𝗛𝗮𝗺𝗶𝗹𝘁𝗼𝗻-𝗝𝗮𝗰𝗼𝗯𝗶 𝗧𝗵𝗲𝗼𝗿𝘆 𝗟𝗶𝗻𝗸𝘀 𝗡𝗲𝘂𝗿𝗮𝗹 𝗔𝗿𝗰𝗵𝗶𝘁𝗲𝗰𝘁𝘂𝗿𝗲𝘀
Neural networks often feel like a collection of separate tricks.
ResNets use skip connections. Transformers use attention. RNNs use recurrence. Each model has its own rules and math. This makes it hard to see the bigger picture.
New research changes this. It shows that ResNets, Transformers, and RNNs are actually the same mathematical object. They all follow Hamilton-Jacobi equations.
Here is how it works:
- Gradient descent is a type of physics evolution.
- Each training step moves weights like a fluid.
- Depth, attention, and recurrence act like time steps in a calculation.
- A single parameter controls how smooth or sparse a model becomes.
This theory connects four different fields: neural networks, tropical algebra, PDEs, and convex optimization.
Why does this matter for you?
Current benchmarks focus mostly on accuracy. This framework suggests a new way to build models. Instead of just adding layers, you can tune the math to balance smoothness and stability.
The theory also predicts how well a model will generalize. It links how much data you need to the specific math used in your architecture.
There are still gaps. Most models use ReLU, but this math works best with log-sum-exp layers. We also need more real-world tests to see if these physics rules improve performance.
We should stop looking at architectures as different types of layers. We should look at them as different ways to solve the same equation.
Source: https://dev.to/olaughter/hamilton-jacobi-view-links-major-neural-architectures-5hln
Optional learning community: https://t.me/GyaanSetuAi