✨ TL;DR
This paper uses Deep Operator Networks (DeepONets) to learn a surrogate for the Riccati differential equation solution operator in finite-horizon LQR problems, enabling fast approximate optimal control without repeated numerical integration. The approach includes theoretical guarantees on stability and performance, and demonstrates significant computational speedups while maintaining high accuracy.
Solving finite-horizon Linear Quadratic Regulator (LQR) problems requires repeatedly solving differential Riccati equations—nonlinear matrix-valued differential equations—for each new system instance or parameter configuration. This repeated numerical integration is computationally expensive and becomes a bottleneck in parametric and real-time optimal control applications where many system configurations must be evaluated or control must be computed rapidly. The challenge is particularly acute for time-varying systems where the Riccati equation must be solved over the entire time horizon for each query.
The authors propose learning an approximation of the solution operator that maps time-dependent system parameters to the entire Riccati solution trajectory using Deep Operator Networks (DeepONets). The computational burden is shifted to a one-time offline learning stage where the operator surrogate is trained, after which online evaluation becomes fast function evaluation rather than differential equation solving. They design specialized DeepONet architectures for matrix-valued, time-dependent problems and introduce a progressive learning strategy to handle scalability as system dimension increases. The framework constructs an operator approximation that can generalize across a wide class of system configurations.
What the paper shows.
Numerical experiments on both time-invariant and time-varying LQR problems demonstrate that the proposed DeepONet-based approach achieves high accuracy in approximating Riccati solutions and strong generalization across diverse system configurations. The method delivers substantial computational speedups compared to classical numerical solvers for the Riccati differential equation. The learned operator surrogate maintains performance across a wide range of parameter values not seen during training, validating both the approximation quality and the practical utility of the approach for parametric optimal control scenarios.
The paper does not explicitly detail limitations, but implicit constraints include: the approach is restricted to LQR problems with their specific structure and may not directly extend to nonlinear optimal control; scalability with system dimension, while addressed through progressive learning, still presents challenges for very high-dimensional systems; the quality of the learned operator depends on the coverage and representativeness of the offline training data; and the theoretical guarantees require sufficiently accurate operator approximation, which may demand substantial training resources for complex or high-dimensional systems.
✨ Generated by Claude · Apr 21, 2026 · Read the PDF for authoritative content.