FORTRAN X-TFC

Extreme Theory of Functional Connections (X-TFC) algorithm implemented in FORTRAN to solve Ordinary Differential Equations (ODEs) as Initial Value Problems with minimal dependencies. It is applied at a simple example, for solving other problems modify problem.f90. This script contains the parameter definitions, the ODEs, and the Jacobian of the loss function. If you want to solve chemical kinetics equations look at the WinNet project for a widely tested and generalized program (minimal input required, no code modifications, you do not have to worry about the Jacobian).

Modules:

• problem. Problem definition: parameters and equations.

• xtf. X-TFC method.

• utils. Utilities for saving the solution and the trained X-TFC network to text files.

• backward_euler. Implicit backward Euler method.

The problem is solved in a log spaced training grid.

main.f90. FORTRAN program that calls the routines defined in the above modules to solve the problem with both solvers. It increases the resolution of the solution provided by the X-TFC making use of the analytical representation obtained.

plot.py. Python script to plot the results.

Installation

Commands for Debian-based Linux distributions.

Intel® oneAPI Math Kernel Library for FORTRAN.

wget -O- https://apt.repos.intel.com/intel-gpg-keys/GPG-PUB-KEY-INTEL-SW-PRODUCTS.PUB | gpg --dearmor | sudo tee /usr/share/keyrings/oneapi-archive-keyring.gpg > /dev/null
echo "deb [signed-by=/usr/share/keyrings/oneapi-archive-keyring.gpg] https://apt.repos.intel.com/oneapi all main" | sudo tee /etc/apt/sources.list.d/oneAPI.list
sudo apt update
sudo apt install intel-oneapi-common-vars intel-oneapi-compiler-fortran intel-oneapi-mkl intel-oneapi-mkl-devel
echo 'source /opt/intel/oneapi/setvars.sh' >> ~/.bashrc
source ~/.bashrc

Python packages: numpy, matplotlib and scipy. Just for the plot.

python3 -m venv venv
source venv/bin/activate
python -m pip install -r requirements.txt

Create /data directory.

mkdir data

Compilation & Execution

Compile,

ifort -c -qmkl -diag-disable=10448 problem.f90 utils.f90 xtfc.f90 backward_euler.f90 && ifort -o main main.f90 -qmkl -diag-disable=10448 utils.o xtfc.o backward_euler.o problem.o

Execute,

./main

And plot.

python plot.py

Compile with extra checks.

ifort -c -qmkl -traceback -check all -diag-disable=10448 problem.f90  utils.f90 xtfc.f90 backward_euler.f90
ifort -o main main.f90 -qmkl -traceback -check all -diag-disable=10448 utils.o xtfc.o backward_euler.o problem.o

Compile with performance profile.

ifort -c -qmkl -pg -diag-disable=10448 problem.f90  utils.f90 xtfc.f90 backward_euler.f90
ifort -o main main.f90 -qmkl -pg -diag-disable=10448 utils.o xtfc.o backward_euler.o problem.o

Execute and then explore the profile (this problem is solved too fast, you will not see anything).

./main
gprof ./main gmon.out > output.txt

Robertson's problem

$$ \begin{cases} A & \hspace{.2cm} \stackrel{k_1}{\longleftrightarrow} \hspace{.2cm} & B \\ B + B & \hspace{.2cm} \stackrel{k_2}{\longleftrightarrow} \hspace{.2cm} & C + B \\ B + C & \hspace{.2cm} \stackrel{k_3}{\longleftrightarrow} \hspace{.2cm} & A + C \\ \end{cases} $$

$$ \begin{cases} k_1 = 4 \times 10^{-2}, \hspace{.2cm} k_2 = 3 \times 10^7, \hspace{.2cm} k_3 = 10^4 \\ y_A(0) = 1, \hspace{.2cm} y_B(0) = 0, \hspace{.2cm} y_C(0) = 0 \\ t \in [10^{-5}, 10^5] \end{cases} $$