Adaptive solution of differential equations for the regularization of artificial neural networks

The authors propose using artificial neural networks to model the solution of the two-dimensional Euler equation which models inviscid and compressible fluid flow. The code thus created should not only have the same accuracy as a more conventional computer code, but should still retain the ability of an ANN to modify itself when exposed to experimental data, thus yielding software that could be specialized with experimental results. To accomplish these objectives, the method of optimal incremental function approximation has been developed for the adaptive solution of differential equations using ANN architecture. Two major attractive features of this approach are that: (1) the developed method is flexible enough to use any of the popular transfer functions and (2) the developed method requires minimal user interaction. The latter is especially advantageous when dealing with complicated physical or computational domains. Numerical results are presented and compared to conventional methods. The accuracy is considered satisfactory.

[1]  Andrew J. Meade,et al.  Integrating experimental data and mathematical models in simulation of physical systems , 1997 .

[2]  Michael Kokkolaras,et al.  Sequential function approximation for the solution of differential equations , 1997 .

[3]  F. Girosi,et al.  Networks for approximation and learning , 1990, Proc. IEEE.

[4]  J. Dennis,et al.  Direct Search Methods on Parallel Machines , 1991 .

[5]  L. Jones.  A Simple Lemma on Greedy Approximation in Hilbert Space and Convergence Rates for Projection Pursuit Regression and Neural Network Training , 1992 .

[6]  Andrew J. Meade,et al.  Approximation properties of local bases assembled from neural network transfer functions , 1998 .

[7]  Andrew R. Barron,et al.  Universal approximation bounds for superpositions of a sigmoidal function , 1993, IEEE Trans. Inf. Theory.

[8]  L. Jones.  Constructive approximations for neural networks by sigmoidal functions , 1990, Proc. IEEE.

[9]  D. Greenspan.  On approximating extremals of functionals—II theory and generalizations related to boundary value problems for nonlinear differential equations , 1967 .

[10]  On the Flow of a Compressible Fluid past a Circular Cylinder , 1938 .

[11]  Christopher M. Bishop,et al.  Curvature-driven smoothing: a learning algorithm for feedforward networks , 1993, IEEE Trans. Neural Networks.

[12]  Leszek Demkowicz,et al.  On a mesh optimization method based on a minimization of interpolation error , 1986 .

[13]  J. Tinsley Oden,et al.  A GENERAL THEORY OF FINITE ELEMENTS II. APPLICATIONS , 1969 .

[14]  Noboru Kikuchi,et al.  A method of grid optimization for finite element methods , 1983 .

[15]  Ernest O. Doebelin,et al.  Engineering Experimentation: Planning, Execution, Reporting , 1995 .

[16]  T. Cherry,et al.  THE VARIATIONAL METHOD IN HYDRODYNAMICS , 1956 .

[17]  Tomaso A. Poggio,et al.  Regularization Theory and Neural Networks Architectures , 1995, Neural Computation.

[18]  D. Greenspan,et al.  Application of a method for approximating extremals of functionals to compressible subsonic fluid flows , 1967 .

[19]  R. Franke.  Scattered data interpolation: tests of some methods , 1982 .