# Data-driven initialization of deep learning solvers for Hamilton-Jacobi-Bellman PDEs

arXiv: https://arxiv.org/abs/2207.09299 · HTML: https://mlsanity.com/paper/2207.09299/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 21 references that are in the map.
- First posted on arXiv 19 Jul 2022.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Semiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation](https://mlsanity.com/paper/2002.08625/) (2020)
- [Pricing options and computing implied volatilities using neural networks](https://mlsanity.com/paper/1901.08943/) (2019)
- [SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems](https://mlsanity.com/paper/2201.05475/) (2022)
- [Towards Robust and Stable Deep Learning Algorithms for Forward Backward Stochastic Differential Equations](https://mlsanity.com/paper/1910.11623/) (2019)
- [Gradient-augmented Supervised Learning of Optimal Feedback Laws Using State-dependent Riccati Equations](https://mlsanity.com/paper/2103.04091/) (2021)
- [An Efficient Policy Iteration Algorithm for Dynamic Programming Equations](https://mlsanity.com/paper/1308.2087/) (2013)

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

- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)
- [GKNet: Graph Kalman Filtering and Model Inference via Model-based Deep Learning](https://mlsanity.com/paper/2506.22004/) (2025)
- [Learning Nash Equilibrial Hamiltonian for Two-Player Collision-Avoiding Interactions](https://mlsanity.com/paper/2503.07013/) (2025)

## Abstract (arXiv)

A deep learning approach for the approximation of the Hamilton-Jacobi-Bellman partial differential equation (HJB PDE) associated to the Nonlinear Quadratic Regulator (NLQR) problem. A state-dependent Riccati equation control law is first used to generate a gradient-augmented synthetic dataset for supervised learning. The resulting model becomes a warm start for the minimization of a loss function based on the residual of the HJB PDE. The combination of supervised learning and residual minimization avoids spurious solutions and mitigate the data inefficiency of a supervised learning-only approach. Numerical tests validate the different advantages of the proposed methodology.