# Physics-informed neural networks via stochastic Hamiltonian dynamics learning

arXiv: https://arxiv.org/abs/2111.08108 · HTML: https://mlsanity.com/paper/2111.08108/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 31 references that are in the map.
- First posted on arXiv 15 Nov 2021.

## Builds on (most cited first, 16 of 16)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [Asynchronous Methods for Deep Reinforcement Learning](https://mlsanity.com/paper/1602.01783/) (2016)
- [An Introduction to Variational Autoencoders](https://mlsanity.com/paper/1906.02691/) (2019)
- [Hyperband: A Novel Bandit-Based Approach to Hyperparameter Optimization](https://mlsanity.com/paper/1603.06560/) (2016)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [You Only Propagate Once: Accelerating Adversarial Training via Maximal Principle](https://mlsanity.com/paper/1905.00877/) (2019)
- [Learning Continuous Control Policies by Stochastic Value Gradients](https://mlsanity.com/paper/1510.09142/) (2015)
- [Benchmarking Model-Based Reinforcement Learning](https://mlsanity.com/paper/1907.02057/) (2019)
- [Continuous Deep Q-Learning with Model-based Acceleration](https://mlsanity.com/paper/1603.00748/) (2016)
- [Maximum Principle Based Algorithms for Deep Learning](https://mlsanity.com/paper/1710.09513/) (2017)
- [Pontryagin Differentiable Programming: An End-to-End Learning and Control Framework](https://mlsanity.com/paper/1912.12970/) (2019)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)

## Built on by (most cited first, 1 of 1)

- [PHAST: Port-Hamiltonian Architecture for Structured Temporal Dynamics Forecasting](https://mlsanity.com/paper/2602.17998/) (2026)

## Abstract (arXiv)

In this paper, we propose novel learning frameworks to tackle optimal control problems by applying the Pontryagin maximum principle and then solving for a Hamiltonian dynamical system. Applying the Pontryagin maximum principle to the original optimal control problem shifts the learning focus to reduced Hamiltonian dynamics and corresponding adjoint variables. Then, the reduced Hamiltonian networks can be learned by going backwards in time and then minimizing loss function deduced from the Pontryagin maximum principle's conditions. The learning process is further improved by progressively learning a posterior distribution of the reduced Hamiltonians. This is achieved through utilizing a variational autoencoder which leads to more effective path exploration process. We apply our learning frameworks called NeuralPMP to various control tasks and obtain competitive results.