# Linear Supervision for Nonlinear, High-Dimensional Neural Control and Differential Games

arXiv: https://arxiv.org/abs/2412.02033 · HTML: https://mlsanity.com/paper/2412.02033/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Dec 2024.

## Built on by (most cited first, 3 of 3)

- [Reachability Barrier Networks: Learning Hamilton-Jacobi Solutions for Smooth and Flexible Control Barrier Functions](https://mlsanity.com/paper/2505.11755/) (2025)
- [Delay compensation of multi-input distinct delay nonlinear systems via neural operators](https://mlsanity.com/paper/2509.17131/) (2025)
- [Forward Trajectory Steering for Hamilton-Jacobi Reachability Analysis](https://mlsanity.com/paper/2608.11480/) (2026)

## Abstract (arXiv)

As the dimension of a system increases, traditional methods for control and differential games rapidly become intractable, making the design of safe autonomous agents challenging in complex or team settings. Deep-learning approaches avoid discretization and yield numerous successes in robotics and autonomy, but at a higher dimensional limit, accuracy falls as sampling becomes less efficient. We propose using rapidly generated linear solutions to the partial differential equation (PDE) arising in the problem to accelerate and improve learned value functions for guidance in high-dimensional, nonlinear problems. We define two programs that combine supervision of the linear solution with a standard PDE loss. We demonstrate that these programs offer improvements in speed and accuracy in both a 50-D differential game problem and a 10-D quadrotor control problem.