# Implemented algorithms¶

The core concept in `adaptive`

is that of a *learner*. A *learner*
samples a function at the best places in its parameter space to get
maximum “information” about the function. As it evaluates the function
at more and more points in the parameter space, it gets a better idea of
where the best places are to sample next.

Of course, what qualifies as the “best places” will depend on your
application domain! `adaptive`

makes some reasonable default choices,
but the details of the adaptive sampling are completely customizable.

The following learners are implemented:

`Learner1D`

, for 1D functions`f: ℝ → ℝ^N`

,`Learner2D`

, for 2D functions`f: ℝ^2 → ℝ^N`

,`LearnerND`

, for ND functions`f: ℝ^N → ℝ^M`

,`AverageLearner`

, For stochastic functions where you want to average the result over many evaluations,`IntegratorLearner`

, for when you want to intergrate a 1D function`f: ℝ → ℝ`

.

Meta-learners (to be used with other learners):

`BalancingLearner`

, for when you want to run several learners at once, selecting the “best” one each time you get more points,`DataSaver`

, for when your function doesn’t just return a scalar or a vector.

In addition to the learners, `adaptive`

also provides primitives for
running the sampling across several cores and even several machines,
with built-in support for
concurrent.futures,
ipyparallel and
distributed.

# Examples¶

Here are some examples of how Adaptive samples vs. homogeneous sampling. Click
on the *Play* button or move the sliders.

```
import itertools
import adaptive
from adaptive.learner.learner1D import uniform_loss, default_loss
import holoviews as hv
import numpy as np
adaptive.notebook_extension(_inline_js=False)
%output holomap='scrubber'
```

`adaptive.Learner1D`

¶

```
%%opts Layout [toolbar=None]
def f(x, offset=0.07357338543088588):
a = 0.01
return x + a**2 / (a**2 + (x - offset)**2)
def plot_loss_interval(learner):
if learner.npoints >= 2:
x_0, x_1 = max(learner.losses, key=learner.losses.get)
y_0, y_1 = learner.data[x_0], learner.data[x_1]
x, y = [x_0, x_1], [y_0, y_1]
else:
x, y = [], []
return hv.Scatter((x, y)).opts(style=dict(size=6, color='r'))
def plot(learner, npoints):
adaptive.runner.simple(learner, lambda l: l.npoints == npoints)
return (learner.plot() * plot_loss_interval(learner))[:, -1.1:1.1]
def get_hm(loss_per_interval, N=101):
learner = adaptive.Learner1D(f, bounds=(-1, 1),
loss_per_interval=loss_per_interval)
plots = {n: plot(learner, n) for n in range(N)}
return hv.HoloMap(plots, kdims=['npoints'])
(get_hm(uniform_loss).relabel('homogeneous samping')
+ get_hm(default_loss).relabel('with adaptive'))
```

`adaptive.Learner2D`

¶

```
def ring(xy):
import numpy as np
x, y = xy
a = 0.2
return x + np.exp(-(x**2 + y**2 - 0.75**2)**2/a**4)
def plot(learner, npoints):
adaptive.runner.simple(learner, lambda l: l.npoints == npoints)
learner2 = adaptive.Learner2D(ring, bounds=learner.bounds)
xs = ys = np.linspace(*learner.bounds[0], learner.npoints**0.5)
xys = list(itertools.product(xs, ys))
learner2.tell_many(xys, map(ring, xys))
return (learner2.plot().relabel('homogeneous grid')
+ learner.plot().relabel('with adaptive')
+ learner2.plot(tri_alpha=0.5).relabel('homogeneous sampling')
+ learner.plot(tri_alpha=0.5).relabel('with adaptive')).cols(2)
learner = adaptive.Learner2D(ring, bounds=[(-1, 1), (-1, 1)])
plots = {n: plot(learner, n) for n in range(4, 1010, 20)}
hv.HoloMap(plots, kdims=['npoints']).collate()
```