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

TL;DR

This paper introduces a Deep Operator Network-based method to approximate Riccati solution operators in time-varying LQR problems, enabling fast, reliable, and scalable optimal control computations.

Contribution

It develops a novel operator surrogate framework for Riccati equations, with theoretical guarantees and tailored DeepONet architectures for efficient, scalable control solutions.

Findings

Achieves high accuracy and generalization in LQR problems.

Provides substantial computational speedups over classical solvers.

Ensures stability preservation under accurate operator approximation.

Abstract

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.