gemseo / algos / ode

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ode_problem module

Ordinary differential equation problem.

class gemseo.algos.ode.ode_problem.ODEProblem(func, initial_state, initial_time, final_time, jac=None, jac_desvar=None, adjoint_wrt_state=None, adjoint_wrt_desvar=None, time_vector=None)

Bases: BaseProblem

First-order ordinary differential equation (ODE).

A first-order ODE is written as

\[\frac{ds(t)}{dt} = f(t, s(t)).\]

where \(f\) is called the right-hand side (RHS) of the ODE and \(s(t)\) is the state vector at time \(t\).

Parameters:

• func (Callable[[float, NumberArray], NumberArray]) – The RHS function \(f\). It will be called as func(time, state) as necessary.

• initial_state (ArrayLike) – The initial state of the ODE.

• initial_time (float) – The start of the integration interval.

• final_time (float) – The end of the integration interval.

• jac (Callable[[float, NumberArray], NumberArray] | NumberArray | None) – The function to compute the Jacobian of \(f\). If Callable, the Jacobian is assumed to be dependent on time and state. It will be called as jac(time, state) as necessary. If NumberArray, the Jacobian is assumed to be constant. If None, the Jacobian will be approximated by finite differences.

• jac_desvar (Callable[[float, NumberArray], NumberArray] | None) – The function to compute the Jacobian of \(f\) relative to the design variables. If None, use a solver that doesn’t require the adjoint.

• adjoint_wrt_state (NumberArray | None) – The adjoint relative to the state. If None, use a solver that doesn’t require the adjoint.

• adjoint_wrt_desvar (NumberArray | None) – The adjoint relative to the design variables. If None, use a solver that doesn’t require the adjoint.

• time_vector (NumberArray | None) – The time vector for the solution. If None, the solver will select times for which the computed solution is stored.

check()

Ensure the parameters of the problem are consistent.

Raises:

ValueError – If the state and time shapes are inconsistent.

Return type:

None

check_jacobian(state_vector, approximation_mode=ApproximationMode.FINITE_DIFFERENCES, step=1e-06, error_max=1e-08)

Check if the Jacobian function is correct.

At a given state, compare the value of the Jacobian computed by the function provided by ther user to an approximated value computed by finite-differences for example.

Parameters:

• state_vector (ArrayLike) – The state at which the Jacobian is checked.

• approximation_mode (ApproximationMode) –

  The approximation mode.

  By default it is set to “finite_differences”.

• step (float) –

  The step used to approximate the gradients.

  By default it is set to 1e-06.

• error_max (float) –

  The error threshold above which the Jacobian is deemed to be incorrect.

  By default it is set to 1e-08.

Raises:

ValueError – When the Jacobian function is wrong.

Return type:

None

adjoint_wrt_desvar: NumberArray

The adjoint of the problem relative to the design variables.

adjoint_wrt_state: NumberArray

The adjoint of the problem relative to the state.

initial_state: NumberArray

The initial conditions \((t_0,s_0)\) of the ODE.

integration_interval: tuple[float, float]

The interval of integration.

jac: Callable[[float, NumberArray], NumberArray] | NumberArray | None

The function to compute the Jacobian of \(f\).

If Callable, the Jacobian is assumed to be dependent on time and state. It will be called as jac(time, state) as necessary. If NumberArray, the Jacobian is assumed to be constant. If None, the Jacobian will be approximated by finite differences.

jac_desvar: Callable[[float, NumberArray], NumberArray]

The function to compute the Jacobian of \(f\) relative to the design variables.

result: ODEResult

The result of the ODE problem.

rhs_function: Callable[[float, NumberArray], NumberArray]

The function \(f\).

property time_vector

The times at which the solution shall be evaluated.

Examples using ODEProblem

Solve an ODE: the Van der Pol problem

Solve an ODE: the Van der Pol problem