A Constrained Backpropagation Approach for the Adaptive Solution of Partial Differential Equations

This paper presents a constrained backpropagation (CPROP) methodology for solving nonlinear elliptic and parabolic partial differential equations (PDEs) adaptively, subject to changes in the PDE parameters or external forcing. Unlike existing methods based on penalty functions or Lagrange multiplier...

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Vydáno v:	IEEE transaction on neural networks and learning systems Ročník 25; číslo 3; s. 571 - 584
Hlavní autoři:	Rudd, Keith, Muro, Gianluca Di, Ferrari, Silvia
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York, NY IEEE 01.03.2014 Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:	Adaptive algorithm Applied sciences Approximation Artificial intelligence Artificial neural networks artificial neural networks (ANNs) Back propagation Boundaries Computer science; control theory; systems Connectionism. Neural networks Constraints Equations Exact sciences and technology Jacobian matrices Linear programming Mathematical analysis Mathematics Neural networks Nonlinearity Optimization Partial differential equations partial differential equations (PDEs) Sciences and techniques of general use scientific computing Training Backpropagation Dimensionality artificial neural networks (ANNs) Adaptive algorithm Initial condition scientific computing Elimination Constraint satisfaction Neural network Boundary condition Adaptive method Partial differential equation Penalty method Constrained optimization Backpropagation algorithm Parabolic equation Lagrange multiplier Elliptic equation Scientific computation partial differential equations (PDEs) Non linear effect Mathematical programming
ISSN:	2162-237X, 2162-2388, 2162-2388
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Popis
Shrnutí:	This paper presents a constrained backpropagation (CPROP) methodology for solving nonlinear elliptic and parabolic partial differential equations (PDEs) adaptively, subject to changes in the PDE parameters or external forcing. Unlike existing methods based on penalty functions or Lagrange multipliers, CPROP solves the constrained optimization problem associated with training a neural network to approximate the PDE solution by means of direct elimination. As a result, CPROP reduces the dimensionality of the optimization problem, while satisfying the equality constraints associated with the boundary and initial conditions exactly, at every iteration of the algorithm. The effectiveness of this method is demonstrated through several examples, including nonlinear elliptic and parabolic PDEs with changing parameters and nonhomogeneous terms.
Bibliografie:	ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 14 content type line 23 ObjectType-Article-1 ObjectType-Feature-2
ISSN:	2162-237X 2162-2388 2162-2388
DOI:	10.1109/TNNLS.2013.2277601