Neural network approach for solving nonlocal boundary value problems

Palade, Vasile; Petrov, Miroslav S.; Todorov, Todor D.

doi:10.1007/s00521-020-04810-0

Cited by 7 publications

(5 citation statements)

References 41 publications

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“…Our objective is to generalize this formula by including Dirichlet vertices. This is important because for example both Dirichlet and Neumann conditions appear in the isoscattering and neural networks, where for the latter ones they appear naturally as a result of learning procedures 59 .…”

Section: The Generalized Euler Characteristic For Quantum Graphs With Dirichlet Boundary Conditionsmentioning

confidence: 99%

A new spectral invariant for quantum graphs

Ławniczak

Kurasov

Bauch

et al. 2021

Sci Rep

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The Euler characteristic i.e., the difference between the number of vertices |V| and edges |E| is the most important topological characteristic of a graph. However, to describe spectral properties of differential equations with mixed Dirichlet and Neumann vertex conditions it is necessary to introduce a new spectral invariant, the generalized Euler characteristic $$\chi _G:= |V|-|V_D|-|E|$$ χ G : = | V | - | V D | - | E | , with $$|V_D|$$ | V D | denoting the number of Dirichlet vertices. We demonstrate theoretically and experimentally that the generalized Euler characteristic $$\chi _G$$ χ G of quantum graphs and microwave networks can be determined from small sets of lowest eigenfrequencies. If the topology of the graph is known, the generalized Euler characteristic $$\chi _G$$ χ G can be used to determine the number of Dirichlet vertices. That makes the generalized Euler characteristic $$\chi _G$$ χ G a new powerful tool for studying of physical systems modeled by differential equations on metric graphs including isoscattering and neural networks where both Neumann and Dirichlet boundary conditions occur.

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Section: The Generalized Euler Characteristic For Quantum Graphs With Dirichlet Boundary Conditionsmentioning

confidence: 99%

A new spectral invariant for quantum graphs

Ławniczak

Kurasov

Bauch

et al. 2021

Sci Rep

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show abstract

“…The majority of the results in the literature involve the development of numerical methods, for which exact solutions for simple cases are used to benchmark the accuracy of numerical solutions [25,27,[30][31][32][33][34][35]. Recent numerical methods also involve the use of neural networks [36].…”

Section: Introductionmentioning

confidence: 99%

Analytical approach to solving linear diffusion–advection–reaction equations with local and nonlocal boundary conditions

Rodrigo,

Thamwattana

2024

Math Methods in App Sciences

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Initial–boundary value problems for a linear diffusion–advection–reaction equation are considered, with general nonhomogeneous linear boundary conditions and general linear nonlocal boundary conditions. Analytical solutions are obtained using an embedding method. The solutions are expressed in terms of time‐varying functions that satisfy coupled linear Volterra integral equations of the first kind. A boundary element method is applied to numerically solve the integral equations. Three examples are given to demonstrate the accuracy of the numerical solutions when compared with the analytical solutions. The embedding method is applicable to problems with bounded and unbounded spatial domains.

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“…Recent research has shown that ML algorithms are more advantageous than traditional statistical methods when constructing predictive models. Artificial neural networks are capable of self-learning, adaptation, fault-tolerance, nonlinearity, and efficient mapping of inputs to outputs [ 25 , 26 ], and have been used effectively to differentiate between mild cognitive impairment and Alzheimer's disease, to develop predictive models for lung cancer diagnosis, and for risk prediction of cardiovascular disease [ [27] , [28] , [29] ]. Decision trees are useful for determining groupings, recognizing connections between groups, and forecasting upcoming occurrences [ 30 ].…”

Section: Introductionmentioning

confidence: 99%