We study the expressivity of shallow polynomial neural networks (PNNs) with monomial activation functions over finite fields. For a given architecture, we define a neuromanifold as the image of the map from all possible network weights into the product of polynomial rings. We quantify the expressivity by the cardinality of the neuromanifold, and derive a natural lower and upper bound. This leads to counting rational points over finite fields, a problem closely linked to the Weil conjectures. Finally, we present an architecture that exhibits a striking difference in the neuromanifolds when considered over a characteristic zero versus a finite-characteristic field, illustrating the critical role of field characteristic in the notion of expressivity.
Closed-loop generative selection has become a workhorse of computational drug discovery: a learned generative model proposes candidate molecules, a fitness oracle scores them, the best are kept, and the model is retrained on this elite set before the next round. Despite its wide…
Selection Hyper-heuristics (HHs) automate algorithmic design by selecting from a set of low-level heuristics which one to apply at each stage of the optimisation process. Several impressive results have been recently rigorously proven regarding the performance of selection hyper-…
Expensive simulation-driven design is widely used in engineering to identify requirement-satisfying designs with as few high-fidelity simulations as possible. Most existing efforts address this challenge by improving optimization algorithms under fixed formulations, yet the formu…
Model merging combines independently trained or fine-tuned models, but pairwise alignability does not imply globally consistent alignment. We formulate merging as a finite descent problem in which checkpoints are local objects, alignment maps are transitions, and cycle products a…
Most explanations of training instability focus on \emph{learning-rate criticality}, typically characterized by the Edge of Stability, beyond which optimization becomes unstable. We argue that, in practical deep neural network training, there is an additional and often overlooked…
AI researchers describe state-of-the-art models as one thing repeated at scale: the Transformer, wired identically for text, pixels, or speech. Neuroscientists describe the cortex as a mosaic - dense Layer 4 in visual cortex for spatial encoding, thick Layers 5/6 in motion cortex…