scientific_julia

Exploring julia packages for data science and other numerical modeling applications.

André Guerra
April, 2023
[email protected]

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Description:
This repository contains a series of scripts that explore various julia packages that may be used for data science and developing numerical solutions to physical models. We start with a simple system of 3 ODEs solved using symbolics packages in ode_sys_sym.jl, then we introduce a parameter to the system of ODEs and solve it for a range of parameter values in ode_sys_num_param.jl.

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Core Contents

1. ode_sys_sym.jl $\rightarrow$ solve a system of 3 ODEs with symbolic packages
2. ode_sys_num_param.jl $\rightarrow$ solve a system of 3 ODEs with one parameter using numerical definition
3. ode_sys_num_param_opt.jl $\rightarrow$ solve for an optimal value of the parameter from (2) using an objective function

References

1. MATLAB example used for the ODE system
2. ModelingToolkit.jl
3. DifferentailEquations.jl
4. Plots.jl

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Below are quick descriptions of the main files in this repo and the problem statements to be solved. More detail is found as comments in the respective scripts files.

ode_sys_sym.jl

The naming convention for this script indicates that we are solving an ODE (ode) system (sys) using symbolics (sym) packages in Julia.

Reproduce the MATALB example of a simple system of ODEs[1] in Julia using ModelingToolkit.jl and DifferentialEquations.jl.

Problem Statement

Consider the nonlinear system:
$x' = -x + 3z$
$y' = -y + 2z$
$z' = x^2 - 2z$

Initial conditions:
$x(t=0) = 0$
$y(t=0) = 1/2$
$z(t=0) = 3$

ode_sys_num_param.jl

In this script, we represent the ODE system numerically (not using the symbolics packages).

Problem Statement

Consider the nonlinear system (the same as above but with a parameter p):
$x' = -x + pz$
$y' = -y + 2z$
$z' = x^2 - 2z$

Initial conditions:
$x(t=0) = 0$
$y(t=0) = 1/2$
$z(t=0) = 3$

In this script, we solve the system with a parameter space p = [0,1,2].

ode_sys_num_opt_param.jl

In this script we optimize the parameter p to find the value that satisfies the data available. In this script we also produce this data given a value of p=1.

Problem Statement

The problem statement here is the same as in ode_sys_num_param.jl above. However, here we use an L2 loss function as the objective cost to be optimized in the search for the value of p.