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====== ➡️ Zusammenhänge erlernen ======

🎯 In diesem Kapitel untersuchst du, wie Zusammenhänge mithilfe eines neuronalen Netzes erlernt werden können.

~~INTOC~~

===== - Kugelstossen =====
[{{ :p:pasted:kugelstoss_small.png|Kugelstoss ((eigene Darstellung, [[https://creativecommons.org/publicdomain/zero/1.0/deed.de|CC0 1.0]])) }}]

Beim Kugelstossen wird eine Kugel (Männer: 7.260 kg, Frauen 4.0 kg) möglichst weit gestossen. Wie genau die Kugel fliegt, lässt sich mit den Bewegungsgleichungen der Physik berechnen (für Details siehe, z.B. [[wpde>Wurfparabel]]). 

In der Abbildung ist der aktuelle Kugelstossweltrekord der Männer 23.56 m dargestellt. Wird der Luftwiderstand nicht berücksichtigt, hängt die Weite nur von der Abwurfgeschwindigkeit, dem Abwurfwinkel und der Abwurfhöhe ab. Unter diesen Annahmen führt eine Abwurfgeschwindigkeit von 14.5 m/s, ein Abwurfwinkel von 40° und eine Abwurfhöhe von 2.3 m (grosser Kugelstösser mit leicht nach oben gestreckten Arm) in etwa zum Weltrekord ((in Anlehnung an: [[https://science-to-go.com/leichtathletik-bessere-weiten-dank-physik/|Leichtathletik: Bessere Weiten dank Physik?]])).

Mit einem neuronalen Netz wollen wir nun lernen, wie die Wurfweite von der der Abwurfgeschwindigkeit, dem Abwurfwinkel und der Abwurfhöhe abhängt. D.h. wir wollen den Zusammenhang Wurfweite, Abwurfgeschwindigkeit, Abwurfwinkel und Abwurfhöhe mit einem neuronalen Netz "modellieren".
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== ✍ Auftrag Kugelstossen 1==
💡 Die Kugelbewegung wurde gefilmt. Aus mehreren Standbildern wurden einzelne Positionen vermessen (Weite und Höhe der Kugel). Diese Messungen bilden die Trainingsdaten für das neuronale Netz. Die Trainingsdaten werden als blaue ☓ Kreuze dargestellt. 
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  - Starte das neuronales Netz Training, indem du auf den  Button ''▶Run'' klickst.\\ 💡 Es kann sein, dass die rote Linie ab und zu "springt". Das liegt daran, dass die Lernrate relativ gross eingestellt ist. Dafür läuft das Training einigermassen flott.
  - Wie gut schätzt das neuronale Netz die Wurfweite ab? Starte das Programm zwei oder drei Mal und notiere das beste Ergebnis. {{gem/plain?0=N4XyA#8da845d6746c1696}}
  - Wenn du Lust hast, kannst du noch mit den neuronalen Netz Parametern herumspielen oder andere Werte für Abwurfgeschwindigkeit, Abwurfwinkel und Abwurfhöhe ausprobieren.
</WRAP>
<WRAP center round box >
== ✍ Auftrag Kugelstossen 2==
💡 In diesem Beispiel werden weniger Trainingsdaten verwendet, wie wirkt sich das aus?
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  - Starte das neuronales Netz Training, indem du auf den  Button ''▶Run'' klickst.
  - Wie gut schätzt dieses Mal das neuronale Netz die Wurfweite ab? Starte das Programm zwei oder drei Mal, notiere das beste Ergebnis und vergleiche es mit dem Ergebnis aus dem Auftrag "Kugelstossen 1". {{gem/plain?0=N4XyA#65e9e45b8906b027}}
  - Wenn du Lust hast, kannst du auch hier mit den neuronalen Netz Parametern herumspielen oder andere Werte für Abwurfgeschwindigkeit, Abwurfwinkel und Abwurfhöhe ausprobieren.
</WRAP>
<WRAP center round box >
== ✍ Auftrag Kugelstossen 3==
💡 In diesem Beispiel werden unvollständige Trainingsdaten verwendet, wie wirkt sich das aus? 
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  - Starte das neuronales Netz Training, indem du auf den  Button ''▶Run'' klickst.
  - Wie gut schätzt hier das neuronale Netz die Wurfweite ab? Was macht das neuronale Netz im Bereich wo keine Trainingsdaten da sind? Was bedeutet das für andere neuronale Netze, wie z.B. einen KI-Chatbot (z.B. ChatGPT)? Halte deine Überlegungen im Textfeld fest. {{gem/plain?0=N4XyA#c3e16c0eab92579f}}
  - Vergleiche deine Überlegungen mit ++unseren Überlegungen|\\ \\ Wo das neuronale Netz keine Daten hat macht es irgendetwas. Wird z.B. in ChatGPT eine Frage eingegeben, welche selten oder nicht in den Trainingstexten vorkommt, so wird irgendetwas ausgegeben.\\ \\ ++.
</WRAP>

===== - Funktionen erraten =====

💡 Ein Zusammenhang kann mathematisch durch eine Funktion beschrieben werden. Wenn $x$ zum Resultat $y$ führt, dann wird das als $y = f(x)$ geschreiben, wobei $x$ eine Eingabe / Aktion / Beobachtung / etc. sein kann und die Funktion $f$ den gesuchten Zusammenhang beschreibt. Wenn wir mit einem neuronalen Netz eine Kurve durch Punkte legen, dann erlernt das neuronale Netz eine Funktion, welche durch alle Punkte geht. In der Fachsprache wird dies als "Funktionsapproximation" bezeichnet. 

<WRAP center round box >
== ✍ Auftrag Lustige Funktion ==

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  - Starte das neuronales Netz Training, indem du auf den  Button ''▶Run'' klickst und etwas wartest.\\ 💡 Es kann sein, dass es das neuronale Netz nicht schafft eine Linie durch alle Punkte zu legen. Starte in diesem Fall das Training einfach noch einmal neu. Da jedes Training mit anderen, zufällig gewählten Anfangsgewichten startet, führt jedes Training zu einem anderen Resultat. \\ 💡 Es kann sein, dass die rote Linie ab und zu "springt". Das liegt daran, dass die Lernrate relativ gross eingestellt ist. Dafür läuft das Training einigermassen flott.
  - Stelle die ''ANZAHL_HIDDEN_NEURONS'' auf ''50'' ein. Siehst du einen Unterschied? Wie kannst du dir das erklären? {{gem/plain?0=N4XyA#d93535b2b7fd4f84}}.
  - Vergleiche deine Überlegungen mit ++unseren Überlegungen|\\ \\ Das neuronale Netz mit fünf Hidden Neuronen erzeugt relativ glatte Kurven. Das neuronale Netz mit 50 Hidden Neuronen erzeugt manchmal besonders in der Nähe der Trainingsdaten (Kreuze) Ecken und Kanten. Für den Fehler ist es jedoch nur wichtig, dass die Trainingsdaten möglichst gut getroffen werden. Ob die rote Kurve neben den Trainingspunkten noch weitere Ecken und Kanten aufweist, spielt für den Fehler keine Rolle. Ein neuronales Netz mit "nur" fünf Hidden Neuronen kann nicht allzu viel lernen, dies genügt gerade um genügend viele Kurven zu erzeugen um die Aufgabe zu lösen, daher sieht diese Lösung "glatt" aus. Mit 50 Hidden Neuronen kann das neuronale Netz noch viel kompliziertere Funktionen erlernen, daher kann es sein, das es zufällig ausserhalb der Kreuze Ecken und Kanten bildet. Wo das neuronale Netz keine Daten hat, macht es irgendetwas...\\ \\ ++.
  - Wenn du Lust hast, darfst du eigene Trainingspunkte eingeben. Es gehören immer eine x-Koordinate und eine y-Koordinate zusammen (die erste x-Koordinate gehört zur ersten y-Koordinate, die zweite zur Zweiten usw.), daher braucht es gleich viele x-Koordinaten wie y-Koordinaten. Viel Spass!
</WRAP>

=== Eigene Notizen ===
{{gem/quill#3c8fe014b815befc}}