A new iterative technique for solving fractal-fractional differential equations based on artificial neural network in the new generalized Caputo sense

Shloof, A. M.; Senu, Norazak; Ahmadian, Ali; Pakdaman, Morteza; Salahshour, Soheil

doi:10.1007/s00366-022-01607-8

Cited by 9 publications

(6 citation statements)

References 28 publications

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“…The weighted residual method is a classical method with the idea of introducing a weighting function for the residuals that forces the residuals to be 0 in some average sense. Based on this idea, we add the weighting functions to equation (19). Specifically, we weight equations (10), ( 11), (12), which are rewritten as follows…”

Section: Lossmentioning

confidence: 99%

“…Xiong et al [18] introduced the gradient-related weight function and sequence-to-sequence learning into PINN to solve four cases of Riemann problems. Recently, neural network-based methods have been used to solve fractional differential equations [19,20]. However, the classical fractional derivatives do not satisfy the chain rule and usually need to be approximated by numerical discretization or truncation, which is at odds with the original intention of the PINN method to use automatic differentiation to calculate the derivative.…”

Section: Introductionmentioning

confidence: 99%

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Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Liu

Fan

et al. 2023

Phys. Scr.

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In this article, we introduce the modiﬁed physics-informed neural network (PINN) method for ﬁnding data-driven solutions of three classes of time-fractional Burgers-type equations under the conformable sense. Since conformable derivative satisﬁes the chain rule, automatic diﬀerentiation can be applied to compute it directly to avoid truncation and other numerical discretization. In addition, the locally adaptive activation function and two effective weighting strategies are introduced to improve solution accuracy. As a result, three numerical examples indicate that the modiﬁed PINN method gives an eﬃcient and reliable solution.

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Section: Lossmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Liu

Fan

et al. 2023

Phys. Scr.

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“…A deep neural network is a type of ML model, and when a deep network is fitted to data, this is referred to as deep learning [31]. Deep learning (DL) has shown very powerful empirical performance for solving very complex real-world problems in areas such as computer vision [32], natural language processing [33,34], speech recognition [35], recommendation systems [36], drug discovery [37], differential equations [38,39], and much more [40][41][42]. In simple words, DL can be seen a neural network [43], composed by many layers, that takes some data set D, input and targets, and learns the rules for forecasting new input data.…”

Section: Introductionmentioning

confidence: 99%

Phase transitions in the mini-batch size for sparse and dense two-layer neural networks

Marino,

Ricci-Tersenghi

2024

Mach. Learn.: Sci. Technol.

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The use of mini-batches of data in training artificial neural networks is nowadays very common. Despite its broad usage, theories explaining quantitatively how large or small the optimal mini-batch size should be are missing. This work presents a systematic attempt at understanding the role of the mini-batch size in training two-layer neural networks.Working in the teacher-student scenario, with a sparse teacher, and focusing on tasks of different complexity, we quantify the effects of changing the mini-batch size $m$.We find that often the generalization performances of the student strongly depend on $m$ and may undergo sharp phase transitions at a critical value $m_c$, such that for $m<m_c$ the training process fails, while for $m>m_c$ the student learns perfectly or generalizes very well the teacher. Phase transitions are induced by collective phenomena firstly discovered in statistical mechanics and later observed in many fields of science. Observing a phase transition by varying the mini-batch size across different architectures raises several questions about the role of this hyperparameter in the neural network learning process.

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“…Furthermore, Reference 49 described a deep autoencoder‐based energy method for analyzing the bending, vibration, and buckling of Kirchhoff plates. Recently, authors in Reference 50 used a pairing of the ANN approach and the generalized PSM (fractional power series) to solve FFDEs.…”

Section: Introductionmentioning

confidence: 99%

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

Shloof

Ahmadian

Senu

et al. 2023

Numerical Meth Engineering

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Artificial neural networks have great potential for learning and stability in the face of tiny input data changes. As a result, artificial intelligence techniques and modeling tools have a growing variety of applications. To estimate a solution for fractal‐fractional differential equations (FFDEs) of high‐order linear (HOL) with variable coefficients, an iterative methodology based on a mix of a power series method and a neural network approach was applied in this study. In the algorithm's equation, an appropriate truncated series of the solution functions was replaced. To tackle the issue, this study uses a series expansion of an unidentified function, where this function is approximated using a neural architecture. Some examples were presented to illustrate the efficiency and usefulness of this technique to prove the concept's applicability. The proposed methodology was found to be very accurate when compared to other available traditional procedures. To determine the approximate solution to FFDEs‐HOL, the suggested technique is simple, highly efficient, and resilient.

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A new iterative technique for solving fractal-fractional differential equations based on artificial neural network in the new generalized Caputo sense

Cited by 9 publications

References 28 publications

Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Phase transitions in the mini-batch size for sparse and dense two-layer neural networks

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

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