Regularization, Bayesian Inference, and Machine Learning Methods for Inverse Problems.

Area of Science:

• Computational Mathematics
• Applied Physics
• Machine Learning

Background:

• Classical inverse problem solutions rely on regularization theory, optimizing data-model matching and regularization terms.
• Bayesian Maximum A Posteriori (MAP) interpretation links these terms to likelihood and prior probability models, offering flexibility but posing computational challenges.

Purpose of the Study:

• To explore the synergy between machine learning (ML) and classical inversion techniques for solving inverse problems.
• To demonstrate how ML methods can provide efficient and practical approximate solutions, overcoming computational burdens of traditional Bayesian approaches.

Main Methods:

• Review of regularization theory and its Bayesian Maximum A Posteriori (MAP) interpretation.
• Introduction of machine learning (ML) methods, including neural networks (NN), deep neural networks (DNN), and physics-informed neural networks (PINN).
• Application of these methods to image denoising, image restoration, and computed tomography (CT) image reconstruction.

Main Results:

• Machine learning methods, particularly neural networks, offer practical and computationally efficient approximate solutions to inverse problems.
• The integration of ML with inversion techniques enhances the ability to handle complex data and model priors.
• Demonstrated success in image denoising, restoration, and CT reconstruction highlights the cooperative potential.

Conclusions:

• Machine learning provides a powerful toolkit for addressing the computational challenges in solving inverse problems.
• The combination of ML and classical inversion methods opens new avenues for practical applications in image processing and reconstruction.
• This tutorial highlights the growing importance of ML in scientific computing and data analysis.