Multilayer Net

Splashery stretches a drumhead over this Splash's outline, twelve inches across, and finds the ways it can ring. Its lowest is 370.5 Hz, 98 cents flat of G4, the nearest tine on the comb. The next fifteen are its overtones, and each one draws its own figure: sand gathers on the lines that stay still.

Tap it to try all four inputs. Hands-on: tap each input to flip it; XOR answers.

One perceptron can only split its inputs with a single straight line, so it cannot learn XOR, “exclusive or”, which is 1 when exactly one of two inputs is 1. Put neurons in layers and it can. Here an OR neuron and a NAND (“not both”) neuron feed an AND neuron, and together they give XOR. Blue wires add to a neuron's sum and red wires subtract.

Tap it and it tries the inputs 00, 01, 10 and 11 in turn, lighting each neuron that fires, and fills in the truth table: 0, 1, 1, 0. Pick the poster or a 3D model in the Splash tab. With ✋ Hands-on on, tap each input lamp to flip it: the OR and NAND neurons and the output show XOR's answer at once, and the truth table marks the row.

The hidden units produce the XOR truth table correctly. The splash demonstrates a fixed nonlinear circuit; it does not learn XOR from examples.

1. Correct

   A single linear decision boundary cannot represent XOR; hidden nonlinear units can.¹

   The fixed hidden units compute OR and NAND; their final threshold computes XOR for all four input pairs.

2. Correct

   The XOR output is one when exactly one input is one.²

   The actual XOR helper returns 0, 1, 1, 0 for 00, 01, 10, 11.

3. Simplified

   The picture illustrates a multilayer network, not a trained model.¹

   The weights and thresholds are stored constants; tapping cycles the four input cases.

1. 1Multilayer perceptrons, Dive into Deep Learning authors
2. 2Half adder, All About Circuits textbook

Simplified

The XOR network uses fixed threshold units. Its truth table is real, but there is no loss calculation or training step.

Made in Splashery. Its code is MIT.

This link opens it in the Playground. To put it on your own page, add the element and one script tag. Every option.