import numpy as np
from probnum.diffeq.ode.ivp import IVP
[docs]def logistic(timespan, initrv, params=(3.0, 1.0)):
r"""Initial value problem (IVP) based on the logistic ODE.
The logistic ODE is defined through
.. math::
f(t, y) = a y \left( 1 - \frac{y}{b} \right)
for some parameters :math:`(a, b)`.
Default is :math:`(a, b)=(3.0, 1.0)`. This implementation includes
the Jacobian :math:`J_f` of :math:`f` as well as a closed form
solution given by
.. math::
f(t) = \frac{b y_0 \exp(a t)}{b + y_0 \left[ \exp(at) - 1 \right]}
where :math:`y_0= y(t_0)` is the initial value.
Parameters
----------
timespan : (float, float)
Time span of IVP.
initrv : RandomVariable,
*(shape=())* -- Scalar-valued RandomVariable that describes the belief
over the initial value. Usually it is a Constant (noise-free or Normal (no
Random Variable isy) with scalar mean and scalar variance.
To replicate "classical" initial values use the Constant distribution.
params : (float, float), optional
Parameters :math:`(a, b)` for the logistic IVP.
Default is :math:`(a, b) = (3.0, 1.0)`.
Returns
-------
IVP
IVP object describing the logistic IVP with the prescribed
configuration.
"""
def rhs(t, y):
return log_rhs(t, y, params)
def jac(t, y):
return log_jac(t, y, params)
def hess(t, y):
return log_hess(t, y, params)
def sol(t):
return log_sol(t, params, initrv.mean)
return IVP(timespan, initrv, rhs, jac, hess, sol)
def log_rhs(t, y, params):
"""RHS for logistic model."""
l0, l1 = params
return l0 * y * (1.0 - y / l1)
def log_jac(t, y, params):
"""Jacobian for logistic model."""
l0, l1 = params
return np.array([l0 - l0 / l1 * 2 * y])
def log_hess(t, y, params):
"""Hessian for logistic model."""
l0, l1 = params
return np.array([[-2 * l0 / l1]])
def log_sol(t, params, y0):
"""Solution for logistic model."""
l0, l1 = params
nomin = l1 * y0 * np.exp(l0 * t)
denom = l1 + y0 * (np.exp(l0 * t) - 1)
return nomin / denom
[docs]def fitzhughnagumo(timespan, initrv, params=(0.0, 0.08, 0.07, 1.25)):
r"""Initial value problem (IVP) based on the FitzHugh-Nagumo model.
The FitzHugh-Nagumo (FHN) model is defined through
.. math::
f(t, y) =
\begin{pmatrix}
y_1 - \frac{1}{3} y_1^3 - y_2 + a \\
\frac{1}{d} (y_1 + b - c y_2)
\end{pmatrix}
for some parameters :math:`(a, b, c, d)`.
Default is :math:`(a, b)=(0.0, 0.08, 0.07, 1.25)`.
This implementation includes the Jacobian :math:`J_f` of :math:`f`.
Parameters
----------
timespan : (float, float)
Time span of IVP.
initrv : RandomVariable,
*(shape=(2, ))* -- Vector-valued RandomVariable that describes the belief
over the initial value. Usually it is a Constant (noise-free) or Normal (noisy)
Random Variable with :math:`2`-dimensional mean vector and
:math:`2 \times 2`-dimensional covariance matrix.
To replicate "classical" initial values use the Constant distribution.
params : (float, float, float, float), optional
Parameters :math:`(a, b, c, d)` for the logistic IVP.
Default is :math:`(a, b, c, d)=(0.0, 0.08, 0.07, 1.25)`.
Returns
-------
IVP
IVP object describing the logistic IVP with the prescribed
configuration.
"""
def rhs(t, y):
return fhn_rhs(t, y, params)
def jac(t, y):
return fhn_jac(t, y, params)
return IVP(timespan, initrv, rhs, jac)
def fhn_rhs(t, y, params):
"""RHS for FitzHugh-Nagumo model."""
y1, y2 = y
a, b, c, d = params
return np.array([y1 - y1 ** 3.0 / 3.0 - y2 + a, (y1 + b - c * y2) / d])
def fhn_jac(t, y, params):
"""Jacobian for FitzHugh-Nagumo model."""
y1, y2 = y
a, b, c, d = params
return np.array([[1.0 - y1 ** 2.0, -1.0], [1.0 / d, -c / d]])
[docs]def lotkavolterra(timespan, initrv, params=(0.5, 0.05, 0.5, 0.05)):
r"""Initial value problem (IVP) based on the Lotka-Volterra model.
The Lotka-Volterra (LV) model is defined through
.. math::
f(t, y) =
\begin{pmatrix}
a y_1 - by_1y_2 \\
-c y_2 + d y_1 y_2
\end{pmatrix}
for some parameters :math:`(a, b, c, d)`.
Default is :math:`(a, b)=(0.5, 0.05, 0.5, 0.05)`.
This implementation includes the Jacobian :math:`J_f` of :math:`f`.
Parameters
----------
timespan : (float, float)
Time span of IVP.
initrv : RandomVariable,
*(shape=(2, ))* -- Vector-valued RandomVariable that describes the belief
over the initial value. Usually it is a Constant (noise-free) or Normal (noisy)
Random Variable with :math:`2`-dimensional mean vector and
:math:`2 \times 2`-dimensional covariance matrix.
To replicate "classical" initial values use the Constant distribution.
params : (float, float, float, float), optional
Parameters :math:`(a, b, c, d)` for the logistic IVP.
Default is :math:`(a, b, c, d)=(0.5, 0.05, 0.5, 0.05)`.
Returns
-------
IVP
IVP object describing the logistic IVP with the prescribed
configuration.
"""
def rhs(t, y):
return lv_rhs(t, y, params)
def jac(t, y):
return lv_jac(t, y, params)
return IVP(timespan, initrv, rhs, jac)
def lv_rhs(t, y, params):
"""RHS for Lotka-Volterra."""
a, b, c, d = params
y1, y2 = y
return np.array([a * y1 - b * y1 * y2, -c * y2 + d * y1 * y2])
def lv_jac(t, y, params):
"""Jacobian for Lotka-Volterra."""
a, b, c, d = params
y1, y2 = y
return np.array([[a - b * y2, -b * y1], [d * y2, -c + d * y1]])
[docs]def seir(timespan, initrv, params=(0.3, 0.3, 0.1)):
r"""Initial value problem (IVP) based on the SEIR model.
The SEIR model with no vital dynamics is defined through
.. math::
f(t, y) =
\begin{pmatrix}
\frac{-\beta y_1 y_3}{N} \\
\frac{\beta y_1 y_3}{N} - \alpha y_2 \\
\alpha y_2 - \gamma y_3 \\
\gamma y_3
\end{pmatrix}
for some parameters :math:`(\alpha, \beta, \gamma)` and population
count :math:`N`. Without taking vital dynamics into consideration,
:math:`N` is constant such that for every time point :math:`t`
.. math::
S(t) + E(t) + I(t) + R(t) = N
holds.
Default parameters are :math:`(\alpha, \beta, \gamma)=(0.3, 0.3, 0.1)`.
The population count is computed from the (mean of the)
initial value random variable.
This implementation includes the Jacobian :math:`J_f` of :math:`f`.
Parameters
----------
timespan : (float, float)
Time span of IVP.
initrv : RandomVariable,
*(shape=(4, ))* -- Vector-valued RandomVariable that describes the belief
over the initial value. Usually it is a Constant (noise-free) or Normal (noisy)
Random Variable with :math:`4`-dimensional mean vector and
:math:`4 \times 4`-dimensional covariance matrix.
To replicate "classical" initial values use the Constant distribution.
params : (float, float, float), optional
Parameters :math:`(\alpha, \beta, \gamma)` for the SEIR model IVP.
Default is :math:`(\alpha, \beta, \gamma)=(0.3, 0.3, 0.1)`.
Returns
-------
IVP
IVP object describing the SEIR model IVP with the prescribed
configuration.
"""
population_count = np.sum(initrv.mean)
params_and_population_count = (*params, population_count)
def rhs(t, y):
return seir_rhs(t, y, params_and_population_count)
def jac(t, y):
return seir_jac(t, y, params_and_population_count)
return IVP(timespan, initrv, rhs, jac=jac)
def seir_rhs(t, y, params):
"""RHS for SEIR model"""
alpha, beta, gamma, population_count = params
y1, y2, y3, y4 = y
y1_next = -beta * y1 * y3 / population_count
y2_next = beta * y1 * y3 / population_count - alpha * y2
y3_next = alpha * y2 - gamma * y3
y4_next = gamma * y3
return np.array([y1_next, y2_next, y3_next, y4_next])
def seir_jac(t, y, params):
"""Jacobian for SEIR model"""
alpha, beta, gamma, population_count = params
y1, y2, y3, y4 = y
d_dy1 = np.array(
[-beta * y3 / population_count, 0.0, -beta * y1 / population_count, 0.0]
)
d_dy2 = np.array(
[beta * y3 / population_count, -alpha, beta * y1 / population_count, 0.0]
)
d_dy3 = np.array([0.0, alpha, -gamma, 0.0])
d_dy4 = np.array([0.0, 0.0, gamma, 0.0])
jac_matrix = np.array([d_dy1, d_dy2, d_dy3, d_dy4])
return jac_matrix
[docs]def rigidbody(timespan, initrv):
r"""Initial value problem (IVP) for rigid body dynamics without external forces
The rigid body dynamics without external forces is defined through
.. math::
f(t, y) =
\begin{pmatrix}
y_2 y_3 \\
-y_1 y_3 \\
-0.51 \cdot y_1 y_2
\end{pmatrix}
The ODE system has no parameters.
This implementation includes the Jacobian :math:`J_f` of :math:`f`.
Parameters
----------
timespan : (float, float)
Time span of IVP.
initrv : RandomVariable,
*(shape=(3, ))* -- Vector-valued RandomVariable that describes the belief
over the initial value. Usually it is a Constant (noise-free) or Normal (noisy)
Random Variable with :math:`3`-dimensional mean vector and
:math:`3 \times 3`-dimensional covariance matrix.
To replicate "classical" initial values use the Constant distribution.
Returns
-------
IVP
IVP object describing the rigid body dynamics IVP with the prescribed
configuration.
"""
def rhs(t, y):
return rigidbody_rhs(t, y)
def jac(t, y):
return rigidbody_jac(t, y)
return IVP(timespan, initrv, rhs, jac=jac)
def rigidbody_rhs(t, y):
y1, y2, y3 = y
return np.array([y2 * y3, -y1 * y3, -0.51 * y1 * y2])
def rigidbody_jac(t, y):
y1, y2, y3 = y
return np.array([[0.0, y3, y2], [-y3, 0.0, -y1], [-0.51 * y2, -0.51 * y1, 0.0]])
[docs]def vanderpol(timespan, initrv, params=0.1):
r"""Initial value problem (IVP) based on the Van der Pol Oscillator.
This function implements the second-order Van-der-Pol Oscillator as a system
of first-order ODEs.
The Van der Pol Oscillator is defined as
.. math::
f(t, y) =
\begin{pmatrix}
y_2 \\
\mu \cdot (1 - y_1^2)y_2 - y_1
\end{pmatrix}
for a constant parameter :math:`\mu`.
:math:`\mu` determines the stiffness of the problem, where
the larger :math:`\mu` is chosen, the more stiff the problem becomes.
Default is :math:`\mu = 0.1`.
This implementation includes the Jacobian :math:`J_f` of :math:`f`.
Parameters
----------
timespan : (float, float)
Time span of IVP.
initrv : RandomVariable,
*(shape=(2, ))* -- Vector-valued RandomVariable that describes the belief
over the initial value. Usually it is a Constant (noise-free) or Normal (noisy)
Random Variable with :math:`2`-dimensional mean vector and
:math:`2 \times 2`-dimensional covariance matrix.
To replicate "classical" initial values use the Constant distribution.
params : (float), optional
Parameter :math:`\mu` for the Van der Pol Equations
Default is :math:`\mu=0.1`.
Returns
-------
IVP
IVP object describing the Van der Pol Oscillator IVP with the prescribed
configuration.
"""
def rhs(t, y):
return vanderpol_rhs(t, y, params)
def jac(t, y):
return vanderpol_jac(t, y, params)
return IVP(timespan, initrv, rhs, jac=jac)
def vanderpol_rhs(t, y, params):
y1, y2 = y
if isinstance(params, float):
mu = params
else:
(mu,) = params
return np.array([y2, mu * (1.0 - y1 ** 2) * y2 - y1])
def vanderpol_jac(t, y, params):
y1, y2 = y
if isinstance(params, float):
mu = params
else:
(mu,) = params
return np.array([[0.0, 1.0], [-2.0 * mu * y2 * y1 - 1.0, mu * (1.0 - y1 ** 2)]])
[docs]def threebody(timespan, initrv, params=0.012277471):
r"""Initial value problem (IVP) based on a three-body problem.
Let the initial conditions be :math:`y = (y_1, y_2, \dot{y}_1, \dot{y}_2)^T`.
This function implements the second-order three-body problem as a system of
first-order ODEs, which is defined as follows: [1]_
.. math::
f(t, y) =
\begin{pmatrix}
\dot{y_1} \\
\dot{y_2} \\
y_1 + 2 \dot{y}_2 - \frac{(1 - \mu) (y_1 + \mu)}{d_1}
- \frac{\mu (y_1 - (1 - \mu))}{d_2} \\
y_2 - 2 \dot{y}_1 - \frac{(1 - \mu) y_2}{d_1} - \frac{\mu y_2}{d_2}
\end{pmatrix}
with
.. math::
d_1 &= ((y_1 + \mu)^2 + y_2^2)^{\frac{3}{2}} \\
d_2 &= ((y_1 - (1 - \mu))^2 + y_2^2)^{\frac{3}{2}}
and a constant parameter :math:`\mu` denoting the standardized moon mass.
Default is :math:`\mu = 0.012277471`.
Parameters
----------
timespan : (float, float)
Time span of IVP.
initrv : RandomVariable,
*(shape=(4, ))* -- Vector-valued RandomVariable that describes the belief
over the initial value. Usually it is a Constant (noise-free) or Normal (noisy)
Random Variable with :math:`4`-dimensional mean vector and
:math:`4 \times 4`-dimensional covariance matrix.
To replicate "classical" initial values use the Constant distribution.
params : (float), optional
Parameter :math:`\mu` for the three-body problem
Default is :math:`\mu = 0.012277471`.
Returns
-------
IVP
IVP object describing a three-body problem IVP with the prescribed
configuration.
References
----------
.. [1] Hairer, E., Norsett, S. and Wanner, G..
Solving Ordinary Differential Equations I.
Springer Series in Computational Mathematics, 1993.
"""
def rhs(t, y):
return threebody_rhs(t, y, params)
return IVP(timespan, initrv, rhs)
def threebody_rhs(t, y, params):
y1, y2, y1_dot, y2_dot = y
if isinstance(params, float):
standardized_moon_mass = params
else:
(standardized_moon_mass,) = params
mu = standardized_moon_mass
mp = 1.0 - mu
d1 = ((y1 + mu) ** 2 + y2 ** 2) ** 1.5
d2 = ((y1 - mp) ** 2 + y2 ** 2) ** 1.5
y1p = y1 + 2.0 * y2_dot - mp * (y1 + mu) / d1 - mu * (y1 - mp) / d2
y2p = y2 - 2.0 * y1_dot - mp * y2 / d1 - mu * y2 / d2
return np.array([y1_dot, y2_dot, y1p, y2p])