Polynomial Model Reduction Dataset
This GitHub repo includes various polynomial structured ordinary differential equations (ODEs) models and its dataset derived from large-scale partial differential equations (PDEs). The polynomial models are defined by
\[\dot{x}(t) = Ax(t) + H(x(t) \otimes x(t)) + G(x(t) \otimes x(t) \otimes x(t)) + Bu(t)\]
where
- $x \in \mathbb{R}^n$: state variable vector
- $u \in \mathbb{R}^m$: control input vector
- $A \in \mathbb{R}^{n \times n}$: linear system matrix
- $H \in \mathbb{R}^{n \times n^2}$: quadratic system matrix
- $G \in \mathbb{R}^{n \times n^3}$: cubic system matrix
- $B \in \mathbb{R}^{n \times m}$: control/input matrix
- $\otimes$: Kronecker product
We call the $A,~F,~H,~B$ matrices the operators of the system, and these operators are constructed using mainly finite difference scheme. This package constructs these operators for a given set of parameters and allows you to numerically integrate the system to collect the solution data for PDEs. The main goal of this package is to accelerate the development of novel model reduction approaches by making these models and dataset available to everyone.
Installation
You can install it using the command
using Pkg
Pkg.add("PolynomialModelReductionDataset")
using PolynomialModelReductionDatasetCurrently Offered Models
| Model | Equation | Boundary Conditions |
|---|---|---|
| 1D Heat | $u_t = \mu u_{xx}$ | periodic, Dirichlet |
| Viscous Burgers' | $u_t = \mu u_{xx} - uu_{x}$ | periodic, Dirichlet |
| FitzHugh-Nagumo | $\begin{align*} u_t &= \epsilon^2 u_{xx} + u(u-0.1)(1-u) - v + g \\ v_t &= hu + \gamma v + g \end{align*}$ | Dirichlet and Neumann |
| Fisher-KPP | $u_t = D u_{xx} + ru(1-u)$ | periodic, Dirichlet, Dirichlet and Neumann (mixed) |
| Allen-Cahn | $u_t = \mu u_{xx} - \epsilon(u - u^3)$ | periodic, Dirichlet, Dirichlet and Neumann (mixed) |
| Kuramoto-Sivashinsky | $u_t = -\mu u_{xxxx} - u_{xx} - uu_x$ | periodic |
| Modified Korteweg-de Vries | $u_t = -\alpha u_{xxx} - \beta u^2u_{xx}$ | periodic, Dirichlet |
| Modified Korteweg-de Vries-Burgers | $u_t = -\alpha u_{xxx} - \beta u^2u_{xx} + \gamma u_{xx}$ | periodic, Dirichlet |
| Gardner | $u_t = -\alpha u_{xxx} + \beta uu_{x} + \gamma u^2u_{x}$ | periodic, Dirichlet |
| Damped Gardner-Burgers' | $u_t = -\alpha u_{xxx} + \beta uu_x + \gamma u^2u_x + \delta u_{xx} + \epsilon u$ | periodic, Dirichlet |
| 2D Heat | $u_t = \mu(u_{xx} + u_{yy})$ | Dirichlet |
Non-Redundant Expressions
In this package, you will see a slightly different version of operators from what is defined above. Those are operators including only the non-redundant terms which eliminate the redundancy coming from the symmetry of Kronecker products. For ease of exposition, we will only introduce the non-redundant operators. For more info, please see this page.
For the quadratic and cubic operators we define the non-redundant counterparts as follows:
- $F \in \mathbb{R}^{n \times n(n+1)/2}$: quadratic operator corresponding to the non-redundant terms of $x \otimes x$
- $E \in \mathbb{R}^{n \times n(n+1)(n+2)/6}$: cubic operator corresponding to the non-redundant terms of $x \otimes x \otimes x$
Then, the ODE is defined by
\[\dot{x}(t) = Ax(t) + F(x(t) \oslash x(t)) + E(x(t) \oslash x(t) \oslash x(t)) + Bu(t)\]
where $\oslash$ is the unique Kronecker product defined in UniqueKronecker.jl.
Contributing
Contributions are welcome! If you find a bug or have a feature request, please open an issue. If you'd like to contribute code, feel free to submit a pull request.
License
This project is licensed under the MIT License.