In [3], the authors proposed a dynamic load balancing mechanism to improve the efficiency and adaptability of the IoT environment. Their research work highlighted how dynamic resource management approaches can provide a solution to fluctuating workloads in IoT environments. Similarly, study [18] proposed an optimized task-scheduling mechanism specifically for fog-assisted IoT environments. It develops this mechanism to introduce preemptive features in scheduling architectures. This concept has been closely aligned with our efforts to improve responsiveness in IoT systems. The study in [30] presented an overview of the secure and adaptive deep learning cloud task scheduling. In their approach, by marrying security with efficiency, there is a developed focus on managing computational tasks efficiently, hence laying down the principles that could echo the ANNDRA-IoT framework objectives. Meanwhile, in [31], the authors have proposed an autonomic framework for workload prediction and resource allocation in industrial IoT systems. Predictive analysis and autonomic allocation strategies in their work share common goals with ANNDRA-IoT and point out the necessity for intelligent resource management.

The authors in [32] conducted a comprehensive review of resource-efficient distributed AI methods for IoT applications. Their review highlighted the increasing demand for AI-driven solutions, laying the foundations for integrating AI to optimize IoT resource allocation-a central theme in our research. On the other hand, study [33] developed an energy management model for wireless sensor networks in IoT systems, emphasizing resource optimization. Finally, reference [34] studied the AI-driven communication-efficient method at the edge of IoT. The contributions of [34] complement our work in minimizing communications overhead in IoT settings. Furthermore, study [35] proposed a multiresource allocation approach with fairness constraints; the ANNDRA-IoT model developed has considered the fairness of coverage consideration for resource allocation.

In [36], an imperialist competitive algorithm was used to implement IoT services in fog computing, focusing on resource utilization. Similarly, study [37] proposed a federated learning technique to achieve energy efficiency and resource optimization within environmentally sustainable IoT edge intelligence systems. These studies align with the goals of ANNDRA-IoT to maintain energy-efficient operations. Lastly, study [38] presented a hybrid authentication architecture for heterogeneous IoT systems, blending centralized and blockchain-based methods. This approach addresses security and efficiency, critical to the secure functioning of IoT environments. Works such as [39] and [40] further contribute to this discourse by exploring distributed IoT service placement and energy-efficient task offloading strategies, respectively.

The management of resources within the IoT has seen significant advancements through the adoption of LSTM neural networks. Table 2 provides a comparison including rule-based systems, classic machine learning techniques, and different neural network designs.

LSTM, a type of recurrent neural network, has the distinct ability to identify and retain temporal patterns, making it essential for managing resources in IoT scenarios characterized by time-series data. In [41], experimental evidence demonstrated that LSTMs exhibited superior predictive performance by effectively capturing temporal dynamics. This capability improves resource allocation strategies, thus improving the responsiveness of IoT systems [13,41,42]. In contrast, rule-based systems perform inadequately in dynamic IoT environments due to their static nature. Their inability to adapt to evolving data patterns renders them ineffective compared to LSTM models, which can seamlessly integrate new information and maintain consistent performance as system configurations evolve.

Conventional machine learning techniques, including support vector machines and decision trees, played a fundamental role in the early stages of IoT resource management. However, their reliance on manual feature selection and significant domain expertise limits their applicability in complex, high-dimensional sequential data scenarios [43,44]. LSTMs address these limitations by processing intricate sequential information with remarkable efficiency, which makes them particularly suited to the sophisticated demands of IoT frameworks. Other neural network architectures, such as convolutional neural networks (CNNs) and gated recurring units (GRUs), have their own strengths in the management of spatial and temporal data [45]. Hybrid models that integrate CNNs with LSTM have proven especially effective in tasks such as solar irradiance prediction and photovoltaic power [46]. These combinations balance computational efficiency with predictive accuracy, overcoming the constraints of single-architecture models in IoT applications [47].

ANNDRA-IoT has been proposed for resource management in highly heterogeneous IoTs. Introduce reconfigurability for adaptation in versatile and complex scenarios related to IoT ecosystems. By incorporating LSTM neural networks with Artificial Neural Networks (ANNs), ANNDRA-IoT enables dynamic allocation of resources, thus performing self-adaptation even under complex operational conditions. This system uses prediction capabilities as an intelligent one, working in real time to manage network and computational resources through device utilization patterns.

The architecture of ANNDRA-IoT includes LSTM units responsible for processing input data streams from a variety of IoT devices. These units analyze resource usage patterns and generate resource allocation decisions. Fig. 1 illustrates the architecture, showcasing the interaction among the system’s components and the data flow within ANNDRA-IoT.

This section introduces a new LSTM neural network that is suitable for real-time IoT resource allocation. The framework is based on stochastic calculus, tensor analysis, variational optimization, differential geometry, and information theory; thus, it can learn sophisticated temporal and multi-dimensional dynamics of an IoT system.

Consider an IoT environment comprising a set of devices 𝒟={d1,d2,…,dN}, each generating multidimensional data streams over continuous time. Let the state of the entire IoT system at time t be represented by a stochastic process X(t)∈Rn, where n denotes the dimensionality of the state space of the system. The objective is to determine an optimal resource allocation policy π∗ that minimizes a cumulative cost functional over a finite time horizon [0,T]: (1)π∗=argminπ∈ΠE[∫0Tℒ(X(t),π(X(t),t),t)dt+Φ(X(T))],where Π is the set of admissible policies, ℒ is an instantaneous cost function capturing resource utilization and performance metrics, and Φ is a terminal cost function. The evolution of the state of the IoT system X(t) is governed by a controlled stochastic differential equation (SDE): (2)dX(t)=f(X(t),π(X(t),t),t)dt+G(X(t),t)dW(t),where f:Rn×Rm×[0,T]→Rn is the drift term, G:Rn×[0,T]→Rn×k is the diffusion term and W(t) is a Wiener process of dimensions k representing environmental uncertainties. To approximate the optimal policy π∗, we construct an LSTM neural network enhanced with tensor operations and higher-order gates. Let xt∈Rp denote the input vector in time t, representing sensor readings and device statuses. The LSTM cell is redefined using the tensor calculus to handle multidimensional interactions: (3)it=σ(𝒲i∗xt+𝒰i∗ht−1+𝒱i∗ct−1+bi), (4)ft=σ(𝒲f∗xt+𝒰f∗ht−1+𝒱f∗ct−1+bf), (5)ct=ft⊙ct−1+it⊙tanh⁡(𝒲c∗xt+𝒰c∗ht−1+bc), (6)ot=σ(𝒲o∗xt+𝒰o∗ht−1+𝒱o∗ct+bo), (7)ht=ot⊙tanh⁡(ct),where ∗ denotes tensor contraction, ⊙ is the element-wise Hadamard product, and σ is the sigmoid activation function. Tensors 𝒲i,𝒰i,𝒱i, etc., are higher-order weight tensors that capture interactions across multiple dimensions. The LSTM network parameters are optimized using variational principles. We define an action functional 𝒮 associated with the policy πθ parameterized by the network weights θ: (8)𝒮[θ]=E[∫0Tℒ(Xθ(t),πθ(Xθ(t),t),t)dt+Φ(Xθ(T))],where Xθ(t) denotes the state trajectory under policy πθ. Optimization proceeds by computing the functional derivative δ𝒮/δθ and updating θ by gradient descent in the function space. Recognizing that the state space of the IoT system may exhibit manifold structures, we introduce differential geometric concepts. Let ℳ be a smooth manifold representing the state space with a Riemannian metric gX. The geodesic distance between states X1 and X2 is given by: (9)dℳ(X1,X2)=infγ∫01gγ(t)(γ˙(t),γ˙(t))dt,where γ:[0,1]→ℳ is a smooth path connecting X1 and X2. We redefine the LSTM input transformation to map inputs onto ℳ: (10)zt=expht−1⁡(𝒲z∗xt+bz),where expht−1 is the Riemannian exponential map at ht−1. To capture non-Gaussian characteristics and higher-order dependencies in the data, we introduce gates based on higher-order statistics: (11)kt=κ(𝒲k∗xt+𝒰k∗ht−1+bk), (12)st=ς(𝒲s∗xt+𝒰s∗ht−1+bs),where κ and ς are functions that extract kurtosis and skewness, respectively. The cell state update now includes these higher-order terms: (13)ct=ft⊙ct−1+it⊙gt+kt⊙(ct−1⊛ct−1)+st⊙(ct−1⊛ct−1⊛ct−1),where ⊛ denotes the element-wise tensor power. An advanced attention mechanism is incorporated using information-theoretic measures. The attention weights are defined on the basis of the mutual information between hidden states and outputs: (14)αt=exp⁡(I(ht;yt))∑s=1Texp⁡(I(hs;yt)),where I(ht;yt) is the mutual information between ht and the output yt. This attention mechanism ensures that the most informative hidden states are emphasized in the output computation: (15)yt=ϕ(∑s=1Tαshs),with ϕ being a nonlinear activation function. The optimal control problem is connected to the Hamilton-Jacobi-Bellman (HJB) equation. The value function V(X,t) satisfies: (16)−∂V∂t=minπ{ℒ(X,π,t)+⟨∂V∂X,f(X,π,t)⟩+12Tr⁡(GG⊤∂2V∂X2)},where ⟨⋅,⋅⟩ denotes the inner product, and Tr is the trace operator. The LSTM network aims to approximate the policy π that minimizes the right-hand side of the HJB equation. To train the LSTM network in this complex setting, we employ Stochastic Gradient Hamiltonian Monte Carlo (SGHMC), which combines stochastic gradient descent with Hamiltonian dynamics. The parameter updates are given by: (17)dθ=−∇θ𝒮[θ]dt+2D(θ)dBt,where D(θ) is a diffusion matrix capturing the curvature of the loss landscape, and Bt is a Brownian motion.

This section presents a novel mathematical framework for training and optimizing the LSTM model within the ANNDRA-IoT architecture. Let 𝒟={(xt,yt)}t=1T denote the training dataset derived from the preprocessed TON_IoT data, where xt∈Rp represents the input features at time t, and yt∈Rq denotes the corresponding target output for resource allocation decisions. The primary objective is to find the optimal network parameters θ∗ that minimize a composite loss function 𝒥(θ), including the empirical risk and regularization terms: (18)θ∗=arg⁡minθ∈Θ𝒥(θ)=arg⁡minθ∈Θ[ℒ(θ)+λℛ(θ)],where Θ is the parameter space, ℒ(θ) is the empirical loss function defined over the training data, ℛ(θ) is a regularization functional, and λ>0 is a regularization coefficient controlling the trade-off between data fitting and model complexity. The empirical loss function is defined using a time-averaged functional that accounts for temporal dependencies. (19)ℒ(θ)=1T∑t=1Tℓ(yt,y^t(θ))+β2∫0T‖ddty^t(θ)‖2dt,where ℓ(⋅,⋅) is a loss function that measures the discrepancy between the true output yt and the predicted output y^t(θ), ‖⋅‖ denotes the Euclidean norm, and β>0 is a smoothing parameter that enforces the temporal smoothness through the Sobolev norm of the predictions. The regularization functional ℛ(θ) is defined using concepts from reproducing kernel Hilbert spaces (RKHS) to promote smoothness and generalization: (20)ℛ(θ)=‖θ‖ℋ2=∑i‖Wi‖ℋ2+‖Ui‖ℋ2+‖bi‖ℋ2,where ‖⋅‖ℋ denotes the RKHS norm, and Wi, Ui, bi are the weight matrices and bias vectors associated with the LSTM gates. This regularization encourages the parameters to lie in a smooth function space, enhancing the model’s ability to generalize to unseen data. To solve the optimization problem, we employ stochastic gradient descent (SGD) enhanced with adaptive moment estimation and preconditioning matrices derived from second-order information: (21)θk+1=θk−ηkPk−1∇θ𝒥(θk),where ηk is the learning rate in iteration k, ∇θ𝒥(θk) is the loss function gradient, and Pk is a preconditioning matrix calculated as: (22)Pk=ϵI+(v^k)1/2,with ϵ being a small constant to ensure numerical stability, I the identity matrix, and v^k the exponentially weighted moving average of the squared gradients: (23)v^k=γv^k−1+(1−γ)(∇θ𝒥(θk)⊙∇θ𝒥(θk)),where γ∈[0,1) is the decay rate and ⊙ denotes element-wise multiplication. This makes the learning rates of each parameter adaptive, which gives a better convergence behavior. Yet another variant of this further incorporates information geometry in order to use the natural gradient that takes into account the Riemannian structure of the parameter space: (24)θk+1=θk−ηkF−1(θk)∇θ𝒥(θk),where F(θk) is the Fisher information matrix in iteration k, defined as: (25)F(θ)=Ext[∇θlog⁡pθ(yt|xt)∇θlog⁡pθ(yt|xt)⊤],with pθ(yt|xt) being the conditional probability of the output given the input under the model parameterized by θ. The natural gradient descent adapts to the curvature of the parameter space, potentially leading to faster convergence. To capture the uncertainty of the model and improve generalization, we adopt a Bayesian learning framework using variational inference. We define a variational posterior q(θ) over the model parameters and minimize the variational free energy ℱ(q): (26)ℱ(q)=∫q(θ)log⁡q(θ)p(𝒟,θ)dθ=KL(q(θ)||p(θ))−Eq(θ)[log⁡p(𝒟|θ)],where p(θ) is the prior distribution over parameters, p(𝒟,θ) is the joint probability of data and parameters, and KL denotes the Kullback-Leibler divergence. The optimization seeks a variational distribution q(θ) that approximates the true posterior, balancing the fit and complexity of the model. This study also introduces entropy regularization to encourage exploration during training. (27)𝒥entropy(θ)=𝒥(θ)−μH[q(θ)],where H[q(θ)] is the differential entropy of the variational posterior: (28)H[q(θ)]=−∫q(θ)log⁡q(θ)dθ,and μ>0 is a coefficient that controls the influence of regularization of the entropy. To analyze the convergence properties of the optimization algorithm, the proposed approach utilizes convex analysis and establishes conditions under which the loss function 𝒥(θ) is convex or satisfies the inequality: (29)12‖∇θ𝒥(θ)‖2≥μPL(𝒥(θ)−𝒥(θ∗)),where μPL>0 is the PL constant. Under this condition, gradient descent methods exhibit linear convergence rates. In addition to standard L2 regularization, we incorporate Total Variation (TV) regularization to penalize rapid changes in the parameter space: (30)ℛTV(θ)=∫‖∇θ(s)‖ds,where θ(s) denotes the parameters as a function over a continuous index s, and ∇θ(s) is the gradient with respect to s. This approach further integrates adaptive learning rates based on the spectral properties of the Hessian matrix H(θ) of the loss function: (31)ηk=2λmax(H(θk))+λmin(H(θk)),where λmax and λmin are the largest and smallest eigenvalues of H(θk), respectively. This choice of learning rate ensures stable convergence by taking into account the curvature of the loss landscape. The advanced training algorithm integrates the concepts mentioned above and is outlined as Algorithm 1.

We incorporate constraints into the optimization problem to ensure feasibility with respect to resource limitations: (32)Find θ∗ such that θ∗=arg⁡minθ∈Θ𝒥(θ),s.t.𝒞(θ)≤Cmax,where 𝒞(θ) is a cost function representing computational or energy resources consumed by the model, and Cmax is the maximum allowable cost. This constrained optimization can be addressed using Lagrangian multipliers or penalty methods. To ensure the global convergence of the training algorithm, we employ Lyapunov stability analysis. We define a Lyapunov function V(θk)=𝒥(θk)−𝒥(θ∗) and show that: (33)V(θk+1)−V(θk)≤−αV(θk),for some α>0, implying exponential convergence to the global minimum.

This framework incorporated decentralized optimization, variational inference, and nonlinear dynamical systems to address the challenges posed by the heterogeneous and dynamic nature of IoT environments.

The development of the integrative ANNDRA-IoT model in existing and future IoT systems is the last significant step towards a deployed model with full capabilities, as shown in Algorithm 2.

IoTSim-Edge simulator emulates the complex IoT ecosystem within the simulation environment. The basis of our training and testing stages comprised data from the TON_IoT Dataset, obtained from UNSW Research. The realization also utilizes IoTSim-Edge’s API for integrating the model within the simulated IoT environment.

This section discusses the key performance metrics to be used to assess the efficacy of the ANNDRA-IoT model in optimal resource allocation within IoT environments. This indicates that 100 training epochs of the ANNDRA-IoT model accuracy metric are tracked. In a comparative analysis, the internal accuracy of the ANNDRA-IoT model over time was benchmarked compared to four important models in this field, namely DEELB, AWPR-FOG, and OSCAR. At epoch 100, the ANNDRA-IoT model outperformed the competing models with a training accuracy of 0.976 and a validation accuracy of 0.968 as illustrated in Table 3. In contrast, the DEELB model reported a training accuracy of 0.925 and a validation accuracy of 0.917, while the OSCAR model achieved a training accuracy of 0.902 and a validation accuracy of 0.895. The CNN-MBO model presented a training accuracy of 0.940 and a validation accuracy of 0.932, and the AWPR-FOG model showed a training accuracy of 0.950 and a validation accuracy of 0.942.

The precision and recall of the ANNDRA-IoT model were observed for the 100 training epochs. The precision and recall of ANNDRA-IoT were compared with other algorithms. It can be seen in Table 4 that upon completion of the training, the ANNDRA-IoT model exhibited a training precision value of 0.97 and a recall value of 0.96. The precision and recall of the ANNDRA-IoT model in the classification states were recorded as the model was trained over 100 training epochs to assess its classification efficacy. At the end of the training, the ANNDRA-IoT model achieved training and validation precision of 0.97 and 0.96, and recall of 0.96 and 0.96, surpassing the corresponding metrics of the algorithms compared as represented in Table 4. The DEELB model was trained to achieve a precision of 0.89 and 0.88 and a recall of 0.87 and 0.86, for training and validation, respectively. OSCAR was trained and validated to reach a precision of 0.85 and 0.84 and a recall of 0.83 and 0.82, respectively. The CNN-MBO approach achieved a training precision of 0.91 and a validation precision of 0.90, a training recall of 0.90 and validation recall of 0.89. Finally, the AWPR-FOG model needed to achieve training precision of 0.93 and recall of 0.92, with corresponding validation measurements of 0.92 and 0.91.

For benchmarking the performance of the ANNDRA-IoT model, the F1 score’s performance was compared to that of DEELB, OSCAR, CNN-MBO, and AWPR-FOG, where this comparison was done against the F1 score at the last epoch, which is 100. Herein, the ANNDRA-IoT model has given the highest performance with 0.965 of the training F1 score and 0.955 of the validation F1 score.

In this comparison, the DEELB model reached 0.88 for a training F1 score and 0.87 for a validation F1 score according to [3]. Besides, OSCAR, which was proposed in [18], reported 0.84 for a training F1 score and 0.83 for a validation F1 score, while the CNN-MBO approach of [30] had 0.91 for a training F1 score and 0.90 for a validation F1 score. The last model, AWPR-FOG, proposed in [31] reached 0.93 F1 score during training, and its F1 score when validated was 0.92.

In a comprehensive way of assessing the effectiveness of ANNDRA-IoT as a framework to optimize resource allocation in IoT environments, a set of custom metrics is used, including efficiency, response time, and overall system performance.