Abstract

The identification of controlled nonlinear dynamical systems from data is central to tasks such as control, prediction, optimization, and fault detection, with applications in robotics, finance, and biology. This talk presents a method for estimating both drift and diffusion in stochastic differential equations with control inputs. The approach applies to general settings involving nonlinear, multidimensional dynamics and unknown, non-uniform diffusion, without requiring restrictive assumptions such as linearity or known diffusion. Assuming Sobolev regularity of the coefficients, we decompose the learning problem into two stages: estimating system dynamics under a finite set of controls, and recovering the governing coefficients via the Fokker–Planck equation. We establish finite-sample learning guarantees in L2, Linf, and risk metrics, with learning rates adaptive to the Sobolev regularity of the coefficients. The method is demonstrated through numerical experiments and released as an open-source Python library. Joint work with Luc Brogat‑Motte, Riccardo Bonalli, and Alessandro Rudi.