} | 3,261.432 | 26,296.15 |

NMSE | 0.0778 | – | – | – | 0.0517 | 0.1047 |

MAE | 60.7313 | 0.0248 | 0.0126 | 0.0093 | 49.286 | 104.04 |

Min Abs Error | 0.9153 | 0.0137 | 0.0044 | 0.0026 | 0.4009 | 0.1678 |

Max Abs Error | 136.1445 | 0.0330 | 0.0204 | 0.0168 | 117.5207 | 528.048 |

R | 0.9629 | – | – | – | 0.9829 | 0.9842 |

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 2,206.416 | 2.79 × 10^{−7} | 0.00012 | 0.00027 | 2,173.492 | 4,750.32 |

NMSE | 0.0350 | – | – | – | 0.034 | 0.0189 |

MAE | 32.9729 | 0.0004 | 0.00887 | 0.0137 | 33.049 | 42.527 |

Min Abs Error | 0.3006 | 1.233 × 10^{−5} | 2.66 × 10^{−5} | 0.0001 | 0.0574 | 0.6218 |

Max Abs Error | 136.8921 | 0.0008 | 0.0208 | 0.0292 | 135.143 | 199.804 |

R | 0.9952 | – | – | – | 0.995 | 0.994 |

(b) | ||||||

MSE | 4,481.87 | 0.0001 | 9.21 × 10^{−5} | 4.65 × 10^{−5} | 4,074.788 | 62,873.20 |

NMSE | 0.0710 | – | – | – | 0.0646 | 0.2503 |

MAE | 59.013 | 0.0118 | 0.0090 | 0.0058 | 56.5728 | 189.57 |

Min Abs Error | 0.0329 | 1.621 × 10^{−5} | 3.74155 × 10^{−5} | 4.2 × 10^{−5} | 3.3458 | 5.8211 |

Max Abs Error | 167.66 | 0.0167 | 0.0136 | 0.0157 | 162.5547 | 700.67 |

R | 0.9784 | – | – | – | 0.9784 | 0.974 |

(c) | ||||||

MSE | 4,907.743 | 0.0006 | 0.0001 | 9.6 × 10^{−5} | 3,261.432 | 26,296.15 |

NMSE | 0.0778 | – | – | – | 0.0517 | 0.1047 |

MAE | 60.7313 | 0.0248 | 0.0126 | 0.0093 | 49.286 | 104.04 |

Min Abs Error | 0.9153 | 0.0137 | 0.0044 | 0.0026 | 0.4009 | 0.1678 |

Max Abs Error | 136.1445 | 0.0330 | 0.0204 | 0.0168 | 117.5207 | 528.048 |

R | 0.9629 | – | – | – | 0.9829 | 0.9842 |

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 0.0002 | 423.6829 | 2.13 × 10^{−5} | 0.0001 | 480.7132 | 2027.12 |

NMSE | – | 0.0603 | – | – | 0.0685 | 0.0095 |

MAE | 0.0122 | 15.1370 | 0.00359 | 0.0094 | 15.5219 | 42.0944 |

Min Abs Error | 3.8 × 10^{−5} | 0.3035 | 1.54294 × 10^{−5} | 1.35 × 10^{−6} | 0.1037 | 5.5711 |

Max Abs Error | 0.0383 | 64.4334 | 0.01051 | 0.0322 | 68.1354 | 82.6840 |

R | – | 0.9841 | – | – | 0.9812 | 0.99 |

(b) | ||||||

MSE | 7.1 × 10^{−5} | 1917.0373 | 6.43 × 10^{−5} | 0.0003 | 3109.76 | 23,499.64 |

NMSE | – | 0.2732 | – | – | 0.4433 | 0.1109 |

MAE | 0.0062 | 34.1655 | 0.006545688 | 0.0181 | 40.3894 | 118.1489 |

Min Abs Error | 9.8 × 10^{−5} | 0.0352 | 0.000154039 | 0.0003 | 0.3941 | 1.9747 |

Max Abs Error | 0.0250 | 107.6675 | 0.020841308 | 0.030 | 129.8926 | 448.3595 |

R | – | 0.9570 | – | – | 0.9600 | 0.9816 |

(c) | ||||||

MSE | 4.3 × 10^{−5} | 2295.4190 | 0.0004 | 0.0005 | 2329.384 | 20,813.30 |

NMSE | – | 0.3272 | – | – | 0.3320 | 0.0982 |

MAE | 0.0048 | 35.7521 | 0.0193 | 0.0220 | 35.8428 | 112.26 |

Min Abs Error | 1.2 × 10^{−6} | 0.4672 | 0.0077 | 0.0010 | 0.1181 | 2.1977 |

Max Abs Error | 0.0151 | 125.7821 | 0.0282 | 0.0298 | 122.1344 | 391.7531 |

R | – | 0.9603 | – | – | 0.95818 | 0.9888 |

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 0.0002 | 423.6829 | 2.13 × 10^{−5} | 0.0001 | 480.7132 | 2027.12 |

NMSE | – | 0.0603 | – | – | 0.0685 | 0.0095 |

MAE | 0.0122 | 15.1370 | 0.00359 | 0.0094 | 15.5219 | 42.0944 |

Min Abs Error | 3.8 × 10^{−5} | 0.3035 | 1.54294 × 10^{−5} | 1.35 × 10^{−6} | 0.1037 | 5.5711 |

Max Abs Error | 0.0383 | 64.4334 | 0.01051 | 0.0322 | 68.1354 | 82.6840 |

R | – | 0.9841 | – | – | 0.9812 | 0.99 |

(b) | ||||||

MSE | 7.1 × 10^{−5} | 1917.0373 | 6.43 × 10^{−5} | 0.0003 | 3109.76 | 23,499.64 |

NMSE | – | 0.2732 | – | – | 0.4433 | 0.1109 |

MAE | 0.0062 | 34.1655 | 0.006545688 | 0.0181 | 40.3894 | 118.1489 |

Min Abs Error | 9.8 × 10^{−5} | 0.0352 | 0.000154039 | 0.0003 | 0.3941 | 1.9747 |

Max Abs Error | 0.0250 | 107.6675 | 0.020841308 | 0.030 | 129.8926 | 448.3595 |

R | – | 0.9570 | – | – | 0.9600 | 0.9816 |

(c) | ||||||

MSE | 4.3 × 10^{−5} | 2295.4190 | 0.0004 | 0.0005 | 2329.384 | 20,813.30 |

NMSE | – | 0.3272 | – | – | 0.3320 | 0.0982 |

MAE | 0.0048 | 35.7521 | 0.0193 | 0.0220 | 35.8428 | 112.26 |

Min Abs Error | 1.2 × 10^{−6} | 0.4672 | 0.0077 | 0.0010 | 0.1181 | 2.1977 |

Max Abs Error | 0.0151 | 125.7821 | 0.0282 | 0.0298 | 122.1344 | 391.7531 |

R | – | 0.9603 | – | – | 0.95818 | 0.9888 |

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 7.1 × 10^{−5} | 0.0001 | 4672.28 | 4.3 × 10^{−5} | 4862.02 | 11,556.10 |

NMSE | – | – | 0.02796 | – | 0.02910 | 0.0400 |

MAE | 0.0068 | 0.0066 | 44.5391 | 0.0056 | 44.8619 | 78.133 |

Min Abs Error | 0.0001 | 1.15 × 10^{−5} | 0.21484 | 9.82 × 10^{−5} | 0.2759 | 0.1942 |

Max Abs Error | 0.0198 | 0.0265 | 232.025 | 0.01454 | 225.264 | 283.233 |

R | – | – | 0.98835 | – | 0.9885 | 0.9970 |

(b) | ||||||

MSE | 0.0003 | 0.0007 | 22,048.51 | 0.0002 | 24,310.4 | 13,625.06 |

NMSE | – | – | 0.1319 | – | 0.1455 | 0.04724 |

MAE | 0.0157 | 0.0271 | 121.5309 | 0.0162 | 126.202 | 99.6741 |

Min Abs Error | 0.0069 | 0.0126 | 2.5753 | 0.0114 | 1.4478 | 0.8670 |

Max Abs Error | 0.0376 | 0.0338 | 388.0812 | 0.0264 | 418.652 | 251.1397 |

R | – | – | 0.9554 | – | 0.9608 | 0.9887 |

(c) | ||||||

MSE | 8.84 × 10^{−5} | 0.0003 | 11,379.90 | 1.5 × 10^{−5} | 19,707.56 | 19,458.30 |

NMSE | – | – | 0.0681 | – | 0.1179 | 0.0674 |

MAE | 0.0088 | 0.0185 | 82.042 | 0.0027 | 104.8487 | 104.71 |

Min Abs Error | 0.0004 | 0.0006 | 0.1881 | 8.8 × 10^{−6} | 0.2549 | 0.22 |

Max Abs Error | 0.0172 | 0.0349 | 337.59 | 0.0110 | 367.049 | 326.006 |

R | – | – | 0.9694 | – | 0.97637 | 0.9911 |

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 7.1 × 10^{−5} | 0.0001 | 4672.28 | 4.3 × 10^{−5} | 4862.02 | 11,556.10 |

NMSE | – | – | 0.02796 | – | 0.02910 | 0.0400 |

MAE | 0.0068 | 0.0066 | 44.5391 | 0.0056 | 44.8619 | 78.133 |

Min Abs Error | 0.0001 | 1.15 × 10^{−5} | 0.21484 | 9.82 × 10^{−5} | 0.2759 | 0.1942 |

Max Abs Error | 0.0198 | 0.0265 | 232.025 | 0.01454 | 225.264 | 283.233 |

R | – | – | 0.98835 | – | 0.9885 | 0.9970 |

(b) | ||||||

MSE | 0.0003 | 0.0007 | 22,048.51 | 0.0002 | 24,310.4 | 13,625.06 |

NMSE | – | – | 0.1319 | – | 0.1455 | 0.04724 |

MAE | 0.0157 | 0.0271 | 121.5309 | 0.0162 | 126.202 | 99.6741 |

Min Abs Error | 0.0069 | 0.0126 | 2.5753 | 0.0114 | 1.4478 | 0.8670 |

Max Abs Error | 0.0376 | 0.0338 | 388.0812 | 0.0264 | 418.652 | 251.1397 |

R | – | – | 0.9554 | – | 0.9608 | 0.9887 |

(c) | ||||||

MSE | 8.84 × 10^{−5} | 0.0003 | 11,379.90 | 1.5 × 10^{−5} | 19,707.56 | 19,458.30 |

NMSE | – | – | 0.0681 | – | 0.1179 | 0.0674 |

MAE | 0.0088 | 0.0185 | 82.042 | 0.0027 | 104.8487 | 104.71 |

Min Abs Error | 0.0004 | 0.0006 | 0.1881 | 8.8 × 10^{−6} | 0.2549 | 0.22 |

Max Abs Error | 0.0172 | 0.0349 | 337.59 | 0.0110 | 367.049 | 326.006 |

R | – | – | 0.9694 | – | 0.97637 | 0.9911 |

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 4.3 × 10^{−5} | 5.10 × 10^{−6} | 6.20 × 10^{−6} | 6213.19 | 6011.07 | 3761.24 |

NMSE | – | – | – | 0.0072 | 0.00700 | 0.0055 |

MAE | 0.0053 | 0.0020 | 0.0021 | 64.1414 | 62.8454 | 53.2630 |

Min Abs Error | 1.6 × 10^{−5} | 3.38172 × 10^{−5} | 1.57 × 10^{−5} | 0.54994 | 0.03488 | 0.1343 |

Max Abs Error | 0.0130 | 0.0048 | 0.0054 | 202.781 | 207.766 | 123.888 |

R | – | – | – | 0.9973 | 0.997 | 0.99911 |

(b) | ||||||

MSE | 7.6 × 10^{−5} | 0.0001 | 0.0001 | 122,791.9 | 145,073.2 | 80,668.77 |

NMSE | – | – | – | 0.14310 | 0.16907 | 0.1200 |

MAE | 0.0060 | 0.0083 | 0.0103 | 308.1076 | 339.253 | 251.4257 |

Min Abs Error | 1.7 × 10^{−6} | 9.52 × 10^{−6} | 5.54 × 10^{−5} | 4.7350 | 0.0090 | 3.83098 |

Max Abs Error | 0.0224 | 0.0265 | 0.0304 | 744.564 | 807.648 | 564.30 |

R | – | – | – | 0.9687 | 0.9510 | 0.9734 |

(c) | ||||||

MSE | 3.4 × 10^{−5} | 0.0003 | 0.0004 | 90,508.8 | 108,300.3 | 30,266.9 |

NMSE | – | – | – | 0.1054 | 0.1262 | 0.0450 |

MAE | 0.0054 | 0.0147 | 0.0160 | 227.888 | 219.009 | 148.994 |

Min Abs Error | 0.0001 | 2.48 × 10^{−5} | 0.0001 | 0.450210 | 0.7565 | 0.0600 |

Max Abs Error | 0.0081 | 0.0404 | 0.0464 | 698.8216 | 797.253 | 310.112 |

R | – | – | – | 0.946 | 0.9438 | 0.9906 |

Performance . | Enfield . | Hilliardstone . | Rockymount . | Tarboro . | Storage rate change . | Average storage . |
---|---|---|---|---|---|---|

(a) | ||||||

MSE | 4.3 × 10^{−5} | 5.10 × 10^{−6} | 6.20 × 10^{−6} | 6213.19 | 6011.07 | 3761.24 |

NMSE | – | – | – | 0.0072 | 0.00700 | 0.0055 |

MAE | 0.0053 | 0.0020 | 0.0021 | 64.1414 | 62.8454 | 53.2630 |

Min Abs Error | 1.6 × 10^{−5} | 3.38172 × 10^{−5} | 1.57 × 10^{−5} | 0.54994 | 0.03488 | 0.1343 |

Max Abs Error | 0.0130 | 0.0048 | 0.0054 | 202.781 | 207.766 | 123.888 |

R | – | – | – | 0.9973 | 0.997 | 0.99911 |

(b) | ||||||

MSE | 7.6 × 10^{−5} | 0.0001 | 0.0001 | 122,791.9 | 145,073.2 | 80,668.77 |

NMSE | – | – | – | 0.14310 | 0.16907 | 0.1200 |

MAE | 0.0060 | 0.0083 | 0.0103 | 308.1076 | 339.253 | 251.4257 |

Min Abs Error | 1.7 × 10^{−6} | 9.52 × 10^{−6} | 5.54 × 10^{−5} | 4.7350 | 0.0090 | 3.83098 |

Max Abs Error | 0.0224 | 0.0265 | 0.0304 | 744.564 | 807.648 | 564.30 |

R | – | – | – | 0.9687 | 0.9510 | 0.9734 |

(c) | ||||||

MSE | 3.4 × 10^{−5} | 0.0003 | 0.0004 | 90,508.8 | 108,300.3 | 30,266.9 |

NMSE | – | – | – | 0.1054 | 0.1262 | 0.0450 |

MAE | 0.0054 | 0.0147 | 0.0160 | 227.888 | 219.009 | 148.994 |

Min Abs Error | 0.0001 | 2.48 × 10^{−5} | 0.0001 | 0.450210 | 0.7565 | 0.0600 |

Max Abs Error | 0.0081 | 0.0404 | 0.0464 | 698.8216 | 797.253 | 310.112 |

R | – | – | – | 0.946 | 0.9438 | 0.9906 |

The results of MISO models ANN are also encouraging, but because they rely on arbitrary flow matching approaches, they do not comply with mass balance continuum flow mechanics. The model performance of TDNN is very similar to the findings of MGNN. Figure 6 shows that the observed values acquired in MGMNN are almost identical to the predicted value. These models are also useful in circumstances where real-time flow forecasting is required.

## DISCUSSION AND CONCLUSION

The MIMO-1 model's results in matching zero flow rates and flow depth for sections other than the forecasting section accurately match the observed flow rate when forecasting that section. This is especially important when there is a provision for matching known flow rates, which gives an extra advantage when assessing forecast accuracy. Connection weights in the MISO ANN model form which are specifically joining the output node through the last hidden node are tapped to zero except for the forecasting section, which incorporates undefined storage and flows variation. Hence in the spatial-temporal domain, these models do not comply with mass balance flow in river flow studies. Although the performances of the MISO and MIMO models are nearly identical, the MISO model's connection weights are less significant. The data presented in this paper suggest that when giving a forecast, storage parameters should be used openly as well as implicitly. The research presented in this paper shows that storage is just as important as other flow parameters like flow rate or flow depth. As a result, when training MIMO and MISO ANN models for flow rate forecasting, including instantaneous and average storage is just as important as observing the continuity norm. Multiple portions of the Tar River Basin in the United States have been forecasted using both static and dynamic ANN. The model's performance is satisfactory when measured using several statistical metrics such as RMSE and CE. For river flow investigations with varying temporal patterns, focused GMNN is appropriate. Other memory parameters, such as Laguaare, should be investigated further. Furthermore, understanding the physics of the model will require an understanding of the connection weights.

## DATA AVAILABILITY STATEMENT

All relevant data are available from https://waterdata.usgs.gov/nc/nwis/current/?type=flow.