SD-DPC: Sparse Dictionary Differentiable Predictive Control

Ali Reza Daneshvar Garmroodi; Jan Drgoňa

eess.SYOct 1, 2026

SD-DPC: Sparse Dictionary Differentiable Predictive Control

Authors: Ali Reza Daneshvar Garmroodi, Jan Drgoňa

Organizations: Concordia University, Montreal, QC H3G1M8, Canada. · Johns Hopkins University, Baltimore, Maryland 21218, USA.

Abstract

We present sparse dictionary differentiable predictive control (SD-DPC), a framework for learning sparse, interpretable feedback policies for nonlinear systems from data. A prediction model is first identified by rollout-based sparse identification of nonlinear dynamics (SINDy), building on gradient-based and multistep formulations. The policy is then parameterized as a sparse combination of dictionary functions and trained by differentiating a constrained finite-horizon predictive-control objective through this model, so that its terms are selected by closed-loop performance rather than by imitating a previously trained controller. The result is an explicit feedback law with only a handful of terms. Across three benchmark control problems, SD-DPC satisfies the constraints in almost all test scenarios, outperforms a policy distilled onto the same terms by up to an order of magnitude, and requires orders of magnitude less memory and online computation than an optimization benchmark, while admitting explicit sensitivity bounds.

Figures & tables

Fig. 1: Pipeline: sparse system identification and differentiable predictive control policy learning.

Fig. 2: Policy sparsity versus

\ell_{1}

weight

\lambda_{\pi}

. Marker shape denotes pruning threshold

\tau_{\pi}

, color shows the loss (excluding the

\ell_{1}

penalty) on the validation scenarios used for selection, and black circles indicate the selected policies.

Fig. 3: Closed-loop trajectories of SD-DPC, NN-DPC, NN-Distilled, and the IPOPT benchmark, with the retained SD-DPC policy coefficients, for (a) the two-tank system, (b) the forced Van der Pol oscillator, and (c) the 2-D integrator with obstacle avoidance.

	Two-Tank	Van der Pol	2-D Int.
Identification
Trajectories $m_{\mathrm{id}}$	$2000$	$2000$	$1500$
Horizon $N_{\mathrm{id}}$	$5$	$3$	$3$
Integrator	RK4	RK4	discrete
Step $\Delta t$ [s]	$1.0$	$0.1$	–
Epochs $n_{e}$	$20\,000$	$6000$	$20\,000$

TABLE I: Identification and policy training parameters.

Method	Task loss $[10^{-3}]$	Control effort	Time $[\mu\mathrm{s}]$ / memory	Violations $[\%]\,/\,$ depth
Two-Tank
IPOPT	$8.8\!\times\!10^{-6}$	–	$8444\,/\,26$ MiB	$0.0\,/\,\text{--}$
NN-DPC	$0.144\pm 0.013$	–	$30\,/\,5.01$ kB	$0.0\,/\,\text{--}$
SD-DPC	$\mathbf{0.012\pm 0.006}$	–	$\mathbf{17\,/\,0.047}$ kB	$0.0\,/\,\text{--}$
NN-Distilled	$0.209\pm 0.039$	–	$17\,/\,0.047$ kB	$0.0\,/\,\text{--}$
Van der Pol

TABLE II: Controller evaluation across case studies.

Policy Learning

Policy learning develops decision rules that choose actions or interventions to maximize outcomes from data, feedback, or interaction.

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Aug 11, 2026J/K move · Enter open · S save

Topological Feasibility Guarantees for Differentiable Predictive Control

Guangyu Wu, Ján Drgoňa

Department of Mathematical Sciences, Chalmers University of Technology, Sweden (Incoming) · Department of Civil and Systems Engineering, Johns Hopkins University, USA

Model Predictive Control Feedback Control

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Differentiable predictive control (DPC), a self-supervised learning approach for approximating explicit model predictive control (MPC) policies, offers significant computational advantages over online optimization-based MPC. However, feasibility guarantees, a core requirement for safe control, are currently provided either probabilistically or via online safety filters. The lack of rigorous feasibility guarantees for offline policy optimization remains an open problem. This paper establishes deterministic feasibility guarantees for DPC using a novel topological analysis of the induced reachable safe set, without requiring online safety filters. By exploiting the inherent model-based nature of DPC, in which differentiable system dynamics are embedded directly into the computational graph, we analyze the properties of the learned control policies and the corresponding system states from topological and geometric perspectives. Inspired by our theoretical analysis, we propose a novel self-supervised offline policy learning strategy that utilizes a proxy loss with Control Barrier Functions (CBFs). Crucially, these properties not only significantly improve policy training but also enable the derivation of strict, deterministic feasibility guarantees from a finite number of training samples. Extensive closed-loop simulations validate our theoretical findings, demonstrating that the empirical constraint violations monotonically decrease to zero as the training sample size increases. Ultimately, this work illustrates that DPC policy optimization yields formal safety certificates that are structurally unattainable with conventional black-box methods, e.g., reinforcement learning (RL) or supervised learning-based approximate MPC, thereby providing a new perspective on feasibility guarantees in learning-based control.

Abstract

Policy Learning

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