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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Bibliographic Details
Published in:IEEE transaction on neural networks and learning systems Vol. 25; no. 3; pp. 571 - 584
Main Authors: Rudd, Keith, Muro, Gianluca Di, Ferrari, Silvia
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.03.2014
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
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
Online Access:Get full text
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Summary: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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ISSN:2162-237X
2162-2388
2162-2388
DOI:10.1109/TNNLS.2013.2277601