Machine learning (ML) computations of increasing sophistication and complexity are being developed to solve complex, data-driven problems in diverse areas. Their output is often subject to undesirable phenomena such as overfitting and hallucinations that are hard to detect, and more generally their scientific rigor is hard to establish. We propose the concept of ML-solvability by combining the theories of learnability, computing, and logic, which characterizes the model space, the learning algorithm that estimates a model using samples, and the inference algorithm that utilizes the model. It provides insights into the applicability and generalization of ML codes and the possibility of incomplete and unsound inferences if the underlying problem is not ML-solvable. In science areas, the long-established laws are synergistically exploited to sharpen and compose powerful ML solutions with provable generalization and correctness properties. We describe a framework for ML-solvability and generalization analyses based on a combination of physical laws that govern systems and information laws that characterize the learning processes. We combine the learning dimension and training error to derive generalization equations to detect and minimize overfitting and utilize physical law violations by learning processes to identify and mitigate inference inadequacies. We briefly describe the uses of smooth, non-smooth, and algebraic forms of laws to develop or analyze ML solutions in the areas of data transport networks, nuclear engineering, and computer system diagnosis.
