Nonlinear Reaction-Diffusion-Convection Equations

Cherniha, Roman; Serov, Mykola; Pliukhin, Oleksii

doi:10.1201/9781315154848

Cited by 40 publications

(11 citation statements)

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“…An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator [52]. In the book of "Nonlinear Reaction-Diffusion-Convection Equations" [52], an important example of integral transform is shown, which is Fourier analysis.…”

Section: E Recurrent Neural Networkmentioning

confidence: 99%

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Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems

Guo¹,

Huang²,

Xu³

2021

Preprint

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In this work, we generalize the reaction-diffusion equation in statistical physics, Schrödinger equation in quantum mechanics, Helmholtz equation in paraxial optics into the neural partial differential equations (NPDE), which can be considered as the fundamental equations in the field of artificial intelligence research. We take finite difference method to discretize NPDE for finding numerical solution, and the basic building blocks of deep neural network architecture, including multi-layer perceptron, convolutional neural network and recurrent neural networks, are generated. The learning strategies, such as Adaptive moment estimation, L-BFGS, pseudoinverse learning algorithms and partial differential equation constrained optimization, are also presented. We believe it is of significance that presented clear physical image of interpretable deep neural networks, which makes it be possible for applying to analog computing device design, and pave the road to physical artificial intelligence.

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Section: E Recurrent Neural Networkmentioning

confidence: 99%

“…This corresponds to diagonalizing an operator [52]. In the book of "Nonlinear Reaction-Diffusion-Convection Equations" [52], an important example of integral transform is shown, which is Fourier analysis. Fourier analysis diagonalizes the heat equation using the eigenbasis of sinusoidal waves.…”

Section: E Recurrent Neural Networkmentioning

confidence: 99%

Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems

Guo¹,

Huang²,

Xu³

2021

Preprint

View full text Add to dashboard Cite

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“…We use them in Section 7 to construct wider families of solutions to (1) than the related family of Lie solutions. Regular reduction operators of reduced equation 1.5 with δ = 1 were described in [15]. Using Lie and nonclassical symmetries, one can construct the following inequivalent solutions for this equation:…”

Section: Lie Reductions Of Codimension Onementioning

confidence: 99%

“…See Section B for known and new exact solutions of equations 1.6 ′ with δ = 0. Lie symmetries, regular reduction operators and exact solutions of classes of diffusion-convection-reaction equations including equations 1.4 ′ and 1.6 ′ were considered in [15]; see also references therein.…”

Section: Lie Reductions Of Codimension Onementioning

confidence: 99%

Extended symmetry analysis of two-dimensional degenerate Burgers equation

Vaneeva¹,

Popovych²,

Sophocleous³

2019

Preprint

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We carry out enhanced symmetry analysis of a two-dimensional degenerate Burgers equation. Its complete point-symmetry group is found using the algebraic method, and its generalized symmetries are proved to be exhausted by its Lie symmetries. We also prove that the space of conservation laws of this equation is infinite-dimensional and is naturally isomorphic to the solution space of the (1+1)-dimensional backward linear heat equation. Lie reductions of the two-dimensional degenerate Burgers equation are comprehensively studied in the optimal way and its new Lie invariant solutions are constructed. We additionally consider solutions that also satisfy an analogous nondegenerate Burgers equation. In total, we construct four families of solutions of two-dimensional degenerate Burgers equation that are expressed in terms of solutions of the (1+1)-dimensional linear heat equation. Various kinds of hidden symmetries and hidden conservation laws (local and potential ones) are discussed as well. As by-product, we exhaustively describe generalized symmetries, cosymmetries and conservation laws of the transport equation, also called the inviscid Burgers equation, and construct new invariant solutions of the nonlinear diffusion equation with power nonlinearity of degree −1/2.

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“…A complete list of these models are given for a finite-dimensional equivalence algebra derived for HESI equation. To obtain these goals we perform algorithms that is explained in references [20,3,12,5], and we use similar works in [13,10,15,16,17,9,22].…”

Section: Introductionmentioning

confidence: 99%

Preliminary Group Classification and some exact solutions of 2Hessian equation

Yourdkhany¹,

Nadjafikhah²,

Toomanian³

2019

Preprint

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We study the class of 3-dimensional nonlinear 2−hessian equations, where f is an arbitrary smooth function of the variables (x, y, z). We perform preliminary group classification on 2−hessian equation. In fact, we find additional equivalence transformation on the space (x, y, z, u, f ), with the aid of N. Bila's method, then we take their projections on the space (x, y, z, f ), so we prove an optimal system of one-dimensional Lie subalgebras of this equation is generated by A 1 , • • • , A 12 , which introduced in theorem (2), ultimately, A number of new interesting nonlinear invariant models are obtained which have non-trivial invariance algebras. The result of these works is a wide class of equations which summarized in table. So at the end of this work, some exact solutions of 2−hessian equation are presented. The paper is one of the few applications of an algebraic approach to the group classification using Lie method.

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Nonlinear Reaction-Diffusion-Convection Equations

Cited by 40 publications

References 0 publications

Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems

Partial Differential Equations is All You Need for Generating Neural Architectures -- A Theory for Physical Artificial Intelligence Systems

Extended symmetry analysis of two-dimensional degenerate Burgers equation

Preliminary Group Classification and some exact solutions of 2Hessian equation

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