Six AI Use Cases with GeodePoly

This page summarizes six practical ways AI can leverage GeodePoly.

1) Differentiable Root Layer (inside neural nets)

• Custom autograd op mapping coefficients → roots.
• Gradients for simple roots: ∂r_i/∂a_k = − r_i^k / p'(r_i).
• Tips: mask/damp gradients if |p'(r)| is tiny; polish roots in forward.

2) GeodeBench: symmetry-generalization tasks

• Dataset from Geode/Hyper‑Catalan arrays.
• Tasks: coefficient prediction, Geode factor recovery (S→G), invariance under shift/scale.

3) Inductive‑bias layers that “speak polynomial”

• Geode convolution (face‑layer accumulation), power‑basis attention, root‑space heads.

4) Training objectives in root space

• Pole placement (half‑plane margins), spectral radius, root‑set matching.

5) Neuro‑symbolic loops & program synthesis

• Generate → verify with GeodePoly → refine.
• Enforce Viète identities as a regularizer.

6) GPU/Compiler target (batched kernels)

• Batched kernels (JAX/XLA, CUDA) and differentiable RootLayer for large batches.

See examples/ai/ for runnable demos and the AI overview page for install and quickstart.