dllaplacian

Laplacian of deep learning data

Since R2024b

Syntax

lap = dllaplacian(u,x,dim)

lap = dllaplacian(u,x,dim,EnableHigherDerivatives=tf)

Description

The Laplacian deep learning operation returns the Laplacian of neural network and model function outputs with respect to the specified input data and operation dimension.

lap = dllaplacian(u,x,dim) returns the Laplacian of the neural network or model function output u with respect to the specified input data x and operation dimension dim.

lap = dllaplacian(u,x,dim,EnableHigherDerivatives=tf) also specifies whether to enable higher derivatives by tracing the backward pass.

Examples

Evaluate Laplacian of Deep Learning Data

Create a neural network.

numInputChannels = 3;
numOutputChannels = 1;

layers = [
    featureInputLayer(numInputChannels)
    fullyConnectedLayer(numOutputChannels)
    tanhLayer];

net = dlnetwork(layers);

Load the training data. For the purposes of this example, generate some random data.

numObservations = 128;

X = rand(numInputChannels,numObservations);
X = dlarray(X,"CB");

T = rand(numOutputChannels,numObservations);
T = dlarray(T,"CB");

Define a model loss function that takes the network and data as input and returns the loss, gradients of the loss with respect to the learnable parameters, and the Laplacian of the predictions with respect to the input data.

function [loss,gradients,lap] = modelLoss(net,X,T)

Y = forward(net,X);
loss = l1loss(Y,T);

X = stripdims(X);
Y = stripdims(Y);

lap = dllaplacian(Y,X,1);
gradients = dlgradient(loss,net.Learnables);

end

Evaluate the model loss function using the dlfeval function.

[loss,gradients,lap] = dlfeval(@modelLoss,net,X,T);

View the size of the Laplacian.

size(lap)

ans = 1×2

     1   128

Input Arguments

u — Input

traced dlarray vector

Input, specified as a traced dlarray vector.

When the software evaluates a function with automatic differentiation enabled, the software traces the input dlarray objects. These are some contexts where the software traces dlarray objects:

• Inside loss functions that the trainnet function evaluates

• Inside forward functions that custom layers evaluate

• Inside model and model loss functions that the dlfeval function evaluates

The sizes of the dimensions not specified by the dim argument must match.

x — Input

traced dlarray matrix

Input, specified as a traced dlarray matrix.

When the software evaluates a function with automatic differentiation enabled, the software traces the input dlarray objects. These are some contexts where the software traces dlarray objects:

• Inside loss functions that the trainnet function evaluates

• Inside forward functions that custom layers evaluate

• Inside model and model loss functions that the dlfeval function evaluates

The sizes of the dimensions not specified by the dim argument must match.

dim — Operation dimension

positive integer

Operation dimension of u, specified as a positive integer.

The dllaplacian function treats the remaining dimensions of the data as independent batch dimensions.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

tf — Flag to enable higher-order derivatives

true or 1 (default) | false or 0

Flag to enable higher-order derivatives, specified as one of the following:

• true — Enable higher-order derivatives. Trace the backward pass so that the returned gradients can be used in further computations for subsequent calls to functions that compute derivatives using automatic differentiation (for example, dlgradient, dljacobian, dldivergence, and dllaplacian).

• false — Disable higher-order derivatives. Do not trace the backward pass. Use this option when you need to compute first-order derivatives only as this is usually quicker and requires less memory.

Output Arguments

lap — Laplacian

unformatted dlarray object

Laplacian, returned as an unformatted 1-by-N dlarray object, where N is the size of the batch dimension of the data. The value of lap(n) is $$\Delta u_n={\displaystyle \sum _{i=1}^K\frac{{\partial }^2{u}_n}{\partial x_{(i,n)}^2}}$$, where i indexes into the operation dimensions, and n indexes into the batch dimensions.

Version History

Introduced in R2024b