A Monte Carlo search for a symbolic model that fits given data. User needs to provide an error function (of how well a model fits the data) and some basic building blocks (math operations, variables, constants), from which random models will be generated. Procedure scales on multicore CPUs by running several independent Monte Carlo loops in parallel and subsequently accumulating obtained results.

A dummy case of fitting sqrt() function using the following - unary operations: log(abs()), sin(), cos(); binary operations: +, -, *, /; constants: e, pi; variables: x.

Evaluation time is set to 10 minutes. Total number of generated models is ~ 6 000 000 (on quad-core intel i7 desktop).

A Mathematica function/module called Error needs to be defined:

Error[model_]:=...

where model is a symbolic expression. Error has to return a non-negative value that is being minimized.

A template is presented below:

Get["...\\my_folder\\sym.m"] (*path to sym.m*)

data=Table[{i, Sqrt[i]}, {i, 0, 1, 0.1}]; (*given data*)
Error[model_]:=Mean@Table[Abs[data[[i,2]]-model/.{x->data[[i,1]]}],{i,Length[data]}] (*mean deviation*)

time=60*10; (*time in seconds for running the procedure*)
ncores=4; (*number of cores to be used*)
uops={Log[Abs[#]]&,Sin[#]&,Cos[#]&}; (*list of unary operations*)
bops={Plus,Subtract,Times,Divide}; (*list of binary operations*)
vars={x,E,Pi}; (*all free variables that are used in Error[], and also constants if needed*)
nops=10; (*max number of operations used for constructing model*)

output=Search[time,ncores,uops,bops,vars,nops]; (*runs procedure*)
output[[1]]//TableForm (*returns accumulated list of best 100 models with corresponding error values*)
output[[2]] (*returns total number of iterations*)

Important:

• Some operations must be protected. Applying Exp and Power on large/small enough numbers, especially when randomly generated model contains nestings like Exp[Exp[Exp[...]]], can cause severe memory leaks. Therefore, arguments of these functions need to be protected by Clip (cuts off large/small values). For instance, instead of listing Exp in uops user needs to list Exp[Clip[#,{10^-6,10^2},{0,Infinity}]]&. Also, it's better to use Log[Abs[#]]& and Sqrt[Abs[#]]& instead of just Log and Sqrt to avoid nasty complex numbers. Arithmetic operations (Plus, Subtract, Times, Divide) are safe, dividing by 0 is OK. When adding new operations, make sure they are safe, otherwise use proper safety!

Paul Knysh