$p$-adic Character Neural Network

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.

This work addresses two fundamental limitations in neural network approximation theory. We demonstrate that a three-dimensional network architecture enables a significantly more efficient representation of sawtooth functions, which serves as the cornerstone in the approximation of analytic and $L^p$ functions. First, we establish substantially improved exponential approximation rates for several important classes of analytic functions and offer a parameter-efficient network design. Second, for the first time, we derive a quantitative and non-asymptotic approximation of high orders for general $L^p$ functions. Our techniques advance the theoretical understanding of the neural network approximation in fundamental function spaces and offer a theoretically grounded pathway for designing more parameter-efficient networks.

We present the first method for native, continuous gradient descent for machine learning models with $p$-adic parameters. Existing native optimizers are discrete, mostly combinatorial searches, as the $p$-adic numbers $\mathbb{Q}_p$ are totally disconnected, with standard losses that are flat away from their minima. To enable continuous optimization, we propose working with $\mathbb{Q}_p$ via its Berkovich affine line: a canonical, path-connected expansion of $\mathbb{Q}_p$ that preserves its isometries and uniquely extends its analytic maps. This hull is a metric tree with interpretable points and local derivatives, which we show enables effective optimizers and backpropagation. We formulate gradient descent and show that its approximations efficiently learn linear models with coefficients in $\mathbb{Q}_p$ to do modular arithmetic, an XOR-like task not expressible by linear models in $\mathbb{R}$. We also demonstrate momentum and Adam variants, linear regression, and classification on binary-encoded hierarchies (Quillian semantic networks), addressing open problems posed by Martins (2025). Library at https://github.com/google-deepmind/padic-ml