qbee 0.7.1

Quadratization of differential equations in python

QBee

Online playground

QBee is a Python library for transforming systems of differential equations into a systems with quadratic right-rand side.

Installation

PyPI

pip install qbee

Manual

1. Clone repository: https://github.com/AndreyBychkov/QBee.git
   • Or, if you want our bleeding edge version, clone https://github.com/AndreyBychkov/QBee/tree/dev
2. Change directory: cd QBee
3. Install package: pip install .

If you use poetry you can alternately install if with poetry install

What is quadratization?

The problem of quadratization is, given a system of ODEs with polynomial right-hand side, reduce the system to a system with quadratic right-hand side by introducing as few new variables as possible. We will explain it using toy example. Consider the system

An example of quadratization of this system will be a new variable

leading to the following ODE

Thus, we attained the system with quadratic right-hand side

We used only one new variable, so we achieved an optimal quadratization.

Qbee usage

QBee implements algorithms that take system of ODEs with elementary functions right-hand side and return optimal monomial quadratization - optimal quadratization constructed from monomial substitutions.

We will demonstrate usage of QBee on the example below. Other interactive examples you can find in examples section.

1. Importing QBee

QBee relies on Sympy for a high-level API.

import sympy
from qbee import *

sympy.init_printing()  # If you work in Jupyter notebook 

2. System definition

For example, we will take the A1 system from Swischuk et al.'2020

The parameters in the system are A, Ea and Ru, and the others are either state variables or inputs. So, let's define them with the system in code:

A, Ea, Ru = parameters("A, Ea, Ru")
c1, c2, c3, c4, T = functions("c1, c2, c3, c4, T")  

eq1 = -A * sp.exp(-Ea / (Ru * T)) * c1 ** 0.2 * c2 ** 1.3
system = [
    (c1, eq1),
    (c2, 2 * eq1),
    (c3, -eq1),
    (c4, -2 * eq1)
]

3. Polynomialization and Quadratization

When we work with ODEs with the right-hand side being a general continuous function, we utilize the following pipeline:

Input system -> Polynomial System -> Quadratic System

and the transformations are called polynomialization and quadratization accordingly.

The example system is not polynomial, so we use the most general method for achieving optimal monomial quadratization.

# {T: 2} means than T can have a derivative of order at most two => T''
quadr_system = polynomialize_and_quadratize(system, input_der_orders={T: 2})
if quadr_system:
    print("Quadratized system:")
    print(quadr_system)

Sample output:

Variables introduced in polynomialization:
w_{0} = c1**(-0.8)
w_{1} = c2**(-0.7)
w_{2} = 1/T
w_{3} = exp(-Ea*w_{2}/Ru)

Elapsed time: 0.139s.
==================================================
Quadratization result
==================================================
Number of introduced variables: 5
Nodes traversed: 117
Introduced variables:
w_{4} = T'*w_{2}
w_{5} = T'*w_{2}**2
w_{6} = c2**2*w_{0}*w_{1}*w_{3}
w_{7} = w_{2}**2
w_{8} = c1*c2*w_{0}*w_{1}*w_{3}

Quadratized system:
c1' = -A*c2*w_{8}
c2' = -2*A*c2*w_{8}
c3' = A*c2*w_{8}
c4' = 2*A*c2*w_{8}
w_{0}' = 4*A*w_{0}*w_{6}/5
w_{1}' = 7*A*w_{1}*w_{8}/5
w_{2}' = -T'*w_{7}
w_{3}' = Ea*w_{3}*w_{5}/Ru
T' = T'
T'' = T''
T''' = 0
w_{4}' = -T'*w_{5} + T''*w_{2}
w_{5}' = T''*w_{7} - 2*w_{4}*w_{5}
w_{6}' = 4*A*w_{6}**2/5 - 13*A*w_{6}*w_{8}/5 + Ea*w_{5}*w_{6}/Ru
w_{7}' = -2*w_{4}*w_{7}
w_{8}' = -A*w_{6}*w_{8}/5 - 3*A*w_{8}**2/5 + Ea*w_{5}*w_{8}/Ru

Process finished with exit code 0

Introduced variables are the optimal monomial quadratization.

4. Work inside of package

1. Configuration

Inside of config.ini you can change the following arguments:

• logging_enable = [True | False]. If enabled, work of algorithm is logged into logging_file and quad_systems_file . Requires memory to work. Is not recommended for long quadratizations.
• logging_file: must be in Apache Arrow .feather format.
• quad_systems_file: dump quadratic systems by using pickle. .pkl file format is recommended.
• progress_bar_enable: enables progress bar during quadratization.

2. Visualization

In order to visualize work of an algorithm you can pass logging data to qbee.visualize.visualize_pyvis:

visualize_pyvis('log.feather', 'quad_systems.pkl')

Papers

• Optimal Monomial Quadratization for ODE systems: arxiv, Springer

Citation

If you find this code useful in your research, please consider citing the above paper that works best for you.