The Theory of Connections: Connecting Points

Mortari, Daniele

doi:10.3390/math5040057

Cited by 79 publications

(130 citation statements)

References 6 publications

(12 reference statements)

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“…(7) Extension to nonlinear DEs. (8) Extension to partial DEs. (9) Extension to nonlinear constraints.…”

Section: Discussionmentioning

confidence: 99%

“…The linear constraints considered in the theory of connections [8] are any linear combination of the function and/or its derivatives evaluated at specified values of the independent variable, t. For an assigned set of n independent functions, h k (t), the n constraints of the DE allow us to derive the unknown vector of coefficients, η. By substituting this expression into Equation (3), we obtain an equation satisfying all the constraints, no matter what g(t) is.…”

Section: Introductionmentioning

confidence: 99%

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Least-Squares Solution of Linear Differential Equations

Mortari

2017

Mathematics

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This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g(t), and they satisfy the constraints, no matter what g(t) is. The second step consists of expressing g(t) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x ∈ [−1, +1]. The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions' accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.

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“…(7) Extension to nonlinear DEs. (8) Extension to partial DEs. (9) Extension to nonlinear constraints.…”

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Least-Squares Solution of Linear Differential Equations

Mortari

2017

Mathematics

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“…It was found that this still solved the differential equation with an accuracy on the order of O(10 −7 ). This can be explained by an equation first presented in the seminal paper on TFC [29]. In this paper an equation (Eq.…”

Section: Discussionmentioning

confidence: 99%

Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections

2020

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This paper shows how to obtain highly accurate solutions of eighth-order boundary-value problems of linear and nonlinear ordinary differential equations. The presented method is based on the Theory of Functional Connections, and is solved in two steps. First, the Theory of Functional Connections analytically embeds the differential equation constraints into a candidate function (called a constrained expression) that contains a function that the user is free to choose. This expression always satisfies the constraints, no matter what the free function is. Second, the free-function is expanded as a linear combination of orthogonal basis functions with unknown coefficients. The constrained expression (and its derivatives) are then substituted into the eighth-order differential equation, transforming the problem into an unconstrained optimization problem where the coefficients in the linear combination of orthogonal basis functions are the optimization parameters. These parameters are then found by linear/nonlinear least-squares. The solution obtained from this method is a highly accurate analytical approximation of the true solution. Comparisons with alternative methods appearing in literature validate the proposed approach.

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“…The Theory of Functional Connections (TFC) is a new technique that analytically derives a constrained expression which satisfies the problem's constraints exactly while maintaining a function that can be freely chosen [7]. This theory, initially called "Theory of Connections", has been renamed for two reasons.…”

Section: Introductionmentioning

confidence: 99%

“…This approach discretizes the domain about collocation points, and the solution of the DE is expressed by a sum of "basis" functions with unknown coefficients that are approximated in order to satisfy the DE as closely as possible. Yet, in order to incorporate boundary conditions, one or more equations must be added to enforce the constraints.The Theory of Functional Connections (TFC) is a new technique that analytically derives a constrained expression which satisfies the problem's constraints exactly while maintaining a function that can be freely chosen [7]. This theory, initially called "Theory of Connections", has been renamed for two reasons.…”

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confidence: 99%

Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines Using the Theory of Functional Connections

Leake

Johnston

Smith

et al. 2019

MAKE

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Differential equations (DEs) are used as numerical models to describe physical phenomena throughout the field of engineering and science, including heat and fluid flow, structural bending, and systems dynamics. While there are many other techniques for finding approximate solutions to these equations, this paper looks to compare the application of the Theory of Functional Connections (TFC) with one based on least-squares support vector machines (LS-SVM). The TFC method uses a constrained expression, an expression that always satisfies the DE constraints, which transforms the process of solving a DE into solving an unconstrained optimization problem that is ultimately solved via least-squares (LS). In addition to individual analysis, the two methods are merged into a new methodology, called constrained SVMs (CSVM), by incorporating the LS-SVM method into the TFC framework to solve unconstrained problems. Numerical tests are conducted on four sample problems: One first order linear ordinary differential equation (ODE), one first order nonlinear ODE, one second order linear ODE, and one two-dimensional linear partial differential equation (PDE). Using the LS-SVM method as a benchmark, a speed comparison is made for all the problems by timing the training period, and an accuracy comparison is made using the maximum error and mean squared error on the training and test sets. In general, TFC is shown to be slightly faster (by an order of magnitude or less) and more accurate (by multiple orders of magnitude) than the LS-SVM and CSVM approaches. IntroductionDifferential equations (DE) and their solutions are important topics in science and engineering. The solutions drive the design of predictive systems models and optimization tools. Currently, these equations are solved by a variety of existing approaches with the most popular based on the Runge-Kutta family [1]. Other methods include those which leverage low-order Taylor expansions, namely Gauss-Jackson [2] and Chebyshev-Picard iteration [3,4,5], which have proven to be highly effective. More recently developed techniques are based on spectral collocation methods [6]. This approach discretizes the domain about collocation points, and the solution of the DE is expressed by a sum of "basis" functions with unknown coefficients that are approximated in order to satisfy the DE as closely as possible. Yet, in order to incorporate boundary conditions, one or more equations must be added to enforce the constraints.The Theory of Functional Connections (TFC) is a new technique that analytically derives a constrained expression which satisfies the problem's constraints exactly while maintaining a function that can be freely chosen [7]. This theory, initially called "Theory of Connections", has been renamed for two reasons. First, the "Theory of Connections" already identifies a specific theory in differential geometry, and second, what this theory is actually doing is "functional interpolation", as it provides all functions satisfying a set of constraints in terms of a function ...

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The Theory of Connections: Connecting Points

Cited by 79 publications

References 6 publications

Least-Squares Solution of Linear Differential Equations

Least-Squares Solution of Linear Differential Equations

Least-Squares Solutions of Eighth-Order Boundary Value Problems Using the Theory of Functional Connections

Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines Using the Theory of Functional Connections

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