This paper derives sufficient conditions for input-dependent contraction and formally establish an input-to-state contraction property under bounded external excitations, and develops a novel deep learning framework that seamlessly incorporates time-varying control inputs while ensuring incremental exponential convergence via input-dependent contraction regularization.
Abstract
Learning continuous-time representations of dynamical systems from observation data has emerged as a cornerstone of data-driven control and scientific machine learning. However, existing neural differential equations either treat external control inputs heuristically without providing strict structural guarantees, or enforce stability properties under the restrictive assumption of constant or vanishing inputs. This paper proposes the Input-Contraction Neural Differential Model (ICNDM), a novel deep learning framework that seamlessly incorporates time-varying control inputs while ensuring incremental exponential convergence via input-dependent contraction regularization. By leveraging an embedded input encoder and a parameterized metric network, the proposed architecture learns both the non-autonomous neural vector fields and a generalized Riemannian contraction metric simultaneously. We derive sufficient conditions for input-dependent contraction and formally establish an input-to-state contraction property under bounded external excitations. Extensive numerical evaluations on highly nonlinear chaotic oscillators and experimental data from a Permanent Magnet Synchronous Motor (PMSM) drive system demonstrate that ICNDM yields substantial reductions in long-horizon rollout errors and exhibits superior structural robustness against input perturbations compared with state-of-the-art neural differential benchmarks.
An adaptive stability-constrained neural differential equation (AS-NDE) for systems with measured controls and unmatched, unknown perturbations and a reproducible evaluation protocol for a forced Duffing oscillator and a permanent-magnet synchronous motor model is given.
The proposed robust invertibility framework is illustrated through a series of application examples in data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows, illustrating its utility for robust control, trajectory optimization, and generative modeling of complex trajectory distributions.
Yurui Zhang, Ruigang Wang, I. Manchester· 0 citations
This letter develops a data-driven control framework for nonlinear ensemble systems using reservoir computing (RC). We consider ensemble control problems, in which the objective is to regulate a large, potentially uncountable, population of systems with unknown dynamics. To address this challenge, we introduce a moment kernelization approach that yields a dual representation and enables a valid finite-dimensional approximation of ensemble dynamics. Building on this reduction, we cast ensemble control synthesis as the approximation of a causal operator that maps moment trajectories to control inputs. We show that continuous-time reservoir systems induce well-defined causal input-output operators with the fading-memory property, providing a principled foundation for learning these feedback operators from moment trajectory data. Based on this theory, we design an RC-based controller trained on input-output moment trajectories and deployed in a closed-loop configuration for tracking and stabilization of nonlinear ensemble systems.
Yuan-Hung Kuan, Lin Tang, Jr-Shin Li· IEEE Control Systems Letters· 0 citations
This paper presents a data-driven stable manifold (DD-SM) method, which integrates Koopman operator representation learning with the geometric stable manifold approach to Hamilton-Jacobi-Bellman (HJB) equations, enabling end-to-end optimal feedback control synthesis from raw trajectory data without prior knowledge of system dynamics. We construct an augmented control system under a unified symmetric subspace decomposition (SSD) and extended dynamic mode decomposition (EDMD) framework for joint approximation of the drift field, control matrix and their spatial derivatives, and derive probabilistic finite-sample error bounds for invariant and non-invariant dictionary spaces to yield a provably accurate approximate characteristic system of HJB equation. Via Lyapunov-Perron operator and ODE perturbation analysis, we prove the data-driven stable manifold achieves monotonically decreasing semi-global error with growing training data. We further establish closed-loop exponential stability and quantify the optimality gap, both tightenable by refining model accuracy. An efficient algorithm pipeline with adaptive data generation and deep neural approximation is developed, outputting control signals within 1 millisecond. Experiments on a modified van der Pol oscillator verify the effectiveness of our method.
This thesis advances the training and scalability of NCDEs through three complementary contributions, building on neural rough differential equations, which reduce the time per training step for an NCDE by up to three orders of magnitude while achieving state-of-the-art performance across diverse time series benchmarks.
Feedback-driven loops support iterative improvement in large language models, reinforcement learning, and autonomous discovery, yet their gains often diminish under repeated internal feedback. We study why closed-loop knowledge systems saturate and what external information can move them beyond their current attractors. We introduce a three-level operational framework in which knowledge states $x_t$ evolve through transition kernels $K_{\theta}$ indexed by a structural parameter $\theta$. The governing structure is defined as the observational equivalence class of $\theta$ induced by these kernels, while attractors and basins are properties of the fixed-$\theta$ dynamics. A structural intervention changes $\theta$ and produces a detectable kernel discrepancy on pre-specified probe states, making structural change falsifiable. Using a Lyapunov drift condition, we show that stable internal dynamics approach bounded stability regions with exponentially attenuated transients and a noise-controlled residual floor. We characterize escape through a metric condition on intervention-induced attractor displacement and a baseline-relative KL lower bound for increasing escape probability. This analysis also explains why conditional mutual information alone cannot certify escape: it measures variation among intervention-conditioned updates rather than departure from the no-intervention law. Case studies in LLM code repair, sparse-reward reinforcement learning, and Bayesian optimization use matched continuation controls to illustrate how feedback strength and alignment affect quality-improving escape. Our contribution is an operational connection among stability tools, measurable intervention effects, and cross-domain diagnostics.