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https://github.com/matlab-deep-learning/constrained-deep-learning

Constrained deep learning is an advanced approach to training deep neural networks by incorporating domain-specific constraints into the learning process.
https://github.com/matlab-deep-learning/constrained-deep-learning

ai-verification convex convex-neural-network deep-learning deep-learning-algorithms lipschitz lipschitz-network monotonic monotonicity neural-networks

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Constrained deep learning is an advanced approach to training deep neural networks by incorporating domain-specific constraints into the learning process.

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README 

        
# AI Verification: Constrained Deep Learning



[![Open in MATLAB

Online](https://www.mathworks.com/images/responsive/global/open-in-matlab-online.svg)](https://matlab.mathworks.com/open/github/v1?repo=matlab-deep-learning/constrained-deep-learning)



Constrained deep learning is an advanced approach to training deep neural

networks by incorporating domain-specific constraints into the learning process.

By integrating these constraints into the construction and training of neural

networks, you can guarantee desirable behaviour in safety-critical scenarios

where such guarantees are paramount.



This project aims to develop and evaluate deep learning models that adhere to

predefined constraints, which could be in the form of physical laws, logical

rules, or any other domain-specific knowledge. In the context of AI

verification, constrained deep learning provides guarantees that certain

desirable properties are present in the trained neural network by design. These

desirable properties could include monotonicity, boundedness, and robustness

amongst others.





    




By bringing together the concepts of monotonicity, convexity, and Lipschitz

continuity, this repository serves as a comprehensive resource for embedding

essential constraints into deep learning models, addressing the complex needs of

safety-critical systems and fostering the convergence of theoretical principles

with practical AI verification applications.



You can learn more about monotonicity and Lipschitz continuity in the context of

aerospace applications in the "Formal Methods Use for Learning Assurance" report

from EASA and Collins Aerospace [1].



## Get Started



Download or clone this repository to your machine and open it in MATLAB®.

Add the conslearn directory and subfolders to the search path. Go to the

location of the repository and run the command: `addpath(genpath("conslearn"))`.



### Requirements



- [MATLAB](http://www.mathworks.com) R2024a or later

- [Deep Learning

  Toolbox™](https://www.mathworks.com/products/deep-learning.html)

- [Parallel Computing

  Toolbox™](https://uk.mathworks.com/products/parallel-computing.html)

  (recommended)

- [Optimization

  Toolbox™](https://www.mathworks.com/products/optimization.html)

- [Reinforcement Learning

  Toolbox™](https://www.mathworks.com/products/reinforcement-learning.html)

- [Image Processing

  Toolbox™](https://www.mathworks.com/products/image-processing.html)

- [Deep Learning Toolbox Verification

  Library](https://uk.mathworks.com/products/deep-learning-verification-library.html)



## Examples



The repository contains several introductory, interactive examples as well as

longer, real-world use case applications of constrained deep learning in the

context of AI verification. In the same directory as the markdown files, you can

find the Live Script (MLX) file that you can open in MATLAB and run

interactively to work through the example.



### Introductory Examples (Short)



Below are links for markdown versions of MATLAB Live Scripts that you can view

in GitHub®.



- [Fully input convex neural networks in

  1-dimension](examples/convex/introductory/PoC_Ex1_1DFICNN.md)

- [Fully input convex neural networks in

  n-dimensions](examples/convex/introductory/PoC_Ex2_nDFICNN.md)

- [Partially input convex neural networks in

  n-dimensions](examples/convex/introductory/PoC_Ex3_nDPICNN.md)

- [Fully input monotonic neural networks in

  1-dimension](examples/monotonic/introductory/PoC_Ex1_1DFMNN.md)

- [Fully input monotonic neural networks in

  n-dimensions](examples/monotonic/introductory/PoC_Ex2_nDFMNN.md)

- [Lipschitz continuous neural networks in

  1-dimensions](examples/lipschitz/introductory/PoC_Ex1_1DLNN.md)



These examples make use of [custom training

loops](https://uk.mathworks.com/help/deeplearning/deep-learning-custom-training-loops.html)

and the

[`arrayDatastore`](https://uk.mathworks.com/help/matlab/ref/matlab.io.datastore.arraydatastore.html)

object. To learn more, click the links.



### Workflow Examples (Long)



- [Dynamical System Modeling Using Convex Neural

ODE](examples/convex/neuralODE/TrainConvexNeuralODENetworkWithEulerODESolverExample.md)

This example works through the modeling of a dynamical system using a neural

ODE, where the underlying dynamics is captured by a fully input convex neural

network and the ODE solver uses a convex update method, for example, the Euler

method. The example shows how the network is expressive enough to capture

nonlinear dynamics and also provides boundedness guarantees on the solution

trajectories owing to the convex constraint of the underlying network and ODE

solver.



- [Train Fully Convex Neural Networks for CIFAR-10 Image

Classification](examples/convex/classificationCIFAR10/TrainICNNOnCIFAR10Example.md)

This example shows the expressive capabilities of fully convex networks by

obtaining high training accuracy on image classification on the natural image

dataset, CIFAR-10.



- [Remaining Useful Life Estimation Using Monotonic Neural

Networks](examples/monotonic/RULEstimateUsingMonotonicNetworks/RULEstimationUsingMonotonicNetworksExample.md)

This example shows how to guarantee monotonic decreasing prediction on a

remaining useful life (RUL) tasks by combining partially and fully monotonic

networks. This example looks at predicting the RUL for turbofan engine

degradation.



- [Train Image Classification Lipschitz Constrained Networks and Measure

Robustness to Adversarial

Examples](examples/lipschitz/classificationDigits/LipschitzClassificationNetworksRobustToAdversarialExamples.md)

This example shows how Lipschitz continuous constrained networks improve the

robustness of neural networks against adversarial attack. In this example, you

use formal verification methods to compute the number of robust images in the

test set against adversarial perturbation for several networks with decreasing

upper bound Lipschitz constants. You find a smaller Lipschitz constant gives a

more robust classification network.



## Functions



This repository introduces the following functions that are used throughout the

examples:



- [`buildConstrainedNetwork`](conslearn/buildConstrainedNetwork.m) - Build a multi-layer perceptron (MLP) with constraints on the architecture and initialization of the weights.

- [`buildConvexCNN`](conslearn/buildConvexCNN.m) - Build a fully-inpt convex convolutional neural network (CNN). 

- [`trainConstrainedNetwork`](conslearn/trainConstrainedNetwork.m) - Train a

  constrained network and maintain the constraint during training.

- [`lipschitzUpperBound`](conslearn/lipschitzUpperBound.m) - Compute an upper

  bound on the Lipschitz constant for a Lipschitz neural network.

- [`convexNetworkOutputBounds`](conslearn/convexNetworkOutputBounds.m) - Compute

  guaranteed upper and lower bounds on hypercubic grids for convex networks.



## Tests



This repository also contains tests for the software in the conslearn package.



As discussed in [1] (see 3.4.1.5), in certain situations, small violations in

the constraints may be admissible. For example, a small violation in

monotonicity may be admissible if the non-monotonic behaviour is kept below a

pre-defined threshold. In the system tests, you will see examples of tests that

incorporate an admissibility constant. This can account for violations owing to

floating point error for instance.



## Technical Articles



This repository focuses on the development and evaluation of deep learning

models that adhere to constraints crucial for safety-critical applications, such

as predictive maintenance for industrial machinery and equipment. Specifically,

it focuses on enforcing monotonicity, convexity, and Lipschitz continuity within

neural networks to ensure predictable and controlled behavior. By emphasizing

constraints like monotonicity, constrained neural networks ensure that

predictions of the Remaining Useful Life (RUL) of components behave intuitively:

as a machine's condition deteriorates, the estimated RUL should monotonically

decrease. This is crucial in applications like aerospace or manufacturing, where

an accurate and reliable estimation of RUL can prevent failures and save costs.

Alongside monotonicity, Lipschitz continuity is also enforced to guarantee model

robustness and controlled behavior. This is essential in environments where

safety and precision are paramount such as control systems in autonomous

vehicles or precision equipment in healthcare. Convexity is especially

beneficial for control systems as it inherently provides boundedness properties.

For instance, by ensuring that the output of a neural network lies within a

convex hull, it is possible to guarantee that the control commands remain within

a safe and predefined operational space, preventing erratic or unsafe system

behaviors. This boundedness property, derived from the convex nature of the

model's output space, is critical for maintaining the integrity and safety of

control systems under various conditions.



These technical articles explain key concepts of AI verification in the context

of constrained deep learning. They include discussions on how to achieve the

specified constraints in neural networks at construction and training time, as

well as deriving and proving useful properties of constrained networks in AI

verification applications. It is not necessary to go through these articles in

order to explore this repository, however, you can find references and more in

depth discussion here.



- [AI Verification: Monotonicity](documentation/AI-Verification-Monotonicity.md) -

  Discussion on fully and partially monotonic neural networks and proveable

  trends. This article introduces monotonic network architectures and

  restrictions on weights to guarantee monotonic behaviour.

- [AI Verification: Convexity](documentation/AI-Verification-Convexity.md) -

  Discussion on fully and partially convex neural networks and proveable

  guarantees of boundedness over hypercubic grids. This article contains proofs

  on how to prove boundedness properties of a convex neural network on

  hypercubic grids by analyzing the network and its derivative at the vertices.

- [AI Verification: Lipschitz

  Continuity](documentation/AI-Verification-Lipschitz.md) - Discussion on

  Lipschitz continuous neural networks and proveable guarantees of robustness.

  This article introduce Lipschitz continuity and how to compute an upper bound

  on the Lipschitz constant for a set of network architectures.



## References



- [1] EASA and Collins Aerospace, Formal Methods use for Learning Assurance

  (ForMuLA), April 2023,

  ,

  

- [2] Amos, Brandon, et al. Input Convex Neural Networks. arXiv:1609.07152,

  arXiv, 14 June 2017. arXiv.org, .

- [3] Gouk, Henry, et al. “Regularisation of Neural Networks by Enforcing

  Lipschitz Continuity.” Machine Learning, vol. 110, no. 2, Feb. 2021, pp.

  393–416. DOI.org (Crossref), 

- [4] Kitouni, Ouail, et al. Expressive Monotonic Neural Networks.

  arXiv:2307.07512, arXiv, 14 July 2023. arXiv.org,

  .



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