Advanced Optimization Techniques in Neural Networks: A Sobolev Space Approach

TMLR Paper3502 Authors

Abstract:

In this article, we explore the concept of Sobolev loss and its advantages over conventional loss functions in neural network training, particularly in the context of approximating smooth functions and their derivatives. Conventional loss functions like Mean Squared Error (MSE) and Mean Absolute Error (MAE) focus solely on minimizing the difference between predicted and true function values. However, they often fail to capture the smoothness and derivative information critical for accurate function approximation in various scientific and engineering applications.

Sobolev loss addresses this limitation by incorporating terms that measure the difference between the derivatives of the predicted and true functions. This not only ensures better function value approximations but also promotes smoother and more accurate representations of the underlying function. The article delves into the theoretical foundations of Sobolev spaces, which provide the mathematical framework for Sobolev loss, and discusses the benefits of using Sobolev loss in terms of improved generalization, stability, and performance.

We illustrate these concepts through a practical example of approximating
$f(x)=sin(x)$ and $f(x)=e^{-x}$ using a neural network. The example demonstrates how Sobolev loss enables the network to learn both the function values and their derivatives, resulting in a more accurate and smooth approximation compared to traditional loss functions. Additionally, we highlight key references for further reading, including foundational texts on Sobolev spaces and research papers that explore the application of Sobolev loss in neural networks.

By integrating derivative information into the training process, Sobolev loss provides a powerful tool for enhancing the quality of neural network approximations, making it particularly valuable for applications requiring smooth and accurate function representations.

Submission Length: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Zhiyu_Zhang1

Submission Number: 3502