ODE-Solvers from Scratch

All the other tutorials show how to use the ODE-solver with the probsolve_ivp function. This is great, though probnum has more customisation to offer.

from probnum import diffeq, filtsmooth, statespace, randvars, randprocs
import numpy as np

import matplotlib.pyplot as plt

plt.style.use("../../probnum.mplstyle")

First we define the ODE problem. As always, we use Lotka-Volterra. Once the ODE functions are defined, they are gathered in an IVP object.

def f(t, y):
    y1, y2 = y
    return np.array([0.5 * y1 - 0.05 * y1 * y2, -0.5 * y2 + 0.05 * y1 * y2])


def df(t, y):
    y1, y2 = y
    return np.array([[0.5 - 0.05 * y2, -0.05 * y1], [0.05 * y2, -0.5 + 0.05 * y1]])


t0 = 0.0
tmax = 20.0
y0 = np.array([20, 20])

ivp = diffeq.IVP((t0, tmax), initrv=randvars.Constant(y0), rhs=f, jac=df)

Next, we define a prior distribution and a measurement model. The former can be any Integrator, which currently restricts the choice to IBM, IOUP, and Matern. We usually recommend IBM. The measurement model requires a choice between EK0, EK1 (extended Kalman filters of order 0 or 1, respectively) and perhaps UK (unscented Kalman filter). The use of the latter is discouraged, because the square-root implementation is not available currently.

The measurement model can either be constructed with DiscreteEKFComponent.from_ode or, perhaps more conveniently, with GaussianIVPFilter.string_to_measurement_model.

prior = statespace.IBM(
    ordint=4,
    spatialdim=ivp.dimension,
    forward_implementation="sqrt",
    backward_implementation="sqrt",
)
initrv = randvars.Normal(mean=np.zeros(prior.dimension), cov=np.eye(prior.dimension))
prior_process = randprocs.MarkovProcess(transition=prior, initrv=initrv, initarg=ivp.t0)
ekf = diffeq.GaussianIVPFilter.string_to_measurement_model(
    "EK1", ivp=ivp, prior_process=prior_process
)

Next, we construct the ODE filter. One choice that has not been made yet is the initialiation strategy. The current default choice is to initialise by fitting the prior to a few steps of a Runge-Kutta solution. An alternative is to use automatic differentiation, which is currently in development. An easy-access version of those initialisation strategies is to use the constructor GaussianIVPFilter.construct_with_rk_init.

diffmodel =statespace.PiecewiseConstantDiffusion(t0=t0)
solver = diffeq.GaussianIVPFilter.construct_with_rk_init(ivp, prior_process=prior_process, measurement_model=ekf, diffusion_model=diffmodel, with_smoothing=True)

Now we can solve the ODE. To this end, define a StepRule, e.g. ConstantSteps or AdaptiveSteps. If you don’t know which firststep to use, the function propose_firststep makes an educated guess for you.

firststep = diffeq.propose_firststep(ivp)
steprule = diffeq.AdaptiveSteps(firststep=firststep, atol=1e-3, rtol=1e-5)
# steprule = diffeq.ConstantSteps(0.1)
odesol = solver.solve(steprule=steprule)

GaussianIVPFilter.solve returns an ODESolution object, which is sliceable and callable. The latter can be used to plot the solution on a uniform grid, even though the solution was computed on an adaptive grid. Be careful: the return values of __call__, etc., are always random variable-like objects. We decide to plot the mean.

evalgrid = np.arange(ivp.t0, ivp.tmax, step=0.1)

Done! This is the solution to the Lotka-Volterra model.

sol = odesol(evalgrid)

plt.plot(evalgrid, sol.mean, "o-", linewidth=1)
plt.ylim((0, 30))
plt.show()