2020
DOI: 10.3390/make2010004
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Deep Theory of Functional Connections: A New Method for Estimating the Solutions of Partial Differential Equations

Abstract: 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 … Show more

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Cited by 64 publications
(61 citation statements)
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“…where g(z) is a free complex function (where the g(z i ) must be defined), can be seen as the most general expression that defines the conformal mapping between the control points in Z and W. In fact, Equation 5represents the TFC functional generalization of Equation 3. Figure 2 shows one effect of the free function in the complex mapping defined by the control points from Equation (4). The free function modifies the domain while the conformal property is still preserved.…”
Section: Complex Tfc Mappingmentioning
confidence: 99%
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“…where g(z) is a free complex function (where the g(z i ) must be defined), can be seen as the most general expression that defines the conformal mapping between the control points in Z and W. In fact, Equation 5represents the TFC functional generalization of Equation 3. Figure 2 shows one effect of the free function in the complex mapping defined by the control points from Equation (4). The free function modifies the domain while the conformal property is still preserved.…”
Section: Complex Tfc Mappingmentioning
confidence: 99%
“…Figure 10 shows the accuracy results for the proposed approximate inverse mapping using the least-squares estimate. In this example, the same control points defined in Equation (4) have been used with N = 121 grid points in the (−1, +1) × (−1, +1) Z-domain. The left figure shows the z i grid of points as little black dots.…”
Section: Approximate Least-squares Inverse Mappingmentioning
confidence: 99%
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