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, randvars, randprocs, problems 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 = 0.0 tmax = 20.0 y0 = np.array([20, 20]) ivp = problems.InitialValueProblem(t0=t0, tmax=tmax, y0=y0, f=f, df=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_process = randprocs.markov.integrator.IntegratedWienerProcess( initarg=ivp.t0, num_derivatives=4, wiener_process_dimension=ivp.dimension, forward_implementation="sqrt", backward_implementation="sqrt", )
Next, we construct the ODE filter. One choice that has not been made yet is the initialisation strategy. The current default choice is to initialise by fitting the prior to a few steps of a Runge-Kutta solution.
diffmodel = randprocs.markov.continuous.PiecewiseConstantDiffusion(t0=t0) rk_init = diffeq.odefiltsmooth.initialization_routines.RungeKuttaInitialization() ode_residual = diffeq.odefiltsmooth.information_operators.ODEResidual(4, ivp.dimension) ek1 = diffeq.odefiltsmooth.approx_strategies.EK1() firststep = diffeq.stepsize.propose_firststep(ivp) steprule = diffeq.stepsize.AdaptiveSteps(firststep=firststep, atol=1e-3, rtol=1e-5) solver = diffeq.odefiltsmooth.GaussianIVPFilter( steprule, prior_process=prior_process, information_operator=ode_residual, approx_strategy=ek1, initialization_routine=rk_init, diffusion_model=diffmodel, 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.
odesol = solver.solve(ivp=ivp)
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()