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Added ListGroup class for better management of a group of objects.

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This commit is contained in:
Alec Helbling
2022-04-18 23:56:31 -04:00
parent 0febbe547d
commit 05f512f185
19 changed files with 424 additions and 71 deletions
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73
manim_ml/list_group.py Normal file
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@ -0,0 +1,73 @@
from manim import *
class ListGroup(Mobject):
"""Indexable Group with traditional list operations"""
def __init__(self, *layers):
super().__init__()
self.items = [*layers]
def __getitem__(self, indices):
"""Traditional list indexing"""
return self.items[indices]
def insert(self, index, item):
"""Inserts item at index"""
self.items.insert(index, item)
self.submobjects = self.items
def remove_at_index(self, index):
"""Removes item at index"""
if index < 0 or index > len(self.items):
raise Exception(f"ListGroup index out of range: {index}")
item = self.items[index]
del self.items[index]
self.submobjects = self.items
return item
def remove_at_indices(self, indices):
"""Removes items at indices"""
items = []
for index in indices:
item = self.remove_at_index(index)
items.append(item)
return items
def remove(self, item):
"""Removes first instance of item"""
self.items.remove(item)
self.submobjects = self.items
return item
def get(self, index):
"""Gets item at index"""
return self.items[index]
def add(self, item):
"""Adds to end"""
self.items.append(item)
self.submobjects = self.items
def replace(self, index, item):
"""Replaces item at index"""
self.items[index] = item
self.submobjects = self.items
def index_of(self, item):
"""Returns index of item if it exists"""
for index, obj in enumerate(self.items):
if item is obj:
return index
return -1
def __len__(self):
"""Length of items"""
return len(self.items)
def set_z_index(self, z_index_value):
"""Sets z index of all values in ListGroup"""
for item in self.items:
item.set_z_index(z_index_value)

4
manim_ml/neural_network/layers/embedding.py
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@ -5,8 +5,8 @@ from manim_ml.neural_network.layers.parent_layers import VGroupNeuralNetworkLaye
class EmbeddingLayer(VGroupNeuralNetworkLayer):
"""NeuralNetwork embedding object that can show probability distributions"""
def __init__(self, point_radius=0.02):
super(EmbeddingLayer, self).__init__()
def __init__(self, point_radius=0.02, **kwargs):
super(EmbeddingLayer, self).__init__(**kwargs)
self.point_radius = point_radius
self.axes = Axes(
tips=False,

6
manim_ml/neural_network/layers/embedding_to_feed_forward.py
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@ -8,8 +8,10 @@ class EmbeddingToFeedForward(ConnectiveLayer):
input_class = EmbeddingLayer
output_class = FeedForwardLayer
def __init__(self, input_layer, output_layer, animation_dot_color=RED, dot_radius=0.03):
super().__init__(input_layer, output_layer, input_class=EmbeddingLayer, output_class=FeedForwardLayer)
def __init__(self, input_layer, output_layer, animation_dot_color=RED, dot_radius=0.03,
**kwargs):
super().__init__(input_layer, output_layer, input_class=EmbeddingLayer, output_class=FeedForwardLayer,
**kwargs)
self.feed_forward_layer = output_layer
self.embedding_layer = input_layer
self.animation_dot_color = animation_dot_color

4
manim_ml/neural_network/layers/feed_forward.py
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@ -7,8 +7,8 @@ class FeedForwardLayer(VGroupNeuralNetworkLayer):
def __init__(self, num_nodes, layer_buffer=SMALL_BUFF/2, node_radius=0.08,
node_color=BLUE, node_outline_color=WHITE, rectangle_color=WHITE,
node_spacing=0.3, rectangle_fill_color=BLACK, node_stroke_width=2.0,
rectangle_stroke_width=2.0, animation_dot_color=RED):
super(VGroupNeuralNetworkLayer, self).__init__()
rectangle_stroke_width=2.0, animation_dot_color=RED, **kwargs):
super(VGroupNeuralNetworkLayer, self).__init__(**kwargs)
self.num_nodes = num_nodes
self.layer_buffer = layer_buffer
self.node_radius = node_radius

6
manim_ml/neural_network/layers/feed_forward_to_embedding.py
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@ -8,8 +8,10 @@ class FeedForwardToEmbedding(ConnectiveLayer):
input_class = FeedForwardLayer
output_class = EmbeddingLayer
def __init__(self, input_layer, output_layer, animation_dot_color=RED, dot_radius=0.03):
super().__init__(input_layer, output_layer, input_class=FeedForwardLayer, output_class=EmbeddingLayer)
def __init__(self, input_layer, output_layer, animation_dot_color=RED, dot_radius=0.03,
**kwargs):
super().__init__(input_layer, output_layer, input_class=FeedForwardLayer, output_class=EmbeddingLayer,
**kwargs)
self.feed_forward_layer = input_layer
self.embedding_layer = output_layer
self.animation_dot_color = animation_dot_color

5
manim_ml/neural_network/layers/feed_forward_to_feed_forward.py
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@ -9,8 +9,9 @@ class FeedForwardToFeedForward(ConnectiveLayer):
def __init__(self, input_layer, output_layer, passing_flash=True,
dot_radius=0.05, animation_dot_color=RED, edge_color=WHITE,
edge_width=0.5):
super().__init__(input_layer, output_layer, input_class=FeedForwardLayer, output_class=FeedForwardLayer)
edge_width=1.5, **kwargs):
super().__init__(input_layer, output_layer, input_class=FeedForwardLayer, output_class=FeedForwardLayer,
**kwargs)
self.passing_flash = passing_flash
self.edge_color = edge_color
self.dot_radius = dot_radius

5
manim_ml/neural_network/layers/feed_forward_to_image.py
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@ -9,8 +9,9 @@ class FeedForwardToImage(ConnectiveLayer):
output_class = ImageLayer
def __init__(self, input_layer, output_layer, animation_dot_color=RED,
dot_radius=0.05):
super().__init__(input_layer, output_layer, input_class=FeedForwardLayer, output_class=ImageLayer)
dot_radius=0.05, **kwargs):
super().__init__(input_layer, output_layer, input_class=FeedForwardLayer, output_class=ImageLayer
**kwargs)
self.animation_dot_color = animation_dot_color
self.dot_radius = dot_radius

5
manim_ml/neural_network/layers/image.py
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@ -5,9 +5,8 @@ from manim_ml.neural_network.layers.parent_layers import NeuralNetworkLayer
class ImageLayer(NeuralNetworkLayer):
"""Single Image Layer for Neural Network"""
def __init__(self, numpy_image, height=1.5):
super().__init__()
self.set_z_index(1)
def __init__(self, numpy_image, height=1.5, **kwargs):
super().__init__(**kwargs)
self.numpy_image = numpy_image
if len(np.shape(self.numpy_image)) == 2:
# Assumed Grayscale

5
manim_ml/neural_network/layers/image_to_feed_forward.py
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@ -9,8 +9,9 @@ class ImageToFeedForward(ConnectiveLayer):
output_class = FeedForwardLayer
def __init__(self, input_layer, output_layer, animation_dot_color=RED,
dot_radius=0.05):
super().__init__(input_layer, output_layer, input_class=ImageLayer, output_class=FeedForwardLayer)
dot_radius=0.05, **kwargs):
super().__init__(input_layer, output_layer, input_class=ImageLayer, output_class=FeedForwardLayer
**kwargs)
self.animation_dot_color = animation_dot_color
self.dot_radius = dot_radius

4
manim_ml/neural_network/layers/paired_query.py
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@ -6,8 +6,8 @@ import numpy as np
class PairedQueryLayer(NeuralNetworkLayer):
"""Paired Query Layer"""
def __init__(self, positive, negative, stroke_width=5):
super().__init__()
def __init__(self, positive, negative, stroke_width=5, **kwargs):
super().__init__(**kwargs)
self.positive = positive
self.negative = negative

5
manim_ml/neural_network/layers/paired_query_to_feed_forward.py
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@ -8,8 +8,9 @@ class PairedQueryToFeedForward(ConnectiveLayer):
input_class = PairedQueryLayer
output_class = FeedForwardLayer
def __init__(self, input_layer, output_layer, animation_dot_color=RED, dot_radius=0.02):
super().__init__(input_layer, output_layer, input_class=PairedQueryLayer, output_class=FeedForwardLayer)
def __init__(self, input_layer, output_layer, animation_dot_color=RED, dot_radius=0.02, **kwargs):
super().__init__(input_layer, output_layer, input_class=PairedQueryLayer, output_class=FeedForwardLayer
**kwargs)
self.animation_dot_color = animation_dot_color
self.dot_radius = dot_radius

10
manim_ml/neural_network/layers/parent_layers.py
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@ -4,9 +4,8 @@ from abc import ABC, abstractmethod
class NeuralNetworkLayer(ABC, Group):
"""Abstract Neural Network Layer class"""
def __init__(self, **kwargs):
def __init__(self, text=None, **kwargs):
super(Group, self).__init__()
self.set_z_index(1)
@abstractmethod
def make_forward_pass_animation(self):
@ -28,8 +27,9 @@ class ConnectiveLayer(VGroupNeuralNetworkLayer):
"""Forward pass animation for a given pair of layers"""
@abstractmethod
def __init__(self, input_layer, output_layer, input_class=None, output_class=None):
super(VGroupNeuralNetworkLayer, self).__init__()
def __init__(self, input_layer, output_layer, input_class=None, output_class=None,
**kwargs):
super(VGroupNeuralNetworkLayer, self).__init__(**kwargs)
self.input_layer = input_layer
self.output_layer = output_layer
self.input_class = input_class
@ -38,8 +38,6 @@ class ConnectiveLayer(VGroupNeuralNetworkLayer):
assert isinstance(input_layer, self.input_class)
assert isinstance(output_layer, self.output_class)
self.set_z_index(-1)
@abstractmethod
def make_forward_pass_animation(self):
pass

5
manim_ml/neural_network/layers/triplet.py
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@ -6,8 +6,9 @@ import numpy as np
class TripletLayer(NeuralNetworkLayer):
"""Shows triplet images"""
def __init__(self, anchor, positive, negative, stroke_width=5):
super().__init__()
def __init__(self, anchor, positive, negative, stroke_width=5,
**kwargs):
super().__init__(**kwargs)
self.anchor = anchor
self.positive = positive
self.negative = negative

5
manim_ml/neural_network/layers/triplet_to_feed_forward.py
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@ -9,8 +9,9 @@ class TripletToFeedForward(ConnectiveLayer):
output_class = FeedForwardLayer
def __init__(self, input_layer, output_layer, animation_dot_color=RED,
dot_radius=0.02):
super().__init__(input_layer, output_layer, input_class=TripletLayer, output_class=FeedForwardLayer)
dot_radius=0.02, **kwargs):
super().__init__(input_layer, output_layer, input_class=TripletLayer, output_class=FeedForwardLayer,
**kwargs)
self.animation_dot_color = animation_dot_color
self.dot_radius = dot_radius

20
manim_ml/neural_network/layers/util.py Normal file
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@ -0,0 +1,20 @@
from manim import *
from ..layers import connective_layers_list
def get_connective_layer(input_layer, output_layer):
"""
Deduces the relevant connective layer
"""
connective_layer = None
for connective_layer_class in connective_layers_list:
input_class = connective_layer_class.input_class
output_class = connective_layer_class.output_class
if isinstance(input_layer, input_class) \
and isinstance(output_layer, output_class):
connective_layer = connective_layer_class(input_layer, output_layer)
if connective_layer is None:
raise Exception(f"Unrecognized class pair {input_layer.__class__.__name__}" + \
" and {output_layer.__class__.__name__}")
return connective_layer

233
manim_ml/neural_network/neural_network.py
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@ -9,24 +9,23 @@ Example:
# Create the object with default style settings
NeuralNetwork(layer_node_count)
"""
from socket import create_connection
from urllib.parse import non_hierarchical
from manim import *
import warnings
import textwrap
from manim_ml.neural_network.layers import \
FeedForwardLayer, FeedForwardToFeedForward, ImageLayer, \
ImageToFeedForward, FeedForwardToImage, EmbeddingLayer, \
EmbeddingToFeedForward, FeedForwardToEmbedding, TripletLayer, \
TripletToFeedForward
from manim_ml.neural_network.layers import connective_layers_list
from manim_ml.neural_network.layers.feed_forward import FeedForwardLayer
from manim_ml.neural_network.layers.util import get_connective_layer
from manim_ml.list_group import ListGroup
class NeuralNetwork(Group):
def __init__(self, input_layers, edge_color=WHITE, layer_spacing=0.8,
animation_dot_color=RED, edge_width=1.5, dot_radius=0.03):
animation_dot_color=RED, edge_width=2.5, dot_radius=0.03):
super(Group, self).__init__()
self.input_layers = Group(*input_layers)
self.input_layers = ListGroup(*input_layers)
self.edge_width = edge_width
self.edge_color = edge_color
self.layer_spacing = layer_spacing
@ -37,6 +36,9 @@ class NeuralNetwork(Group):
# and make it have explicit distinct subspaces
self._place_layers()
self.connective_layers, self.all_layers = self._construct_connective_layers()
# Place layers at correct z index
self.connective_layers.set_z_index(2)
self.input_layers.set_z_index(3)
# Center the whole diagram by default
self.all_layers.move_to(ORIGIN)
self.add(self.all_layers)
@ -50,17 +52,14 @@ class NeuralNetwork(Group):
for layer_index in range(1, len(self.input_layers)):
previous_layer = self.input_layers[layer_index - 1]
current_layer = self.input_layers[layer_index]
current_layer.move_to(previous_layer)
shift_vector = np.array([(previous_layer.get_width()/2 + current_layer.get_width()/2) + 0.2, 0, 0])
current_layer.shift(shift_vector)
# Handle layering
self.input_layers.set_z_index(2)
def _construct_connective_layers(self):
"""Draws connecting lines between layers"""
connective_layers = Group()
all_layers = Group()
connective_layers = ListGroup()
all_layers = ListGroup()
for layer_index in range(len(self.input_layers) - 1):
current_layer = self.input_layers[layer_index]
all_layers.add(current_layer)
@ -72,28 +71,196 @@ class NeuralNetwork(Group):
if isinstance(next_layer, NeuralNetwork):
# First layer of the next layer
next_layer = next_layer.all_layers[0]
# Find connective layer with correct layer pair
connective_layer = None
for connective_layer_class in connective_layers_list:
input_class = connective_layer_class.input_class
output_class = connective_layer_class.output_class
if isinstance(current_layer, input_class) \
and isinstance(next_layer, output_class):
connective_layer = connective_layer_class(current_layer, next_layer)
connective_layers.add(connective_layer)
all_layers.add(connective_layer)
if connective_layer is None:
raise Exception(f"Unrecognized class pair {current_layer.__class__.__name__} and {next_layer.__class__.__name__}")
connective_layer = get_connective_layer(current_layer, next_layer)
connective_layers.add(connective_layer)
all_layers.add(connective_layer)
# Add final layer
all_layers.add(self.input_layers[-1])
# Handle layering
return connective_layers, all_layers
def insert_layer(self, layer, insert_index):
"""Inserts a layer at the given index"""
layers_before = self.all_layers[:insert_index]
layers_after = self.all_layers[insert_index:]
# Make connective layers and shift animations
# Before layer
if len(layers_before) > 0:
before_connective = get_connective_layer(layers_before[-1], layer)
before_shift = np.array([-layer.width/2, 0, 0])
# Shift layers before
before_shift_animation = Group(*layers_before).animate.shift(before_shift)
else:
before_connective = AnimationGroup()
# After layer
if len(layers_after) > 0:
after_connective = get_connective_layer(layer, layers_after[0])
after_shift = np.array([layer.width/2, 0, 0])
# Shift layers after
after_shift_animation = Group(*layers_after).animate.shift(after_shift)
else:
after_connective = AnimationGroup
# Make animation group
shift_animations = AnimationGroup(
before_shift_animation,
after_shift_animation
)
insert_animation = Create(layer)
animation_group = AnimationGroup(
shift_animations,
insert_animation,
lag_ratio=1.0
)
return animation_group
def remove_layer(self, layer):
"""Removes layer object if it exists"""
# Get layer index
layer_index = self.all_layers.index_of(layer)
if layer_index == -1:
raise Exception("Layer object not found")
# Get the layers before and after
before_layer = None
after_layer = None
if layer_index - 2 >= 0:
before_layer = self.all_layers[layer_index - 2]
if layer_index + 2 < len(self.all_layers):
after_layer = self.all_layers[layer_index + 2]
# Remove the layer
self.all_layers.remove(layer)
# Remove surrounding connective layers from self.all_layers
before_connective = None
after_connective = None
if layer_index - 1 >= 0:
# There is a layer before
before_connective = self.all_layers.remove_at_index(layer_index - 1)
if layer_index + 1 < len(self.all_layers):
# There is a layer after
after_connective = self.all_layers.remove_at_index(layer_index + 1)
# Make animations
# Fade out the removed layer
fade_out_removed = FadeOut(layer)
# Fade out the removed connective layers
fade_out_before_connective = Animation()
if not before_connective is None:
fade_out_before_connective = FadeOut(before_connective)
fade_out_after_connective = Animation()
if not after_connective is None:
fade_out_after_connective = FadeOut(after_connective)
# Create new connective layer
new_connective = None
if not before_layer is None and not after_layer is None:
new_connective = get_connective_layer(before_layer, after_layer)
before_layer_index = self.all_layers.index_of(before_layer)
self.all_layers.insert(before_layer_index, new_connective)
# Place the new connective
new_connective.move_to(layer)
# Animate the creation of the new connective layer
create_new_connective = Animation()
if not new_connective is None:
create_new_connective = Create(new_connective)
# Collapse the neural network to fill the empty space
removed_width = layer.width + before_connective.width + after_connective.width - new_connective.width
shift_right_amount = np.array([removed_width / 2, 0, 0])
shift_left_amount = np.array([-removed_width / 2, 0, 0])
move_before_layer = Animation()
if not before_layer is None:
move_before_layer = before_layer.animate.shift(shift_right_amount)
move_after_layer = Animation()
if not after_layer is None:
move_after_layer = after_layer.animate.shift(shift_left_amount)
# Make the final AnimationGroup
fade_out_group = AnimationGroup(
fade_out_removed,
fade_out_before_connective,
fade_out_after_connective
)
move_group = AnimationGroup(
move_before_layer,
move_after_layer
)
animation_group = AnimationGroup(
fade_out_group,
move_group,
create_new_connective,
lag_ratio=1.0
)
return animation_group
"""
remove_layer = list(self.all_layers)[remove_index]
if remove_index > 0:
connective_before = list(self.all_layers)[remove_index - 1]
else:
connective_before = None
if remove_index < len(list(self.all_layers)) - 1:
connective_after = list(self.all_layers)[remove_index + 1]
else:
connective_after = None
# Collapse the surrounding layer
layers_before = list(self.all_layers)[:remove_index]
layers_after = list(self.all_layers)[remove_index+1:]
before_group = Group(*layers_before)
after_group = Group(*layers_after)
before_shift_amount = np.array([remove_layer.width/2, 0, 0])
after_shift_amount = np.array([-remove_layer.width/2, 0, 0])
# Remove the layers from the neural network representation
self.all_layers.remove(remove_layer)
if not connective_before is None:
self.all_layers.remove(connective_before)
if not connective_after is None:
self.all_layers.remove(connective_after)
# Connect the layers before and layers after
pre_index = remove_index - 1
pre_layer = None
if pre_index >= 0:
pre_layer = list(self.all_layers)[pre_index]
post_index = remove_index
post_layer = None
if post_index < len(list(self.all_layers)):
post_layer = list(self.all_layers)[post_index]
if not pre_layer is None and not post_layer is None:
connective_layer = get_connective_layer(pre_layer, post_layer)
self.all_layers = Group(
*self.all_layers[:remove_index],
connective_layer,
*self.all_layers[remove_index:]
)
# Make animations
fade_out_animation = FadeOut(remove_layer)
shift_animations = AnimationGroup(
before_group.animate.shift(before_shift_amount),
after_group.animate.shift(after_shift_amount)
)
animation_group = AnimationGroup(
fade_out_animation,
shift_animations,
lag_ratio=1.0
)
return animation_group
"""
def replace_layer(self, old_layer, new_layer):
"""Replaces given layer object"""
remove_animation = self.remove_layer(insert_index)
insert_animation = self.insert_layer(layer, insert_index)
# Make the animation
animation_group = AnimationGroup(
FadeOut(self.all_layers[insert_index]),
FadeIn(layer),
lag_ratio=1.0
)
return animation_group
def make_forward_pass_animation(self, run_time=10, passing_flash=True):
"""Generates an animation for feed forward propogation"""
"""Generates an animation for feed forward propagation"""
all_animations = []
for layer_index, layer in enumerate(self.input_layers[:-1]):
layer_forward_pass = layer.make_forward_pass_animation()
@ -126,19 +293,11 @@ class NeuralNetwork(Group):
return animation_group
def remove_layer(self, layer_index):
"""Removes layer at given index and returns animation for removing the layer"""
raise NotImplementedError()
def add_layer(self, layer):
"""Adds layer and returns animation for adding action"""
raise NotImplementedError()
def __repr__(self):
"""Print string representation of layers"""
inner_string = ""
for layer in self.all_layers:
inner_string += f"{repr(layer)},\n"
inner_string += f"{repr(layer)} {layer.z_index} ,\n"
inner_string = textwrap.indent(inner_string, " ")
string_repr = "NeuralNetwork([\n" + inner_string + "])"

1
manim_ml/probability.py
View File

@ -16,7 +16,6 @@ class GaussianDistribution(VGroup):
self.cov = np.array([[3, 0], [0, 3]])
# Make the Gaussian
self.ellipses = self.construct_gaussian_distribution(self.mean, self.cov)
self.ellipses.set_z_index(2)
@override_animation(Create)
def _create_gaussian_distribution(self):

95
tests/test_neural_network.py
View File

@ -1,6 +1,10 @@
from cv2 import exp
from manim import *
from manim_ml.neural_network.layers.embedding import EmbeddingLayer
from manim_ml.neural_network.layers.embedding_to_feed_forward import EmbeddingToFeedForward
from manim_ml.neural_network.layers.feed_forward import FeedForwardLayer
from manim_ml.neural_network.layers.feed_forward_to_embedding import FeedForwardToEmbedding
from manim_ml.neural_network.layers.feed_forward_to_feed_forward import FeedForwardToFeedForward
from manim_ml.neural_network.layers.image import ImageLayer
from manim_ml.neural_network.neural_network import NeuralNetwork, FeedForwardNeuralNetwork
from PIL import Image
@ -11,6 +15,72 @@ config.pixel_width = 1280
config.frame_height = 6.0
config.frame_width = 6.0
"""
Unit Tests
"""
def assert_classes_match(all_layers, expected_classes):
assert len(list(all_layers)) == 5
for index, layer in enumerate(all_layers):
expected_class = expected_classes[index]
assert isinstance(layer, expected_class), f"Wrong layer class {layer.__class__} expected {expected_class}"
def test_embedding_layer():
embedding_layer = EmbeddingLayer()
neural_network = NeuralNetwork([
FeedForwardLayer(5),
FeedForwardLayer(3),
embedding_layer
])
expected_classes = [
FeedForwardLayer,
FeedForwardToFeedForward,
FeedForwardLayer,
FeedForwardToEmbedding,
EmbeddingLayer
]
assert_classes_match(neural_network.all_layers, expected_classes)
def test_remove_layer():
embedding_layer = EmbeddingLayer()
neural_network = NeuralNetwork([
FeedForwardLayer(5),
FeedForwardLayer(3),
embedding_layer
])
expected_classes = [
FeedForwardLayer,
FeedForwardToFeedForward,
FeedForwardLayer,
FeedForwardToEmbedding,
EmbeddingLayer
]
assert_classes_match(neural_network.all_layers, expected_classes)
print("before removal")
print(list(neural_network.all_layers))
neural_network.remove_layer(embedding_layer)
print("after removal")
print(list(neural_network.all_layers))
expected_classes = [
FeedForwardLayer,
FeedForwardToFeedForward,
FeedForwardLayer,
]
print(list(neural_network.all_layers))
assert_classes_match(neural_network.all_layers, expected_classes)
class FeedForwardNeuralNetworkScene(Scene):
def construct(self):
@ -92,6 +162,31 @@ class RecursiveNNScene(Scene):
self.play(Create(nn))
class LayerInsertionScene(Scene):
def construct(self):
pass
class LayerRemovalScene(Scene):
def construct(self):
image = Image.open('images/image.jpeg')
numpy_image = np.asarray(image)
layer = FeedForwardLayer(5),
layers = [
ImageLayer(numpy_image, height=1.4),
FeedForwardLayer(3),
layer,
FeedForwardLayer(3),
FeedForwardLayer(6)
]
nn = NeuralNetwork(layers)
self.play(Create(nn))
self.play(nn.remove_layer(layer))
if __name__ == "__main__":
"""Render all scenes"""
# Feed Forward Neural Network