Genetic Programming Approaches for Solving Elliptic Partial Differential Equations

Abstract

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.

A Universal Concept Based on Cellular Neural Networks for Ultrafast and Flexible Solving of Differential Equations

“…However, despite the interesting and extensive involvement of complex differential equation models in different areas of science and engineering the traditional methods (algorithms and computing schemes) to solve or simulate them fail to cope with a series of issues related to the high complexity. The relevant literature clearly suggests that the traditional solving and computing approaches (for stiff/stochastic ODEs and PDEs) are very slow [11], generally less precise [12], and not always robust enough [13]. Furthermore, those traditional paradigms are computationally expensive and not capable of realizing real-time computation at an acceptable and realistic performance/cost ratio.…”

Solving Ordinary Differential Equations With Adaptive Differential Evolution

“…Tsoulos et al [17] proposed a novel method based on the grammatical evolution to solve ODEs with the aid of genetic programming (GP). Motivated by the method in [17], a technique combined GP and RBF network was presented in [18]. Chaquet et al [19] further applied RBF network into the basis function.…”

Closed-Form Solutions to Differential Equations via Differential Evolution

“…In [11], genetic algorithms are used to solve some differential equations appearing in economic sciences. In [12] a variational approach has been used in order to solve elliptic partial differential equations, and a genetic algorithm is used as the optimization method. In all the previously referenced articles-deterministic or not-the solution is given in a numerical approximated form.…”

“…For instance, operf2(12,[ñan,sindual(̃),ñan,1]) will return sindual(̃) +1 and then vecval = [ñan, sindual(̃),ñan,1] is changed to [ñan, nan,ñan,ñan]. The Fortran code for the parsv function is shown in Algorithm 1.Let us analyze the above code for the case when x =[12,17,13,2,11,20,18,14,11,5] andxval = [1.1, 1, 0]. In this particular case, the dimension length of the vector x is length = 10 and x represents the function ( ) = sin(2 )+ log(cos( /5)).…”