An inverse problem is the process of calculating the hidden causes or factors that produced a set of observed effects. This is the direct opposite of a forward problem, which uses known causes and physical laws to predict future states or effects. Inverse problems are fundamental across scientific disciplines because they allow researchers to determine parameters that cannot be directly observed, effectively revealing hidden mechanisms from observable phenomena.

A defining characteristic of most inverse problems is that they are mathematically "ill-posed" according to Jacques Hadamard's criteria. This means they often violate at least one of three conditions: the existence of a solution, the uniqueness of the solution, or the continuous stability of the solution. The lack of stability is particularly challenging; because measurement operators often filter out high-frequency information, arbitrarily small errors or noise in the collected data can lead to indefinitely large and unphysical discrepancies in the calculated causes.

To solve ill-posed inverse problems, mathematicians employ a technique called regularization, such as Tikhonov regularization. Regularization stabilizes the problem by introducing additional prior information, constraints, or assumptions—such as penalizing highly complex solutions to prefer smoother models—which prevents the solution from drifting into unphysical oscillations driven by noise.

Inverse problems have ubiquitous real-world applications, including X-ray computed tomography (CT), seismic exploration, radar, and acoustics. Recently, inverse problem theory has converged with machine learning and causal inference to uncover unobserved confounders (hidden variables) in complex social, biological, and economic systems. Advanced deep learning architectures, such as Variational Autoencoders (VAEs), are now used to simultaneously estimate these unknown latent spaces and their corresponding causal effects. Furthermore, recent research has demonstrated that the fundamental theoretical and practical limits of reconstructing these hidden causal factors are strictly governed by the geometric and algebraic symmetries (group-theoretic structures) of the systems themselves.