---
title: Research Approach
allDay: true
date: 2024-10-10
completed: 
type: single
endDate: 2024-10-22
---

5 Pages
**TARGET: 2800-3000 words**
# Outline
1. What are unperturbed perturbations and why are they important?
2. What is a generative diffusion model?
3. Where have generative diffusion models been used?
4. How will generative diffusion models generate new plants?
5.  How can we know if these new plants are in the set of 'controllable' plants?
6. How are we going to get training data?

Something to justify, why diffusion model as opposed to other generative AI

# First Draft
## Story
1. Generating unstructured perturbations is pretty hard
2. They need to be 'random'
3. diffusion model is good at creating novel examples (cite!)
4. we can use diffusion models to solve this problem of generating examples
5. But how does a diffusion model work?
6. Diffusion model uses two processes
7. a forward process that introduces noise into an 'input'
8. this forward process does this using several small steps of noise
9. These are called timesteps
10. This noise is a gaussian noise
11. over time this forward process degrades the input until it is unrecognizable
12. This takes several several iterations, but is dependent on the amount of noise in each step
13. The markov chain that this creates is also gaussian at every step, until the noise at the end of the day is some gaussian distribution. Usually mean 0 std. dev beta
14. A reverse process that tries to remove the noise
15. But if we destroy the input how can we do this?
16. Well we train a neural network as a denoiser.
17. Because the diffusion model forward steps are small and gaussian, we can know the reverse step is also a gaussian distribution. 
18. So for our neural network, what we're trying to learn is the mean and standard deviation of the reverse steps for a given timestep. 
19. What does this mean for perturbed plants?
20. Well, we can use a diffusion model to generate new perturbed plants that are similar to the original plant
21. How? We train a diffusion model on a structured set. 
22. We need to pick a uncertainty function W_2
23. Do this using the usual means. What is the worst we expect to handle?
24. We create a structured set by picking random scalar gains for $\Delta$
25. This is our training data. We train the diffusion model to learn to this data
26. Then, we take the nominal plant, and take some amount of steps forward.
27. We heuristically go a certain amount forward to introduce a certain amount of noise
28. then we go backwards. The diffusion model will try to remove the noise, but not knowing the orignial nominal plant will introduce a perturbation.
29. **This is outcome number 3 and number 2**
30. We can use this frequency response data to find the 'worst case' distance to the critical point -1. We plot this location in the complex plane. 
31. Why do we care about the complex plane?
32. This is where the nyquist robust stability and performance criterion live.
33. Our 'valid' perturbations live inside this circle of radius -1
34. We repeat this several times, and only include examples in our set of unstructured perturbations that are within or robustness circle 
35. **This is outcome number 1**
36. We generate enough examples to populate this circle until we're comfortable.

## Writin some stuff

The purpose of this proposal is to suggest that using a generative network to create unstructured perturbations can be a viable way to advance the state of the art. But to do this, the current state of diffusion models and their place must be introduced. The generative diffusion model is a recent breakthrough in generative models [@sohl-dicksteinDeepUnsupervisedLearning2015]. Diffusion generative models are the state of the art for image and video generation, and have demonstrated promise for audio generation and noise removal  [@kongDiffWaveVersatileDiffusion2020] [@SoraCreatingVideo].  A diffusion generative model, AlphaFold 3, won the Nobel Prize in Chemistry [@AlphaFold3Predicts2024] Diffusion models do this through a forward noise-inducing process, and a learned backwards noise-removing process. 

The forward diffusion process works by introducing small amounts of noise into 

## ChatGPT Analysis
Similar to the [[2. QE State of the Art]] I had some help creating an outline:
**Objective: Applying Diffusion Models for Unstructured Perturbations**
    
### **3.1 Research Goals and Hypothesis**
Restate the main goal: to create a diffusion model that can generate unstructured perturbations, assisting in robustness verification. Clearly outline the hypothesis that diffusion models can reliably and efficiently generate realistic perturbations within the desired uncertainty bounds.

### **Methodology**
    
**4.1 Model Training with Structured Perturbations**: Explain the training process, using a structured set of perturbed plants. Detail how parameters for the uncertainty function W2W_2W2​ will be chosen based on robustness requirements, and why training on structured perturbations is effective.
**4.2 Diffusion Model Forward Process for Noise Introduction**: Describe the forward process, introducing Gaussian noise in small increments to structured perturbations. Discuss how the amount of noise relates to the uncertainty level.
**4.3 Diffusion Model Reverse Process for Perturbation Generation**: Explain the reverse process, where a neural network denoises the plant sample to produce novel, unstructured perturbations.
    - **4.4 Experimental Design for Verifying Perturbation Validity**: Outline how you’ll validate that the generated plants belong to the allowable set, ensuring they meet robust stability/performance criteria by checking Nyquist stability.
###  **Expected Outcomes and Verification of Results**
 ***5.1 Outcome 1**: A set of valid unstructured perturbations that satisfy robustness requirements.
**5.2 Outcome 2**: Quantitative assessment of the model’s ability to approximate the uncertainty bounds in W2W_2W2​.
**5.3 Outcome 3**: Reduced effort in generating perturbations for robustness verification in controller implementation.
###  **Implications and Potential Applications**
Discuss broader implications for robust control, such as reducing the cost and complexity of verification processes for infrastructure systems. Explain how success could advance diffusion models’ applications beyond robust control, perhaps influencing fields requiring resilient system validation.