Differentiable Model Predictive Control (Differentiable MPC) is a framework that embeds a Model Predictive Control optimization solver as a differentiable layer within a neural network, enabling end-to-end gradient-based learning of the dynamics model and cost function that drive the controller — combining MPC's constraint satisfaction and safe planning guarantees with deep learning's ability to learn complex system models from data — making it possible to learn interpretable, physically-grounded control policies for robotics, autonomous vehicles, and industrial systems where constraint satisfaction is non-negotiable.

What Is Differentiable MPC?

• MPC Background: Model Predictive Control solves a finite-horizon optimization problem at each timestep — finding the sequence of K actions that minimizes a cost function subject to dynamics constraints, then executes only the first action and re-plans (receding horizon).
• Differentiable Extension: By differentiating through the MPC optimization (using implicit differentiation or differentiable QP solvers), gradients of the task loss can flow backward through the entire control pipeline — updating the learned dynamics model and cost function jointly.
• Learning the Model: Rather than manually engineering a physics model, the agent learns a neural dynamics model f(s, a) → s' that is used inside the MPC optimizer.
• Learning the Cost: Rather than manually specifying the cost function, it can be learned from demonstrations or task reward — the optimizer finds the action sequence minimizing this learned cost.

Why Differentiability Matters

• End-to-End Training: The controller, dynamics model, and cost function can all be updated together with a single backward pass — standard autoML optimization replaces manual system identification.
• Safety by Design: Unlike black-box neural policies, MPC enforces explicit state/action constraints at every step — critical for physical systems where constraint violation causes hardware damage or safety incidents.
• Interpretability: The learned dynamics model is explicit and inspectable — engineers can examine what the system predicts and diagnose failure modes.
• Data Efficiency: Physics priors encoded in the MPC structure reduce the amount of data needed to learn a competent controller compared to pure model-free methods.

Key Technical Approaches

OptNet (Amos & Kolter, 2017):

• Embeds quadratic programming (QP) solvers as differentiable layers via implicit differentiation through KKT conditions.
• First general framework for differentiable constrained optimization in neural networks.
• Foundation for differentiable MPC implementations.

DMPC (Amos et al., 2018):

• Applies OptNet's QP differentiation to the MPC setting — linear dynamics with quadratic cost.
• Demonstrated learning dynamics and cost from demonstrations with analytical gradients.

Neural MPC / CausalMPC:

• Replaces linear dynamics assumption with learned neural dynamics model.
• Combines uncertainty-aware ensemble models with MPC for robust control under model error.

Applications

Differentiable MPC is the union of physics-aware planning and data-driven learning — enabling AI systems that respect hard real-world constraints while continuously improving their understanding of complex dynamics from experience, bridging the gap between classical control theory and modern deep learning.