Further reading

Categorical accounts of learning provide the closest conceptual background. Backpropagation as a functor appears in Fong et al. [ 2019 ] ; reverse derivative categories and categorical gradient learning are developed in Cockett et al. [ 2020 ] and Cruttwell et al. [ 2022 ] . The categorical-deep-learning program [ Gavranović et al. , 2024 ] broadens this agenda from optimizers to algebraic descriptions of architectures and their constraints.

Natural gradient begins with Amari [ 1998 ] ; Amari [ 2016 ] develops its information-geometric setting. For scalable deep networks, Kronecker-factored approximate curvature provides an influential structured approximation [ Martens and Grosse , 2015 ] . Natural policy gradient [ Kakade , 2001 ] and natural actor–critic [ Peters and Schaal , 2008 ] show how the same geometric principle enters reinforcement learning. Natural DLINCS uses these methods to choose intrinsic repair directions while retaining typed obstruction and admission records.

On the machine-learning side, neural ordinary differential equations [ Chen et al. , 2018 ] , normalizing flows [ Papamakarios et al. , 2021 ] , diffusion models [ Ho et al. , 2020 ] , and flow matching [ Lipman et al. , 2023 ] are useful examples of architectures whose semantics depend on paths, transports, or continuous dynamics. Deep LINCS does not subsume these models; it supplies a common way to declare which of their computational routes should compose, differentiate those promises, and decide whether a detected discrepancy should influence training.