Summary of Convergence Of Implicit Gradient Descent For Training Two-layer Physics-informed Neural Networks, by Xianliang Xu et al.
Convergence of Implicit Gradient Descent for Training Two-Layer Physics-Informed Neural Networks
by Xianliang Xu, Ting Du, Wang Kong, Ye Li, Zhongyi Huang
First submitted to arxiv on: 3 Jul 2024
Categories
- Main: Machine Learning (cs.LG)
- Secondary: Optimization and Control (math.OC)
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| Summary difficulty | Written by | Summary |
|---|---|---|
| High | Paper authors | High Difficulty Summary Read the original abstract here |
| Medium | GrooveSquid.com (original content) | Medium Difficulty Summary The paper presents a comprehensive study on implicit gradient descent (IGD) for training over-parameterized physics-informed neural networks (PINNs). Compared to traditional gradient descent (GD), IGD is shown to outperform GD in handling certain multi-scale problems. The authors provide convergence analysis for IGD in training two-layer PINNs, demonstrating the positive definiteness of Gram matrices for various smooth activation functions. Over-parameterization enables a globally optimal solution at a linear convergence rate, and the learning rate can be selected independently of sample size and least eigenvalue. Empirical results validate theoretical findings. |
| Low | GrooveSquid.com (original content) | Low Difficulty Summary This paper is about finding better ways to train special kinds of artificial intelligence models called physics-informed neural networks (PINNs). PINNs are important because they can help us solve complex problems in fields like physics, engineering, and computer science. The researchers compared two different methods for training these models: one called gradient descent, which is commonly used, and another called implicit gradient descent (IGD). They found that IGD works better than GD for certain types of problems. The team also studied how IGD converges to a solution and showed that it can find the best answer in some cases. Finally, they tested their ideas on real data and saw that they worked. |
Keywords
* Artificial intelligence * Gradient descent
