brainpy.integrators.sde.normal.Euler#

class brainpy.integrators.sde.normal.Euler(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]#

Euler method for the Ito and Stratonovich integrals.

For Ito schema, the Euler method (also called as Euler-Maruyama method) is given by:

\[\begin{split}\begin{aligned} Y_{n+1} &=Y_{n}+f\left(Y_{n}\right) h_{n}+g\left(Y_{n}\right) \Delta W_{n} \\ \Delta W_{n} &=\left[W_{t+h}-W_{t}\right] \sim \sqrt{h} \mathcal{N}(0,1) \end{aligned}\end{split}\]

As the order of convergence for the Euler-Maruyama method is low (strong order of convergence 0.5, weak order of convergence 1), the numerical results are inaccurate unless a small step size is used. In fact, Euler-Maruyama represents the order 0.5 strong Taylor scheme.

For Stratonovich scheme, the Euler-Heun method has to be used instead of the Euler-Maruyama method

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

See also

Heun

__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.