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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| Veröffentlicht in: | IEEE transaction on neural networks and learning systems Jg. 25; H. 3; S. 571 - 584 |
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| Hauptverfasser: | , , |
| Format: | Journal Article |
| Sprache: | Englisch |
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New York, NY
IEEE
01.03.2014
Institute of Electrical and Electronics Engineers The Institute of Electrical and Electronics Engineers, Inc. (IEEE) |
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| ISSN: | 2162-237X, 2162-2388, 2162-2388 |
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| Abstract | 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. |
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| AbstractList | 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.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. 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. |
| Author | Rudd, Keith Ferrari, Silvia Muro, Gianluca Di |
| Author_xml | – sequence: 1 givenname: Keith surname: Rudd fullname: Rudd, Keith email: keith.rudd@duke.edu organization: Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA – sequence: 2 givenname: Gianluca Di surname: Muro fullname: Muro, Gianluca Di organization: Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA – sequence: 3 givenname: Silvia surname: Ferrari fullname: Ferrari, Silvia email: sferrari@duke.edu organization: Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA |
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| SubjectTerms | 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 |
| Title | A Constrained Backpropagation Approach for the Adaptive Solution of Partial Differential Equations |
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