# Learning the Riccati solution operator for time-varying LQR via Deep Operator Networks

arXiv: https://arxiv.org/abs/2604.18507 · HTML: https://mlsanity.com/paper/2604.18507/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 35 references that are in the map.
- First posted on arXiv 20 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [LQR-Assisted Whole-Body Control of a Wheeled Bipedal Robot with Kinematic Loops](https://mlsanity.com/paper/2005.11431/) (2020)
- [Modeling and LQR Control of Insect Sized Flapping Wing Robot](https://mlsanity.com/paper/2406.20061/) (2024)
- [A constraint-separation principle in model predictive control](https://mlsanity.com/paper/2005.01296/) (2020)

## Abstract (arXiv)

We propose a computational framework for replacing the repeated numerical solution of differential Riccati equations in finite-horizon Linear Quadratic Regulator (LQR) problems by a learned operator surrogate. Instead of solving a nonlinear matrix-valued differential equation for each new system instance, we construct offline an approximation of the associated solution operator mapping time-dependent system parameters to the Riccati trajectory. The resulting model enables fast online evaluation of approximate optimal feedbacks across a wide class of systems, thereby shifting the computational burden from repeated numerical integration to a one-time learning stage. From a theoretical perspective, we establish control-theoretic guarantees for this operator-based approximation. In particular, we derive bounds quantifying how operator approximation errors propagate to feedback performance, trajectory accuracy, and cost suboptimality, and we prove that exponential stability of the closed-loop system is preserved under sufficiently accurate operator approximation. These results provide a framework to assess the reliability of data-driven approximations in optimal control. On the computational side, we design tailored DeepONet architectures for matrix-valued, time-dependent problems and introduce a progressive learning strategy to address scalability with respect to the system dimension. Numerical experiments on both time-invariant and time-varying LQR problems demonstrate that the proposed approach achieves high accuracy and strong generalization across a wide range of system configurations, while delivering substantial computational speedups compared to classical solvers. The method offers an effective and scalable alternative for parametric and real-time optimal control applications.