brainpy.integrators.sde.normal.Milstein#

class brainpy.integrators.sde.normal.Milstein(f, g, dt=None, name=None, show_code=False, var_type=None, intg_type=None, wiener_type=None, state_delays=None, dyn_vars=None)[source]#

Milstein method for Ito or Stratonovich integrals.

The Milstein scheme represents the order 1.0 strong Taylor scheme. For the Ito integral,

\[\begin{split}\begin{aligned} &Y_{n+1}=Y_{n}+f_{n} h+g_{n} \Delta W_{n}+\frac{1}{2} g_{n} g_{n}^{\prime}\left[\left(\Delta W_{n}\right)^{2}-h\right] \\ &\Delta W_{n}=\left[W_{t+h}-W_{t}\right] \sim \sqrt{h} \mathcal{N}(0,1) \end{aligned}\end{split}\]

where \(g_{n}^{\prime}=\frac{d g\left(Y_{n}\right)}{d Y_{n}}\) is the first derivative of \(g_n\).

For the Stratonovich integral, the Milstein method is given by

\[\begin{split}\begin{aligned} &Y_{n+1}=Y_{n}+f_{n} h+g_{n} \Delta W_{n}+\frac{1}{2} g_{n} g_{n}^{\prime}\left(\Delta W_{n}\right)^{2} \\ &\Delta W_{n}=\left[W_{t+h}-W_{t}\right] \sim \sqrt{h} \mathcal{N}(0,1) \end{aligned}\end{split}\]
__init__(f, g, dt=None, name=None, show_code=False, var_type=None, intg_type=None, wiener_type=None, state_delays=None, dyn_vars=None)[source]#

Methods

__init__(f, g[, dt, name, show_code, ...])

load_states(filename[, verbose])

Load the model states.

nodes([method, level, include_self])

Collect all children nodes.

register_implicit_nodes(*nodes, **named_nodes)

register_implicit_vars(*variables, ...)

save_states(filename[, variables])

Save the model states.

set_integral(f)

Set the integral function.

step(*args, **kwargs)

train_vars([method, level, include_self])

The shortcut for retrieving all trainable variables.

unique_name([name, type_])

Get the unique name for this object.

vars([method, level, include_self])

Collect all variables in this node and the children nodes.

Attributes

arguments

All arguments when calling the numer integrator of the differential equation.

dt

The numerical integration precision.

integral

The integral function.

name

Name of the model.

parameters

The parameters defined in the differential equation.

state_delays

State delays.

variables

The variables defined in the differential equation.