There are many operator methods that explain how deep neural networks can approximate operators, not just functions. This concept is important because operators, like those found in differential equations, map functions to functions. This means that we can use neural networks to solve problems in physics, biology, actuarial sciences, statistical analysis, and financial analysis, because this approach isn’t limited to just differential equations, any number of other mathematical equations for other scientific and numerical analysis can be leveraged. For this discussion, we will just discuss Fourier Neural Operators. I have published my first book, "What Everone Should Know about the Rise of AI" is live now on google play books at Google Play Books and Audio, check back with us at https://theapibook.com for the print versions, go to Barnes and Noble at Barnes and Noble Print Books!
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How do Fourier Neural Operators work?
Fourier Neural Operator (FNO) is very useful for image-to-image problems and comparisons. All that is required is to replace convolutional layers with Fourier layers, and this is how you establish your FNO’s. Fourier transforms are well-suited for representing physical phenomena and thus we can use it to capture the underlying physics on the objects or data sources to the AI model. A great example is to monitor amplitude frequencies and power spectral entropy or the smoothing effect on accelerometer and gyroscope data. Whats really neat is your able to calculate the optimal frequency and data population sizes to reduce overlapping windows, overfitting. One promising application use-case is zero-shot super resolution, which is where your data is trained on low-resolution data and then used to generate high-resolution solutions, essentially upscaling the results. Super resolution likely works when the low-resolution data captures enough essential features of the physics. Pushing the limits of down sampling could lead to inaccurate results.
Generalizing Neural Operators: Customization, Flexibility and Kernels
The FNO is a specific instance of a more general neural operator framework. This framework allows for customization by specifying different kernel functions in the neural operator layers. This flexibility enables users to tailor neural operators to their specific physics problems. You can leverage various linear problem and logistic regression algorithms by setting a range of K values to visualizing clusters in a 3D scatter plot, or star constellation maps. There are different fourier kernels that are suitable for periodic boundary conditions, like those of fluid flow problems or heat transfer equations. However, for complex geometries, there are a number of different kernels that can be used for different applications. Understanding the underlying physics and boundary conditions are very important on the onset of any FNO project, or you will end up with useless outputs.
Mesh Invariance
Mesh invariance enable discretization, which allows for flexible mesh resolution. This feature enables the refinement of solutions and the capture of intricate details like shock waves. Neural operators offer several advantages over traditional methods, including their mesh invariance, ability to learn complex relationships, and potential for zero-shot super resolution. However, it's crucial to evaluate their performance carefully and understand their limitations. You can also use Laplace Neural Operators which generalize Fourier Neural Operators to handle exponential growth and decay problems. The possibilities from there are endless.
Use Case: Predicting Network Equipment Failure, Congestion and Packet Loss
Data Collection
Typical large ISPs like ATT, Charter/Spectrum, and Verizon collects a continuous stream of data from their massive scaled networks. This data can be processed with Fourier transforms to create output data on their equipment health, failure rates, and performance and use it as inputs to an AI/ML logistical regression model. We can use WebRTC clients to capture a level of metrics that are deeper insights into the network. These clients gather real-time metrics related to WebRTC sessions, including packet loss, jitter, round-trip time, local hardware details, and bandwidth estimations. By collecting this data, we can start building a complete picture of network performance, and unprecedented observability.
Forier Transform Application
Once this data is collected, the next step involves applying Fourier transforms to the time series data. This technique is essential because it converts the data from the time domain (where metrics vary over time) to the frequency domain. In the frequency domain, instead of tracking values as they change over time, the data reveals the strength of different frequencies. This allows us to better analyze patterns and trends that may not be obvious in the raw time series data and enable additional causation, and correlations with the actual un-expected failures. By comparing unexpected failures and their context to the prediction model created by the webRTC network_test tool data, we can predict with pretty accurate cadence, which equipment will fail, and when.
Feature Extraction, Historical Metrics and Logistical Regression Prep
When the data is transformed into the frequency domain, we are able to extract key features that are crucial for our predictive analysis. Dominant frequencies, data center temperature, and traffic capacity patterns all will be highlight periodic patterns of network health markers, which then is correlated with the MOS score of 1-5 {1 being terrible network performance, 5 being excellent). We can then, with that data, focus on the amplitude of these frequencies, the network segments involved, which can indicate how severe these congestion patterns are. Finally, phase information is extracted to help pinpoint shifts in network behavior over time.
We can leverage historical network failure data to analyze times when packet loss and jitter exceeded certain thresholds, asymmetric packet arrival delay, or when network outages occurred. By labeling this historical data, we can prepare the dataset for training machine learning models that identify pattern correlation and causations that lead to network equipment failures. At this stage, we can even take into account weather alminacs and news prediction feed to identify hurricanes, thunder storms, earth quakes, and various natural disasters.
Logistic Regression Model Training
Once the training model has consumed these training datasets and is grounded with output boundaries, we can implement a logistic regression model using the Fourier features extracted from the phase one model training data to gather dominant frequencies, amplitudes, and phase information—along with other relevant contextual data like time of day and overall network load. Wecan then train the model to predict the probability of network equipment failure, network congestion or packet loss surpassing a predefined threshold.
Conclusion
Fourier Neural Operators and Neural Operators represent a powerful approach to operator learning and physics-informed machine learning. Their mesh invariance and flexibility make them promising tools for solving complex problems.
With this trained model, we can continuously refine its real-time predictions outputs by grounding and comparing them to actual real world results. Based on the incoming WebRTC MOS scores of any session data, the model can assess the likelihood of congestion or packet loss, generate alerts or trigger proactive adjustments and replacements to the network. This helps mitigate potential problems before they impact the user experience, ensuring smoother and more reliable network.
This approach leverages AI to not only monitor network health but also to anticipate and address issues before they escalate, leading to more robust and resilient network.
Check out this youtube video by Steve Brunton summarizing these concepts here:
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