The Multivariate Theory of Connections

Mortari, Daniele; Leake, Carl

doi:10.3390/math7030296

Cited by 42 publications

(57 citation statements)

References 6 publications

(17 reference statements)

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“…The previous example derived the constrained expression by creating and solving a series of simultaneous algebraic equations. This technique works well for constrained expressions in one dimension; however, it can become needlessly complicated when deriving these expression in n dimensions for constraints on the value and arbitrary order derivative of n − 1 dimensional manifolds [9]. Fortunately, a different, more mechanized formalism exists that is useful for this case.…”

Section: N-dimensional Constrained Expressionsmentioning

confidence: 99%

“…A mathematical proof that this form of the constrained expression satisfies the boundary constraints is given in Ref. [9]. The remainder of this section discusses how to construct the n-th order tensor M and the v vectors shown in Equation (5).…”

Section: N-dimensional Constrained Expressionsmentioning

confidence: 99%

“…[5,6]. Luckily, a framework that can embed higher-dimensional or more complex constraints has already been invented: The Theory of Functional Connections (TFC) [8,9]. In Ref.…”

Section: Introductionmentioning

confidence: 99%

“…TFC has already been used to solve ordinary differential equations with initial value constraints, boundary value constraints, relative constraints, integral constraints, and linear combinations of constraints [8,[11][12][13]. Recently, the framework was extended to n-dimensions [9] for constraints on the value and arbitrary order derivative of (n − 1)-dimensional manifolds. This means the TFC framework can now generate constrained expressions that satisfy the boundary constraints of multidimensional PDEs [14].…”

Section: Introductionmentioning

confidence: 99%

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Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations

Leake

Mortari

2020

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This article presents a new methodology called Deep Theory of Functional Connections (TFC) that estimates the solutions of partial differential equations (PDEs) by combining neural networks with the TFC. The TFC is used to transform PDEs into unconstrained optimization problems by analytically embedding the PDE’s constraints into a “constrained expression” containing a free function. In this research, the free function is chosen to be a neural network, which is used to solve the now unconstrained optimization problem. This optimization problem consists of minimizing a loss function that is chosen to be the square of the residuals of the PDE. The neural network is trained in an unsupervised manner to minimize this loss function. This methodology has two major differences when compared with popular methods used to estimate the solutions of PDEs. First, this methodology does not need to discretize the domain into a grid, rather, this methodology can randomly sample points from the domain during the training phase. Second, after training, this methodology produces an accurate analytical approximation of the solution throughout the entire training domain. Because the methodology produces an analytical solution, it is straightforward to obtain the solution at any point within the domain and to perform further manipulation if needed, such as differentiation. In contrast, other popular methods require extra numerical techniques if the estimated solution is desired at points that do not lie on the discretized grid, or if further manipulation to the estimated solution must be performed.

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Section: N-dimensional Constrained Expressionsmentioning

confidence: 99%

Section: N-dimensional Constrained Expressionsmentioning

confidence: 99%

“…[5,6]. Luckily, a framework that can embed higher-dimensional or more complex constraints has already been invented: The Theory of Functional Connections (TFC) [8,9]. In Ref.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations

Leake

Mortari

2020

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“…ij ∂x 2 ). Using the Multivariate TFC[23], the constrained expression for this problem can be written as,z(x, y) = Aijvivj + wjϕj(x, y) − w k B ijk vivj Aij =   0 c1(x, 0) c3(x, 1) c2(0, y) −c1(0, 0) −c3(0, 1) c4(1, y) −c1(1, 0) −c3(1, 1) k (x, 0) ϕ k (x, 1) ϕ k (0, y) −ϕ k (0, 0) −ϕ k (0, 1) ϕ k (1, y) −ϕ k (1, 0) −ϕ k (1, 1)   vi = 1 1 − x x vj = 1 1 − y y .whereẑ will satisfy the boundary constraints c k (x, y) regardless of the choice of w and ϕ k . Now, the Lagrange multiplies are added in to form L, L(w, α, e) = 1 2 wiwi + γ 2 eiei − αI (ẑ xx I +ẑ yy I − fI − eI ), whereẑI is the vector composed of the elementsẑ(xn, yn) where n = 1, ..., Np and there are Np training points.…”

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

Leake

Johnston

Smith

et al. 2019

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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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