paraheat_gaussian_parameters, a keras code which reads data for the heat equation with a parameterized Gaussian diffusivity, and uses a neural network to estimate four parameters from solution values at sensor locations.
The steady state heat equation to be solved is:
- del ( k(x,y) * grad u ) = f(x,y)
over the unit square 0 < x, y < 1.
Zero Dirichlet boundary conditions are applied. The right hand side function is set as:
f(x,y) = 1000 * x * ( 1 - x ) * y * ( 1 - y );
The diffusivity is represented by a gaussian function:
k(x,y) = 0.5 + vc * exp ( - 0.5 * ((x-xc)^2+(y-yc)^2)/sc^2 )
with parameters xc, yc, sc, vc.
The equation is solved using a finite element method. Values of the solution are sampled at 50 sensor locations.
For this code, the problem is to be solved many (500) times. Each time, the values of xc, yc, sc and vc were picked uniformaly at random from given intervals. A data file is created, containing, for each case, the random number seed, the parameter values (xc,yc,sc,vc), and the 50 solution values at sensor locations.
The task of the neural network is a kind of regression. Given the solution values at the sensor locations, estimate the values of the parameter xc, yc, sc, vc used to define the heat equation.
With 500 sets of data, the results were not brilliant, although we tried variations in the number of levels, nodes, and epochs, without getting good matches to the data. Moving to 2000 sets of data made a huge improvement in the results.
The information on this web page is distributed under the MIT license.
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John Burkardt.
