RungeKuttaInitialization¶

class probnum.diffeq.odefilter.initialization_routines.RungeKuttaInitialization(dt=0.01, method='DOP853')

Initialize a probabilistic ODE solver by fitting the prior process to a few steps of an approximate ODE solution computed with Scipy’s Runge-Kutta methods.

Parameters
• dt (Union[float, Real, floating, None]) – Maximum step-size to use for computing the approximate ODE solution. The smaller, the more accurate, but also, the smaller, the less stable. The best value here depends on the ODE problem, and probably the chosen method. Optional. Default is 1e-2.

• method (Optional[str]) – Which solver to use. This is communicated as a string that is compatible with scipy.integrate.solve_ivp(..., method=method). Optional. Default is DOP853.

Examples

>>> import numpy as np
>>> from probnum.randvars import Normal
>>> from probnum.problems.zoo.diffeq import vanderpol
>>> from probnum.randprocs.markov.integrator import IntegratedWienerProcess

Compute the initial values of the van-der-Pol problem as follows. First, we set up the ODE problem and the prior process.

>>> ivp = vanderpol()
>>> print(ivp.y0)
[2. 0.]
>>> prior_process = IntegratedWienerProcess(initarg=ivp.t0, num_derivatives=3, wiener_process_dimension=2)

Next, we call the initialization routine.

>>> rk_init = RungeKuttaInitialization()
>>> improved_initrv = rk_init(ivp=ivp, prior_process=prior_process)
>>> print(prior_process.transition.proj2coord(0) @ improved_initrv.mean)
[2. 0.]
>>> print(np.round(improved_initrv.mean, 1))
[    2.      0.     -2.     58.2     0.     -2.     60.  -1745.7]
>>> print(np.round(np.log10(improved_initrv.std), 1))
[-13.8 -11.3  -9.   -1.5 -13.8 -11.3  -9.   -1.5]

Attributes Summary

 is_exact Exactness of the computed initial values. requires_jax Whether the implementation of the routine relies on JAX.

Methods Summary

 __call__(ivp, prior_process) Compute the initial distribution.

Attributes Documentation

is_exact

Exactness of the computed initial values.

Some initialization routines yield the exact initial derivatives, some others only yield approximations.

Return type

bool

requires_jax

Whether the implementation of the routine relies on JAX.

Return type

bool

Methods Documentation

__call__(ivp, prior_process)[source]

Compute the initial distribution.

For Runge-Kutta initialization, it goes as follows:

1. The ODE integration problem is set up on the interval [t0, t0 + (2*order+1)*h0] and solved with a call to scipy.integrate.solve_ivp. The solver is uses adaptive steps with atol=rtol=1e-12, but is forced to pass through the events (t0, t0+h0, t0 + 2*h0, ..., t0 + (2*order+1)*h0). The result is a vector of time points and states, with at least (2*order+1). Potentially, the adaptive steps selected many more steps, but because of the events, fewer steps cannot have happened.

1. A prescribed prior is fitted to the first (2*order+1) (t, y) pairs of the solution. order is the order of the prior.

3. The value of the resulting posterior at time t=t0 is an estimate of the state and all its derivatives. The resulting marginal standard deviations estimate the error. This random variable is returned.

Parameters
Returns

Estimated (improved) initial random variable. Compatible with the specified prior.

Return type

Normal