# SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems

arXiv: https://arxiv.org/abs/2201.05475 · HTML: https://mlsanity.com/paper/2201.05475/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 94 references that are in the map.
- First posted on arXiv 14 Jan 2022.

## Builds on (most cited first, 30 of 38)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [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)
- [DeepReach: A Deep Learning Approach to High-Dimensional Reachability](https://mlsanity.com/paper/2011.02082/) (2020)
- [GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems](https://mlsanity.com/paper/2109.00092/) (2021)
- [Adaptive Deep Learning for High-Dimensional Hamilton-Jacobi-Bellman Equations](https://mlsanity.com/paper/1907.05317/) (2019)
- [Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks](https://mlsanity.com/paper/2012.03133/) (2020)
- [Actor-Critic Method for High Dimensional Static Hamilton--Jacobi--Bellman Partial Differential Equations based on Neural Networks](https://mlsanity.com/paper/2102.11379/) (2021)
- [Algorithms for Overcoming the Curse of Dimensionality for Certain Hamilton-Jacobi Equations Arising in Control Theory and Elsewhere](https://mlsanity.com/paper/1605.01799/) (2016)
- [Deep neural networks algorithms for stochastic control problems on finite horizon: convergence analysis](https://mlsanity.com/paper/1812.04300/) (2018)
- [Overcoming the curse of dimensionality for some Hamilton--Jacobi partial differential equations via neural network architectures](https://mlsanity.com/paper/1910.09045/) (2019)
- [Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications](https://mlsanity.com/paper/1812.05916/) (2018)
- [Mitigating the Curse of Dimensionality: Sparse Grid Characteristics Method for Optimal Feedback Control and HJB Equations](https://mlsanity.com/paper/1507.04769/) (2015)
- [Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems](https://mlsanity.com/paper/1903.06652/) (2019)
- [Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs](https://mlsanity.com/paper/1702.04400/) (2017)
- [Perspectives on characteristics based curse-of-dimensionality-free numerical approaches for solving Hamilton-Jacobi equations](https://mlsanity.com/paper/1711.03314/) (2017)
- [Linear Hamilton Jacobi Bellman Equations in High Dimensions](https://mlsanity.com/paper/1404.1089/) (2014)
- [On Decomposition Models in Imaging Sciences and Multi-time Hamilton-Jacobi Partial Differential Equations](https://mlsanity.com/paper/1906.09502/) (2019)
- [QRnet: optimal regulator design with LQR-augmented neural networks](https://mlsanity.com/paper/2009.05686/) (2020)
- [Deep backward schemes for high-dimensional nonlinear PDEs](https://mlsanity.com/paper/1902.01599/) (2019)
- [Robust feedback control of nonlinear PDEs by numerical approximation of high-dimensional Hamilton-Jacobi-Isaacs equations](https://mlsanity.com/paper/1905.06276/) (2019)
- [Using Neural Networks to Compute Approximate and Guaranteed Feasible Hamilton-Jacobi-Bellman PDE Solutions](https://mlsanity.com/paper/1611.03158/) (2016)
- [Neural network architectures using min-plus algebra for solving certain high dimensional optimal control problems and Hamilton-Jacobi PDEs](https://mlsanity.com/paper/2105.03336/) (2021)
- [Safe Sequential Path Planning of Multi-Vehicle Systems via Double-Obstacle Hamilton-Jacobi-Isaacs Variational Inequality](https://mlsanity.com/paper/1412.7223/) (2014)
- [Minimum-time trajectory generation for quadrotors in constrained environments](https://mlsanity.com/paper/1706.06478/) (2017)
- [Generating Robust Supervision for Learning-Based Visual Navigation Using Hamilton-Jacobi Reachability](https://mlsanity.com/paper/1912.10120/) (2019)
- [Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems](https://mlsanity.com/paper/2110.02541/) (2021)
- [Error analysis for POD Approximations of infinite horizon problems via the Dynamic Programming approach](https://mlsanity.com/paper/1511.00265/) (2015)
- [Reachability-Based Safety and Goal Satisfaction of Unmanned Aerial Platoons on Air Highways](https://mlsanity.com/paper/1602.08150/) (2016)

## Built on by (most cited first, 9 of 9)

- [Anant-Net: Breaking the Curse of Dimensionality with Scalable and Interpretable Neural Surrogate for High-Dimensional PDEs](https://mlsanity.com/paper/2505.03595/) (2025)
- [Symplectic Neural Networks Based on Dynamical Systems](https://mlsanity.com/paper/2408.09821/) (2024)
- [Data-driven initialization of deep learning solvers for Hamilton-Jacobi-Bellman PDEs](https://mlsanity.com/paper/2207.09299/) (2022)
- [Offline Supervised Learning V.S. Online Direct Policy Optimization: A Comparative Study and A Unified Training Paradigm for Neural Network-Based Optimal Feedback Control](https://mlsanity.com/paper/2211.15930/) (2022)
- [Solving Elliptic Optimal Control Problems via Neural Networks and Optimality System](https://mlsanity.com/paper/2308.11925/) (2023)
- [PINNs in PDE Constrained Optimal Control Problems: Direct vs Indirect Methods](https://mlsanity.com/paper/2604.04920/) (2026)
- [Neural Network Optimal Feedback Control with Guaranteed Local Stability](https://mlsanity.com/paper/2205.00394/) (2022)
- [Deep learning methods for inverse problems using connections between proximal operators and Hamilton-Jacobi equations](https://mlsanity.com/paper/2512.23829/) (2025)
- [Hessian-augmented Supervised Learning for Hamilton-Jacobi-Bellman PDEs](https://mlsanity.com/paper/2606.23827/) (2026)

## Abstract (arXiv)

Solving high-dimensional optimal control problems in real-time is an important but challenging problem, with applications to multi-agent path planning problems, which have drawn increased attention given the growing popularity of drones in recent years. In this paper, we propose a novel neural network method called SympOCnet that applies the Symplectic network to solve high-dimensional optimal control problems with state constraints. We present several numerical results on path planning problems in two-dimensional and three-dimensional spaces. Specifically, we demonstrate that our SympOCnet can solve a problem with more than 500 dimensions in 1.5 hours on a single GPU, which shows the effectiveness and efficiency of SympOCnet. The proposed method is scalable and has the potential to solve truly high-dimensional path planning problems in real-time.