Genetic programming approaches for solving elliptic partial differential equations

Description/Abstract

In this paper, we propose a technique based on
genetic programming (GP) for meshfree solution of elliptic partial
differential equations. We employ the least-squares collocation
principle to define an appropriate objective function, which is
optimized using GP. Two approaches are presented for the repair
of the symbolic expression for the field variables evolved by the
GP algorithm to ensure that the governing equations as well as
the boundary conditions are satisfied. In the case of problems
defined on geometrically simple domains, we augment the solution
evolved by GP with additional terms, such that the boundary
conditions are satisfied by construction. To satisfy the boundary
conditions for geometrically irregular domains, we combine the
GP model with a radial basis function network. We improve
the computational efficiency and accuracy of both techniques
with gradient boosting, a technique originally developed by the
machine learning community. Numerical studies are presented
for operator problems on regular and irregular boundaries to
illustrate the performance of the proposed algorithms.