paraheat_gaussian_parameter, a keras script which reads data for the heat equation with a parameterized gaussian diffusivity, and uses a neural network to estimate the parameter 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 value of vc is picked uniformaly at random from the interval [0.5,5.0], while xc, yc, and sc are held fixed. 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 value of the parameter vc that was used to define the heat equation.
The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.
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John Burkardt.
