Growing Cellular Automata

Differentiable Model of Morphogenesis

So I have a goal of implementing this work in Mathematica.

I’ll use the lizard image used in paper :

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lizard=

;

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//ImageDimensions

Out[]=

{1024,1024}

I create a function for building multi-channel convolution layers of arbitrary depth and kernel.

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convolution[kernel_,depth_Integer,bias_]:=ConvolutionLayerdepth,3,"Weights"TableRotateRightAppendConstantArray

0	0	0
0	0	0
0	0	0

,depth-1,kernel,i,{i,depth},"Biases"ConstantArray[bias,depth],PaddingSize1

I build the neural network for a single step evolution. It must only consist of convolutional or element-wise layers, as the rule for the evolution of the state of a cell must depend only on the state of the cell itself or of the state of its immediate neighbors.

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step[numberOfStates_Integer:16,hiddenConvolutionLayerSize_Integer:128]:=NetGraph<|"P"CatenateLayer[],"Σ"TotalLayer[],"∇x"convolution

1

4

-1	0	1
-2	0	2
-1	0	1

,numberOfStates,.5,"∇y"convolution

1

4

-1	-2	-1
0	0	0
1	2	1

,numberOfStates,.5,"I"convolution

0	0	0
0	1	0
0	0	0

,numberOfStates,0.,"D1"ConvolutionLayer[hiddenConvolutionLayerSize,1],"ReLU"Ramp,"D2"ConvolutionLayer[numberOfStates,1,"Weights"ConstantArray[0.,{numberOfStates,hiddenConvolutionLayerSize,1,1}],"Biases"ConstantArray[0.,numberOfStates]]|>,{NetPort["Input"]{"∇x","∇y","I","Σ"},"∇x"NetPort["P",1],"∇y"NetPort["P",2],"I"NetPort["P",3],"P""D1""ReLU""D2"->"Σ"}

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