Machine Learning and AI for Inverse Problems

Inverse problems occur throughout science and engineering when hidden parameters or structures are inferred from indirect, noisy data, with applications ranging from medical imaging and geophysics to structural health monitoring. These problems are typically ill-posed, lacking uniqueness or stability, and require regularization or Bayesian methods to produce meaningful solutions. Classical approaches impose assumptions such as smoothness or piecewise constancy, but these priors may not capture real-world complexity. Recent advances in machine learning, particularly deep learning, have enabled data-driven solutions that approximate inverse mappings, learn surrogate forward models, and embed implicit priors from training data. In medical imaging, for example, learning-based reconstructions can outperform analytic and variational methods by reducing noise and artifacts. The integration of rigorous inverse theory with flexible data-driven models defines a new frontier, offering both theoretical insights and practical methodologies for solving ill-posed problems.