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 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
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. tmax = 20. 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
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
prior = statespace.IBM(ordint=4, spatialdim=ivp.dimension, forward_implementation="sqrt", backward_implementation="sqrt") ekf = diffeq.GaussianIVPFilter.string_to_measurement_model("EK1", ivp=ivp, prior=prior)
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
solver = diffeq.GaussianIVPFilter.construct_with_rk_init(ivp, prior=prior, measurement_model=ekf, with_smoothing=True)
Now we can solve the ODE. To this end, define a
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-2, rtol=1e-2) 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=0.2)
Done! This is the solution to the Lotka-Volterra model.
sol = odesol(evalgrid) plt.plot(evalgrid, sol.mean, "o-", linewidth=1) plt.show()