dynabench.equation

Module for representing partial differential equations.

Functions

expand(e[, deep, modulus, power_base, ...])	Expand an expression using methods given as hints.
sympify(a[, locals, convert_xor, strict, ...])	Converts an arbitrary expression to a type that can be used inside SymPy.

Classes

AdvectionEquation([c_x, c_y, evolution_rate])	Advection equation in 2D.
BaseEquation([equations, parameters, ...])	Base class for all equations.
CahnHilliardEquation([D, gamma, evolution_rate])	Cahn-Hilliard equation in 2D.
DiffusionEquation([D, evolution_rate])	Diffusion equation in 2D.
Eq	alias of Equality
FitzhughNagumoEquation([stimulus, a, b, ...])	FitzHugh–Nagumo model with diffusive coupling given by the following equations:
Function(*args)	Base class for applied mathematical functions.
KuramotoSivashinskyEquation([evolution_rate])	Kuramoto-Sivashinsky equation in 2D.
PDE(rhs, *[, bc, bc_ops, user_funcs, ...])	PDE defined by mathematical expressions
SimpleBurgersEquation([nu, evolution_rate])	Simple Burgers equation in 2D.
Symbol(name, **assumptions)	Assumptions:
WaveEquation([c, evolution_rate])	Wave equation in 2D.

class dynabench.equation.AdvectionEquation(c_x: float = 1.0, c_y: float = 1.0, evolution_rate: float = 1.0, **kwargs)

Bases: BaseEquation

Advection equation in 2D. The equation is given by:

\[\frac{\partial u}{\partial t} = -c_x \frac{\partial u}{\partial x} - c_y \frac{\partial u}{\partial y}\]

where c_x and c_y are the speeds of the advection in the x and y directions respectively.

Parameters:

parameters (dict, default {c_x: 1, c_y: 1}) – Dictionary of parameters for the equations.

linear_terms

List of linear terms in the equation.

Type:

List[str]

nonlinear_terms

List of nonlinear terms in the equation.

Type:

List[str]

class dynabench.equation.BaseEquation(equations: List[str] | str = ['dt(u) = 0'], parameters: Dict[str, float] = {}, evolution_rate: float = 1.0, **kwargs)

Bases: object

Base class for all equations. Represents the equation in the form of a dictionary which can be used with the py-pde library.

Parameters:

• equations (List[str] | str, default ["u_t = 0"]) – List of equations to be represented.

• parameters (dict, default {}) – Dictionary of parameters for the equations.

linear_terms

List of linear terms in the equation.

Type:

List[str]

nonlinear_terms

List of nonlinear terms in the equation.

Type:

List[str]

property equations

Get the equations of the equation.

export_as_pypde_equation()

Export the equation as a py-pde equation.

property lhs

Get the left-hand side of the equation.

property name

Get the name of the equation.

property num_variables

Get the number of variables in the equation.

property rhs

Get the right-hand side of the equation.

simplify_equation(eq: str) → str

Simplify the equation.

Parameters:

eq (str) – The equation to be simplified.

Returns:

The simplified equation.

Return type:

str

property variables

Get the variables of the equation.

class dynabench.equation.CahnHilliardEquation(D: float = 1.0, gamma: float = 1.0, evolution_rate: float = 1.0, **kwargs)

Bases: BaseEquation

Cahn-Hilliard equation in 2D. The equation is given by:

\[\frac{\partial u}{\partial t} = D \nabla^2(u^3 - u - \gamma \nabla^2(u))\]

where D and gamma are parameters of the equation.

Parameters:

parameters (dict, default {D: 1, gamma: 1}) – Dictionary of parameters for the equations.

linear_terms

List of linear terms in the equation.

Type:

List[str]

nonlinear_terms

List of nonlinear terms in the equation.

Type:

List[str]

class dynabench.equation.DiffusionEquation(D: float = 1.0, evolution_rate: float = 1.0, **kwargs)

Bases: BaseEquation

Diffusion equation in 2D. The equation is given by:

\[\frac{\partial u}{\partial t} = D \nabla^2 u\]

where D is the diffusion coefficient.

Parameters:

parameters (dict, default {D: 1}) – Dictionary of parameters for the equations.

linear_terms

List of linear terms in the equation.

Type:

List[str]

nonlinear_terms

List of nonlinear terms in the equation.

Type:

List[str]

class dynabench.equation.FitzhughNagumoEquation(stimulus: float = 0.5, τ: float = 10, a: float = 0, b: float = 0, evolution_rate: float = 1.0, **kwargs)

Bases: BaseEquation

FitzHugh–Nagumo model with diffusive coupling given by the following equations:

\[\frac{\partial v}{\partial t} = \nabla^2 v + v - \frac{v^3}{3} - w + stimulus \frac{\partial w}{\partial t} = \frac{v + a - b * w}{\tau}\]

where v is the membrane potential and w is the recovery variable, and stimulus, a, b, and tau are parameters of the equation.

class dynabench.equation.KuramotoSivashinskyEquation(evolution_rate: float = 1.0, **kwargs)

Bases: BaseEquation

Kuramoto-Sivashinsky equation in 2D. The equation is given by:

\[\frac{\partial u}{\partial t} = -\nabla^2 u - \nabla^4 u - \frac{1}{2}|\nabla u|^2\]

Parameters:

parameters (dict, default {}) – Dictionary of parameters for the equations.

linear_terms

List of linear terms in the equation.

Type:

List[str]

nonlinear_terms

List of nonlinear terms in the equation.

Type:

List[str]

class dynabench.equation.SimpleBurgersEquation(nu: float = 1.0, evolution_rate: float = 1.0, **kwargs)

Bases: BaseEquation

Simple Burgers equation in 2D. The equation is given by:

\[\frac{\partial u}{\partial t} = -u (\frac{\partial u}{\partial x}+\frac{\partial u}{\partial y}) + \nu \nabla^2 u\]

where nu is the viscosity of the fluid.

Parameters:

parameters (dict, default {nu: 1}) – Dictionary of parameters for the equations.

linear_terms

List of linear terms in the equation.

Type:

List[str]

nonlinear_terms

List of nonlinear terms in the equation.

Type:

List[str]

class dynabench.equation.WaveEquation(c: float = 1.0, evolution_rate: float = 1.0, **kwargs)

Bases: BaseEquation

Wave equation in 2D. The equation is given by:

\[\frac{\partial^2 u}{\partial t^2} = c^2 \nabla^2 u\]

where c is the speed of the wave.

Parameters:

parameters (dict, default {c: 1}) – Dictionary of parameters for the equations.

linear_terms

List of linear terms in the equation.

Type:

List[str]

nonlinear_terms

List of nonlinear terms in the equation.

Type:

List[str]