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.

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