- The paper introduces WSINDYc, which leverages weak-form integration to produce noise-robust, sparse models for nonlinear model predictive control.
- Empirical evaluations demonstrate that WSINDYc achieves superior tracking performance and safety in challenging domains such as plasma regulation, drone navigation, and chaotic systems.
- The approach outperforms traditional SINDy and neural network methods by combining computational efficiency, interpretability, and robust performance under high noise.
WSINDy for Model Predictive Control: Advancing Data-Driven Nonlinear Control under Noisy Dynamics
Introduction
The challenge of discovering interpretable, robust models for high-dimensional, nonlinear, and noisy dynamical systems remains central in physics and engineering, especially for the purpose of closed-loop control. While the sparse identification of nonlinear dynamics (SINDy) framework has seen success in model discovery, its reliance on explicit numerical differentiation impedes its reliability under high noise—a critical limitation for real-world control scenarios. The work “WSINDy for Model Predictive Control with Applications to Fusion, Drones, and Chaos” (2604.23269) introduces the integration of the weak-form SINDy with control (WSINDYc) into nonlinear model predictive control (MPC), providing a compelling methodology for robust, data-driven feedback control. The paper presents extensive empirical evidence comparing WSINDY--MPC with SINDYc, its ensemble variants, DMDc, and neural networks across challenging domains including plasma boundary regulation, quadrotor navigation, and chaotic systems.
Unlike traditional SINDy methods that perform regression on derivative estimates, the WSINDYc paradigm employs a weak-formulation by integrating the governing equations against smooth, localized test functions and transferring differential operators onto these test functions via integration by parts. This approach enhances noise robustness due to its natural low-pass filtering effect and avoids instability from numerical differentiation. The WSINDYc framework extends naturally to the controlled setting, formulating regression problems that incorporate actuation input and facilitating identification of control-dependent nonlinearities. Solution sparsity is enforced via thresholding regression algorithms (e.g., MSTLS), with additional robustness or uncertainty quantification achieved by ensemble methods (E-WSINDYc).
Integration with Model Predictive Control
MPC is particularly sensitive to the fidelity of the predictive model in the optimization loop. WSINDYc models, by virtue of their noise-robust identification and parsimonious representations, serve as compelling candidates for MPC in settings with noisy measurements and limited data. The WSINDY--MPC workflow involves:
- Offline identification: State and actuation data are collected, models are identified using WSINDYc, and system update operators are constructed via standard numerical integration (e.g., RK4).
- Online control: At each sampling instant, the current noisy state is sensed; the learned WSINDYc model is used within the MPC optimization to compute a receding-horizon control sequence; only the first input is applied; this process is repeated online.
A crucial aspect of the framework is that online noise levels used in identification and feedback are matched, leading to realistic benchmarks for controller robustness.
Empirical Results and Comparative Evaluations
Plasma Boundary Regulation
Using high-fidelity, noisy simulation data from SOLPS-ITER, the study demonstrates that WSINDYc-derived models capture the essential plasma boundary dynamics with accuracy comparable to SINDYc in mild noise, but exhibit significantly enhanced robustness as noise levels escalate. Under increasing synthetic noise (up to order-unity amplitude relative to signal), both WSINDYc and its ensemble variants substantially outperform SINDYc and ensemble SINDYc (E-SINDYc) in achieving low trajectory tracking error and higher success rates in producing stable controllers. The results further confirm that while ensemble weak-form models (E-WSINDYc) may yield further marginal gains in some regimes, standard WSINDYc typically provides a favorable trade-off between accuracy and computational cost.
Drone Trajectory Tracking and Obstacle Avoidance
In a quadrotor MPC benchmark with aggressive maneuvers and obstacle constraints, WSINDYc-based controllers exhibit consistently lower trajectory tracking MSE and maintain critical obstacle clearance over a wide range of noise magnitudes. Competing methods, including SINDYc and its ensemble variant, are prone to trajectory divergence and violation of safety margins as sensor noise increases; in contrast, WSINDY--MPC and E-WSINDY--MPC maintain control reliability and safety until much higher noise amplitudes.
Chaotic System: Lorenz Attractor
For the canonical Lorenz 63 system under control, WSINDYc models yield superior predictive horizons and lower control cost in the presence of both moderate and substantial measurement noise. The method demonstrates robust performance even with limited training data, in contrast to neural network (NN) and DMDc baselines, which deteriorate rapidly under data scarcity or noise. While ensemble methods (E-WSINDYc) occasionally provide slight improvements, their increased computational overhead is offset by the already strong robustness of standard WSINDYc.
F-8 Aircraft Model
Similar findings are reported for a non-affine, highly nonlinear F-8 aircraft flight control task (details in Appendix D). WSINDY--MPC achieves tracking performance comparable to or exceeding that of baseline methods, with the additional advantage of reduced data and computational requirements relative to NNs.
Theoretical and Practical Implications
The integration of a weak-form regression approach with MPC decisively addresses the longstanding trade-off between data-driven model interpretability, parsimony, and robustness to experimental noise. Unlike black-box neural architectures, WSINDYc and its ensemble variants produce closed-form, sparse ODE models amenable to analysis and certification—properties of paramount importance in safety-critical physical systems (e.g., plasma control for fusion or autonomous flight). The documented noise tolerance arises from the mathematical properties of weak-form regression and is theoretically justified by recent analyses on unconditional and conditional asymptotic consistency [MessengerBortz2025IMAJNumerAnal].
On the practical side, WSINDYc models are shown to be computationally efficient both in training (often scaling linearly with data) and at run-time (due to sparse library evaluations), making them attractive for embedded and real-time control.
Limitations and Future Research Directions
A noted limitation of WSINDYc is its minimum data requirement—test functions must be supported on intervals sufficiently covered by training data, limiting applicability in extreme low-data regimes where classical SINDYc may still operate. Developing adaptive test function strategies tailored for data-limited settings is an open direction [TranBortz2025arXiv250703206]. Moreover, further validation on experimental datasets, especially with non-Gaussian or correlated noise, and in scenarios with partially observed or unknown actuation input, is warranted. Integration with online learning—enabling continuous refinement of models during operation—also presents a promising avenue for closing the loop between system identification and adaptive control. Extending the framework to systems governed by PDEs and incorporating mixed strong/weak formulations for hybrid modeling are theoretically feasible and likely to be impactful in large-scale, distributed control tasks.
Conclusion
The WSINDY--MPC framework (2604.23269) presents a comprehensive, principled approach for leveraging weak-form sparse regression to build interpretable, noise-robust models for nonlinear MPC. Empirical results across multiple domains establish its superior reliability and practicality relative to classical and black-box alternatives, particularly in realistic, noise-corrupted environments. The methodology holds significant promise for advancing data-driven control in complex physical systems, with ongoing research poised to address remaining challenges in data efficiency, adaptive learning, and generalization to broader classes of dynamical processes.