Tutorial `Learner1D`#

Note

Because this documentation consists of static html, the live_plot and live_info widget is not live. Download the notebook in order to see the real behaviour. [1]

scalar output: `f:ℝ → ℝ`#

We start with the most common use-case: sampling a 1D function f: ℝ → ℝ.

We will use the following function, which is a smooth (linear) background with a sharp peak at a random location:

offset = random.uniform(-0.5, 0.5)


def f(x, offset=offset, wait=True):
    from random import random
    from time import sleep

    a = 0.01
    if wait:
        sleep(random() / 10)
    return x + a**2 / (a**2 + (x - offset) ** 2)

We start by initializing a 1D “learner”, which will suggest points to evaluate, and adapt its suggestions as more and more points are evaluated.

learner = adaptive.Learner1D(f, bounds=(-1, 1))

Next we create a “runner” that will request points from the learner and evaluate ‘f’ on them.

By default on Unix-like systems the runner will evaluate the points in parallel using local processes concurrent.futures.ProcessPoolExecutor.

On Windows systems the runner will use a loky.get_reusable_executor. A ProcessPoolExecutor cannot be used on Windows for reasons.

# The end condition is when the "loss" is less than 0.01. In the context of the
# 1D learner this means that we will resolve features in 'func' with width 0.01 or wider.
runner = adaptive.Runner(learner, loss_goal=0.01)

When instantiated in a Jupyter notebook the runner does its job in the background and does not block the IPython kernel. We can use this to create a plot that updates as new data arrives:

runner.live_info()

runner.live_plot(update_interval=0.1)

We can now compare the adaptive sampling to a homogeneous sampling with the same number of points:

if not runner.task.done():
    raise RuntimeError(
        "Wait for the runner to finish before executing the cells below!"
    )

learner2 = adaptive.Learner1D(f, bounds=learner.bounds)

xs = np.linspace(*learner.bounds, len(learner.data))
learner2.tell_many(xs, map(partial(f, wait=False), xs))

learner.plot() + learner2.plot()

vector output: `f:ℝ → ℝ^N`#

Sometimes you may want to learn a function with vector output:

random.seed(0)
offsets = [random.uniform(-0.8, 0.8) for _ in range(3)]

# sharp peaks at random locations in the domain


def f_levels(x, offsets=offsets):
    a = 0.01
    return np.array(
        [offset + x + a**2 / (a**2 + (x - offset) ** 2) for offset in offsets]
    )

adaptive has you covered! The Learner1D can be used for such functions:

learner = adaptive.Learner1D(f_levels, bounds=(-1, 1))
runner = adaptive.Runner(
    learner, loss_goal=0.01
)  # continue until `learner.loss()<=0.01`

runner.live_info()

runner.live_plot(update_interval=0.1)

Looking at curvature#

By default adaptive will sample more points where the (normalized) euclidean distance between the neighboring points is large. You may achieve better results sampling more points in regions with high curvature. To do this, you need to tell the learner to look at the curvature by specifying loss_per_interval.

from adaptive.learner.learner1D import (
    curvature_loss_function,
    default_loss,
    uniform_loss,
)

curvature_loss = curvature_loss_function()
learner = adaptive.Learner1D(f, bounds=(-1, 1), loss_per_interval=curvature_loss)
runner = adaptive.Runner(learner, loss_goal=0.01)

runner.live_info()

runner.live_plot(update_interval=0.1)

We may see the difference of homogeneous sampling vs only one interval vs including the nearest neighboring intervals in this plot. We will look at 100 points.

def sin_exp(x):
    from math import exp, sin

    return sin(15 * x) * exp(-(x**2) * 2)


learner_h = adaptive.Learner1D(sin_exp, (-1, 1), loss_per_interval=uniform_loss)
learner_1 = adaptive.Learner1D(sin_exp, (-1, 1), loss_per_interval=default_loss)
learner_2 = adaptive.Learner1D(sin_exp, (-1, 1), loss_per_interval=curvature_loss)

# adaptive.runner.simple is a non parallel blocking runner.
adaptive.runner.simple(learner_h, npoints_goal=100)
adaptive.runner.simple(learner_1, npoints_goal=100)
adaptive.runner.simple(learner_2, npoints_goal=100)

(
    learner_h.plot().relabel("homogeneous")
    + learner_1.plot().relabel("euclidean loss")
    + learner_2.plot().relabel("curvature loss")
).cols(2)

More info about using custom loss functions can be found in Custom adaptive logic for 1D and 2D.

Exporting the data#

We can view the raw data by looking at the dictionary learner.data. Alternatively, we can view the data as NumPy array with

learner.to_numpy()

array([[-1.00000000e+00, -9.99837596e-01],
       [-8.33333333e-01, -8.33071541e-01],
       [-6.66666667e-01, -6.66175921e-01],
       [-5.00000000e-01, -4.98767199e-01],
       [-4.16666667e-01, -4.14204927e-01],
       [-3.33333333e-01, -3.26198694e-01],
       [-2.91666667e-01, -2.74779579e-01],
       [-2.70833333e-01, -2.39352608e-01],
       [-2.60416667e-01, -2.13457265e-01],
       [-2.50000000e-01, -1.73045260e-01],
       [-2.39583333e-01, -9.39052558e-02],
       [-2.34375000e-01, -1.76035090e-02],
       [-2.29166667e-01,  1.15139783e-01],
       [-2.23958333e-01,  3.51362445e-01],
       [-2.21354167e-01,  5.14753480e-01],
       [-2.18750000e-01,  6.78540041e-01],
       [-2.16145833e-01,  7.77820386e-01],
       [-2.13541667e-01,  7.54224320e-01],
       [-2.10937500e-01,  6.25058076e-01],
       [-2.08333333e-01,  4.60704768e-01],
       [-2.03125000e-01,  1.97100177e-01],
       [-1.97916667e-01,  4.92997743e-02],
       [-1.92708333e-01, -2.96830566e-02],
       [-1.87500000e-01, -7.34168001e-02],
       [-1.77083333e-01, -1.13210733e-01],
       [-1.66666667e-01, -1.26208580e-01],
       [-1.45833333e-01, -1.25569456e-01],
       [-1.25000000e-01, -1.12902457e-01],
       [-8.33333333e-02, -7.76297414e-02],
       [-5.55111512e-17,  2.15133258e-03],
       [ 8.33333333e-02,  8.44528819e-02],
       [ 1.66666667e-01,  1.67351366e-01],
       [ 3.33333333e-01,  3.33665370e-01],
       [ 5.00000000e-01,  5.00195370e-01],
       [ 6.66666667e-01,  6.66795188e-01],
       [ 1.00000000e+00,  1.00006769e+00]])

If Pandas is installed (optional dependency), you can also run

df = learner.to_dataframe()
df

	x	y	function.offset	function.wait
0	-1.000000e+00	-0.999838	-0.215367	True
1	-8.333333e-01	-0.833072	-0.215367	True
2	-6.666667e-01	-0.666176	-0.215367	True
3	-5.000000e-01	-0.498767	-0.215367	True
4	-4.166667e-01	-0.414205	-0.215367	True
5	-3.333333e-01	-0.326199	-0.215367	True
6	-2.916667e-01	-0.274780	-0.215367	True
7	-2.708333e-01	-0.239353	-0.215367	True
8	-2.604167e-01	-0.213457	-0.215367	True
9	-2.500000e-01	-0.173045	-0.215367	True
10	-2.395833e-01	-0.093905	-0.215367	True
11	-2.343750e-01	-0.017604	-0.215367	True
12	-2.291667e-01	0.115140	-0.215367	True
13	-2.239583e-01	0.351362	-0.215367	True
14	-2.213542e-01	0.514753	-0.215367	True
15	-2.187500e-01	0.678540	-0.215367	True
16	-2.161458e-01	0.777820	-0.215367	True
17	-2.135417e-01	0.754224	-0.215367	True
18	-2.109375e-01	0.625058	-0.215367	True
19	-2.083333e-01	0.460705	-0.215367	True
20	-2.031250e-01	0.197100	-0.215367	True
21	-1.979167e-01	0.049300	-0.215367	True
22	-1.927083e-01	-0.029683	-0.215367	True
23	-1.875000e-01	-0.073417	-0.215367	True
24	-1.770833e-01	-0.113211	-0.215367	True
25	-1.666667e-01	-0.126209	-0.215367	True
26	-1.458333e-01	-0.125569	-0.215367	True
27	-1.250000e-01	-0.112902	-0.215367	True
28	-8.333333e-02	-0.077630	-0.215367	True
29	-5.551115e-17	0.002151	-0.215367	True
30	8.333333e-02	0.084453	-0.215367	True
31	1.666667e-01	0.167351	-0.215367	True
32	3.333333e-01	0.333665	-0.215367	True
33	5.000000e-01	0.500195	-0.215367	True
34	6.666667e-01	0.666795	-0.215367	True
35	1.000000e+00	1.000068	-0.215367	True

and load that data into a new learner with

new_learner = adaptive.Learner1D(learner.function, (-1, 1))  # create an empty learner
new_learner.load_dataframe(df)  # load the pandas.DataFrame's data
new_learner.plot()

Tutorial Learner1D#

scalar output: f:ℝ → ℝ#

vector output: f:ℝ → ℝ^N#

Looking at curvature#

Exporting the data#

Tutorial `Learner1D`#

scalar output: `f:ℝ → ℝ`#

vector output: `f:ℝ → ℝ^N`#