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

• 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
• DOI: 10.1109/TNNLS.2013.2277601

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.