Sharp Minimax Rates for Equation Discovery from Dynamical Systems Data

We establish information-theoretic minimax bounds for learning nonlinear differential equations from trajectory observations. For the class of r-smooth vector fields on a p-dimensional compact domain, the minimax risk of equation identification is \Theta(n^{-r/(2r+p)}) in L^{2}, achieved by local polynomial regression and matched by a Fano-based lower bound. Reaching accuracy \epsilon requires at least n\ge\epsilon^{-(2r+p)/r} trajectories, a burden that grows exponentially with p/r this is the fundamental sample-complexity barrier for high-dimensional equation discovery, which we call the curse of discovery. For polynomial vector fields of degree d the minimax rate sharpens to \Theta(n^{-1/(2d+2)}), where higher degree worsens the rate because of the ill-posedness of inverting the trajectory map. When the polynomial is additionally s-sparse, sparsity reduces the rate to \Theta(\sqrt{s \log(N_{d,p}/s)/n}), giving a quantitative justification for sparse regression approaches to equation discovery. We also establish a strict separation between solution operator learning and equation identification: there exist systems for which the solution operator is learnable at rate O(n^{-1/2}), requiring roughly \epsilon^{-2} trajectories, while identifying the governing equation requires \epsilon^{-6} trajectories. Accurate prediction therefore does not imply identifiability of the governing equation. Proofs combine Fano's inequality with localized packings, trajectory sensitivity via Grönwall's inequality, and explicit KL divergence calculations along noisy trajectory distributionsm.