Reconstructing physical phenomena from indirect observations lies at the heart of scientific measurement and discovery, and is a unifying challenge across medical imaging, geophysical exploration, remote sensing, astronomy, and nondestructive testing. Classical approaches to such inverse problems have drawn on optimization, signal processing, and careful exploitation of physical forward models, yielding principled methods with well-understood theoretical properties. Machine learning offers transformative new capabilities in this space, yet off-the-shelf data-driven approaches fail to leverage our collective, if partial, understanding of the underlying physics. Such approaches can be brittle, data-hungry, and difficult to interpret.
In this talk, we explore how physical structure can be systematically integrated into learned reconstruction pipelines to yield methods that are faster, more accurate, and more data-efficient than either purely classical or purely data-driven alternatives. We develop these ideas through a progression of linear and nonlinear inverse problems, using nonlinear inverse scattering — where one seeks to recover an inhomogeneous medium from scattered acoustic or electromagnetic wave measurements — as a rich example that illustrates the challenges of nonlinearity, non-locality, and ill-posedness at scale. We show how classical algorithmic insights, such as frequency-continuation strategies and recursive linearization, can inspire neural network architectures and training procedures that respect the structure of the problem. Looking ahead, we identify key open problems and opportunities for the imaging community: the need for uncertainty quantification and robustness to distribution shift, theoretical foundations for learned solvers, and scalability to high-fidelity forward models.
