probsolve_ivp

probnum.diffeq.probsolve_ivp(f, t0, tmax, y0, df=None, method='EK0', dense_output=True, algo_order=2, adaptive=True, atol=0.01, rtol=0.01, step=None, diffusion_model='dynamic')[source]

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)

\[\dot x(t) = f(t, x(t)), \quad x(t_0) = x_0, \quad t \in [t_0, T]\]

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 ODEPrior, a method-string into a filter/smoother of class GaussFiltSmooth, creates a GaussianIVPFilter object and calls the 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.

Parameters
  • f – ODE vector field.

  • t0 – Initial time point.

  • tmax – Final time point.

  • y0 – Initial value.

  • df – Jacobian of the ODE vector field.

  • adaptive – Whether to use adaptive steps or not. Default is True.

  • 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 propose_firststep().

  • 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'.

Returns

solution – Solution of the ODE problem.

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

tnp.ndarray, shape=(N,)

Mesh used by the solver to compute the solution. It includes the initial time \(t_0\) but not necessarily the final time \(T\).

ylist of RandomVariable, length=N

Discrete-time solution at times \(t_1, ..., t_N\), as a list of random variables. The means and covariances can be accessed with solution.y.mean and solution.y.cov.

Return type

KalmanODESolution

See also

GaussianIVPFilter

Solve IVPs with Gaussian filtering and smoothing

KalmanODESolution

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.. Calibrated Adaptive Probabilistic ODE Solvers. 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]]