Deep Reinforcement Learning for Fano Hypersurfaces
This work addresses a long-standing classification problem in algebraic geometry, providing new examples that can test and generalize theory, though it is incremental in applying existing methods to a new domain.
The paper tackled the problem of discovering Fano 4-fold hypersurfaces with terminal singularities, which are fundamental in algebraic geometry but have been difficult to classify due to combinatorial intractability, and the result was that their deep reinforcement learning algorithm yielded thousands of previously unknown examples, with hundreds inaccessible to known methods.
We design a deep reinforcement learning algorithm to explore a high-dimensional integer lattice with sparse rewards, training a feedforward neural network as a dynamic search heuristic to steer exploration toward reward dense regions. We apply this to the discovery of Fano 4-fold hypersurfaces with terminal singularities, objects of central importance in algebraic geometry. Fano varieties with terminal singularities are fundamental building blocks of algebraic varieties, and explicit examples serve as a vital testing ground for the development and generalisation of theory. Despite decades of effort, the combinatorial intractability of the underlying search space has left this classification severely incomplete. Our reinforcement learning approach yields thousands of previously unknown examples, hundreds of which we show are inaccessible to known search methods.