# Source code for probnum.diffeq._probsolve_ivp

"""Convenience functions for Gaussian filtering and smoothing.

References
----------
.. [1] https://arxiv.org/pdf/1610.05261.pdf
.. [2] https://arxiv.org/abs/1807.09737
.. [3] https://arxiv.org/abs/1810.03440
.. [4] https://arxiv.org/pdf/2004.00623.pdf
"""

import numpy as np

from probnum import problems, randprocs
from probnum.diffeq import odefilter, stepsize

__all__ = ["probsolve_ivp"]

[docs]def probsolve_ivp(
f,
t0,
tmax,
y0,
df=None,
method="EK0",
dense_output=True,
algo_order=2,
atol=1e-2,
rtol=1e-2,
step=None,
diffusion_model="dynamic",
time_stops=None,
):
r"""Solve an initial value problem with a filtering-based probabilistic ODE solver.

Numerically computes a Gauss-Markov process which solves numerically
the initial value problem (IVP) based on a system of first order
ordinary differential equations (ODEs)

.. math:: \dot x(t) = f(t, x(t)), \quad x(t_0) = x_0,

by regarding it as a (nonlinear) Gaussian filtering (and smoothing)
problem [3]_. For some configurations it recovers certain multistep
methods [1]_.
Convergence rates of filtering [2]_ and smoothing [4]_ are
comparable to those of methods of Runge-Kutta type.

This function turns a prior-string into an :class:ODEPrior, a
method-string into a filter/smoother of class :class:GaussFiltSmooth, creates a
:class:ODEFilter object and calls the :meth:solve() method. For
advanced usage we recommend to do this process manually which
enables advanced methods of tuning the algorithm.

This function supports the methods:
extended Kalman filtering based on a zero-th order Taylor
approximation (EKF0),
extended Kalman filtering (EKF1),
unscented Kalman filtering (UKF),
extended Kalman smoothing based on a zero-th order Taylor
approximation (EKS0),
extended Kalman smoothing (EKS1), and
unscented Kalman smoothing (UKS).

For adaptive step-size selection of ODE filters, we implement the
scheme proposed by Schober et al. (2019), and further examined by Bosch et al (2021),
where the local error estimate is derived from the local, calibrated
uncertainty estimate.

Arguments
---------
f :
ODE vector field.
t0 :
Initial time point.
tmax :
Final time point.
y0 :
Initial value.
df :
Jacobian of the ODE vector field.
Whether to use adaptive steps or not. Default is True.
If False, a step needs to be specified.
atol : float
Absolute tolerance  of the adaptive step-size selection scheme.
Optional. Default is 1e-4.
rtol : float
Relative tolerance   of the adaptive step-size selection scheme.
Optional. Default is 1e-4.
step :
Step size. If atol and rtol are not specified, this step-size is used for a fixed-step ODE solver.
If they are specified, this only affects the first step. Optional.
Default is None, in which case the first step is chosen as prescribed by :meth:propose_firststep.
Is required only when adaptive=False.
algo_order
Order of the algorithm. This amounts to choosing the number of derivatives of an integrated Wiener process prior.
For too high orders, process noise covariance matrices become singular.
For integrated Wiener processes, this maximum seems to be num_derivatives=11 (using standard float64).
It is possible that higher orders may work for you.
The type of prior relates to prior assumptions about the derivative of the solution.
The higher the order of the algorithm, the faster the convergence, but also, the higher-dimensional (and thus the costlier) the state space.
method : str, optional
Which method is to be used. Default is EK0 which is the
method proposed by Schober et al.. The available
options are

================================================  ==============
Extended Kalman filtering/smoothing (0th order)  'EK0'
Extended Kalman filtering/smoothing (1st order)  'EK1'
================================================  ==============

First order extended Kalman filtering and smoothing methods (EK1)
require Jacobians of the RHS-vector field of the IVP.
That is, the argument df needs to be specified.
They are likely to perform better than zeroth order methods in
terms of (A-)stability and "meaningful uncertainty estimates".
While we recommend to use correct capitalization for the method string,
lower-case letters will be capitalized internally.
dense_output : bool
Whether we want dense output. Optional. Default is True. For the ODE filter,
dense output requires smoothing, so if dense_output is False, no smoothing is performed;
but when it is True, the filter solution is smoothed.
diffusion_model : str
Which diffusion model to use. The choices are 'constant' and 'dynamic',
which implement different styles of
online calibration of the underlying diffusion [5]_.
Optional. Default is 'dynamic'.
time_stops: np.ndarray
Time-points through which the solver must step. Optional. Default is None.

Returns
-------
solution : ODEFilterSolution
Solution of the ODE problem.

Can be evaluated at and sampled from at arbitrary grid points.
Further, it contains fields:

t : :obj:np.ndarray, shape=(N,)
Mesh used by the solver to compute the solution.
It includes the initial time :math:t_0 but not necessarily the
final time :math:T.
y : :obj:list of :obj:RandomVariable, length=N
Discrete-time solution at times :math:t_1, ..., t_N,
as a list of random variables.
The means and covariances can be accessed with solution.y.mean
and solution.y.cov.

--------
ODEFilter : Solve IVPs with Gaussian filtering and smoothing
ODEFilterSolution : Solution of ODE problems based on Gaussian filtering and smoothing.

References
----------
.. [1] Schober, M., Särkkä, S. and Hennig, P..
A probabilistic model for the numerical solution of initial
value problems.
Statistics and Computing, 2019.
.. [2] Kersting, H., Sullivan, T.J., and Hennig, P..
Convergence rates of Gaussian ODE filters.
2019.
.. [3] Tronarp, F., Kersting, H., Särkkä, S., and Hennig, P..
Probabilistic solutions to ordinary differential equations as
non-linear Bayesian filtering: a new perspective.
Statistics and Computing, 2019.
.. [4] Tronarp, F., Särkkä, S., and Hennig, P..
Bayesian ODE solvers: the maximum a posteriori estimate.
2019.
.. [5] Bosch, N., and Hennig, P., and Tronarp, F..
2021.

Examples
--------
>>> from probnum.diffeq import probsolve_ivp
>>> import numpy as np

Solve a simple logistic ODE with fixed steps.

>>>
>>> def f(t, x):
...     return 4*x*(1-x)
>>>
>>> y0 = np.array([0.15])
>>> t0, tmax = 0., 1.5
>>> solution = probsolve_ivp(f, t0, tmax, y0, step=0.1, adaptive=False)
>>> print(np.round(solution.states.mean, 2))
[[0.15]
[0.21]
[0.28]
[0.37]
[0.47]
[0.57]
[0.66]
[0.74]
[0.81]
[0.87]
[0.91]
[0.94]
[0.96]
[0.97]
[0.98]
[0.99]]

Other methods are easily accessible.

>>> def df(t, x):
...     return np.array([4. - 8 * x])
>>> solution = probsolve_ivp(f, t0, tmax, y0, df=df, method="EK1", algo_order=2, step=0.1, adaptive=False)
>>> print(np.round(solution.states.mean, 2))
[[0.15]
[0.21]
[0.28]
[0.37]
[0.47]
[0.57]
[0.66]
[0.74]
[0.81]
[0.87]
[0.91]
[0.93]
[0.96]
[0.97]
[0.98]
[0.99]]

"""

# Create IVP object
ivp = problems.InitialValueProblem(t0=t0, tmax=tmax, y0=np.asarray(y0), f=f, df=df)

# Create steprule
if atol is None or rtol is None:
raise ValueError(
)
firststep = step if step is not None else stepsize.propose_firststep(ivp)
else:
if step is None:
raise ValueError("Constant steps require a 'step' argument.")
steprule = stepsize.ConstantSteps(step)

# Construct diffusion model.
diffusion_model = diffusion_model.lower()
if diffusion_model not in ["constant", "dynamic"]:
raise ValueError("Diffusion model is not supported.")

choose_diffusion_model = {
"constant": randprocs.markov.continuous.ConstantDiffusion(),
"dynamic": randprocs.markov.continuous.PiecewiseConstantDiffusion(t0=ivp.t0),
}
diffusion = choose_diffusion_model[diffusion_model]

# Create solver
prior_process = randprocs.markov.integrator.IntegratedWienerProcess(
initarg=ivp.t0,
num_derivatives=algo_order,
wiener_process_dimension=ivp.dimension,
diffuse=True,
forward_implementation="sqrt",
backward_implementation="sqrt",
)

info_op = odefilter.information_operators.ODEResidual(
num_prior_derivatives=prior_process.transition.num_derivatives,
ode_dimension=prior_process.transition.wiener_process_dimension,
)

choose_method = {
"EK0": odefilter.approx_strategies.EK0(),
"EK1": odefilter.approx_strategies.EK1(),
}
method = method.upper()
if method not in choose_method.keys():
raise ValueError("Method is not supported.")
approx_strategy = choose_method[method]

rk_init = odefilter.initialization_routines.RungeKuttaInitialization()

solver = odefilter.ODEFilter(
steprule,
prior_process,
information_operator=info_op,
approx_strategy=approx_strategy,
with_smoothing=dense_output,
initialization_routine=rk_init,
diffusion_model=diffusion,
)

return solver.solve(ivp=ivp, stop_at=time_stops)