Merge branch 'master' of github.com:KindXiaoming/pykan

README.md

@@ -2,12 +2,45 @@
 
 # Kolmogorov-Arnold Newtworks (KANs)
 
-This the github repo for the paper "KAN: Kolmogorov-Arnold Networks" [link]. Find the documentation [here](https://kindxiaoming.github.io/pykan/).
+This the github repo for the paper ["KAN: Kolmogorov-Arnold Networks"](https://arxiv.org/abs/2404.19756). Find the documentation [here](https://kindxiaoming.github.io/pykan/).
 
-Kolmogorov-Arnold Networks (KANs) are promising alternatives of Multi-Layer Perceptrons (MLPs). KANs have strong mathematical foundations just like MLPs: MLPs are based on the universal approximation theorem, while KANs are based on Kolmogorov-Arnold representation theorem. KANs and MLPs are dual: KANs have activation functions on edges, while MLPs have activation functions on nodes. This simple change makes KANs better (sometimes much better!) than MLPs in terms of both model accuracy and interpretability. A quick intro of KANs [here](https://kindxiaoming.github.io/pykan/intro.html).
+Kolmogorov-Arnold Networks (KANs) are promising alternatives of Multi-Layer Perceptrons (MLPs). KANs have strong mathematical foundations just like MLPs: MLPs are based on the universal approximation theorem, while KANs are based on Kolmogorov-Arnold representation theorem. KANs and MLPs are dual: KANs have activation functions on edges, while MLPs have activation functions on nodes. This simple change makes KANs better (sometimes much better!) than MLPs in terms of both model **accuracy** and **interpretability**. A quick intro of KANs [here](https://kindxiaoming.github.io/pykan/intro.html).
 
 <img width="1163" alt="mlp_kan_compare" src="https://github.com/KindXiaoming/pykan/assets/23551623/695adc2d-0d0b-4e4b-bcff-db2c8070f841">
 
+## Accuracy
+**KANs have faster scaling than MLPs. KANs have better accuracy than MLPs with fewer parameters.**
+
+**Example 1: fitting symbolic formulas**
+<img width="1824" alt="Screenshot 2024-04-30 at 10 55 30" src="https://github.com/KindXiaoming/pykan/assets/23551623/e1fc3dcc-c1f6-49d5-b58e-79ff7b98a49b">
+
+**Example 2: fitting special functions**
+<img width="1544" alt="Screenshot 2024-04-30 at 11 07 20" src="https://github.com/KindXiaoming/pykan/assets/23551623/b2124337-cabf-4e00-9690-938e84058a91">
+
+**Example 3: PDE solving**
+<img width="1665" alt="Screenshot 2024-04-30 at 10 57 25" src="https://github.com/KindXiaoming/pykan/assets/23551623/5da94412-c409-45d1-9a60-9086e11d6ccc">
+
+**Example 4: avoid catastrophic forgetting**
+<img width="1652" alt="Screenshot 2024-04-30 at 11 04 36" src="https://github.com/KindXiaoming/pykan/assets/23551623/57d81de6-7cff-4e55-b8f9-c4768ace2c77">
+
+## Interpretability
+**KANs can be intuitively visualized. KANs offer interpretability and interactivity that MLPs cannot provide. We can use KANs to potentially discover new scientific laws.**
+
+**Example 1: Symbolic formulas**
+<img width="1510" alt="Screenshot 2024-04-30 at 11 04 56" src="https://github.com/KindXiaoming/pykan/assets/23551623/3cfd1ca2-cd3e-4396-845e-ef8f3a7c55ef">
+
+**Example 2: Discovering mathematical laws of knots**
+<img width="1443" alt="Screenshot 2024-04-30 at 11 05 25" src="https://github.com/KindXiaoming/pykan/assets/23551623/80451ac2-c5fd-45b9-89a7-1637ba8145af">
+
+**Example 3: Discovering physical laws of Anderson localization**
+<img width="1295" alt="Screenshot 2024-04-30 at 11 05 53" src="https://github.com/KindXiaoming/pykan/assets/23551623/8ee507a0-d194-44a9-8837-15d7f5984301">
+
+**Example 4: Training of a three-layer KAN**
+
+![kan_training_low_res](https://github.com/KindXiaoming/pykan/assets/23551623/e9f215c7-a393-46b9-8528-c906878f015e)
+
+
+
 ## Installation
 There are two ways to install pykan, through pypi or github.