Solving Fredholm Integral Equations Using Deep Learning.

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Artificial Intelligence

Background:

• High-dimensional Fredholm integral equations pose significant computational challenges.
• Traditional numerical methods often struggle with the
• curse of dimensionality
• making them inefficient for complex problems.

Purpose of the Study:

• To develop and present a novel deep learning-based method for solving high-dimensional Fredholm integral equations.
• To demonstrate the efficiency and scalability of the proposed deep learning approach.

Main Methods:

• A deep residual neural network (DRNN) is constructed using a fixed number of randomly selected collocation points within the integration domain.
• The DRNN's loss function is formulated as a linear least-square problem based on the integral equation at the training set's collocation points.
• Training involves iterative parameter updates across different training datasets.

Main Results:

• The deep learning method demonstrates efficiency in solving high-dimensional Fredholm integral equations.
• Numerical experiments indicate a moderate generalization error across all evaluated points.
• The computational cost is shown to be unaffected by the "curse of dimensionality".

Conclusions:

• The proposed deep learning method offers an effective solution for high-dimensional Fredholm integral equations.
• The approach provides a computationally efficient alternative to traditional methods, overcoming the "curse of dimensionality".
• This work highlights the potential of deep learning in addressing complex mathematical and scientific challenges.