Selected Projects

High dimensional control with SympNet

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 demonstrate that our proposed method can solve a problem with more than 500 dimensions on a single GPU. The proposed method is scalable and has the potential to solve truly high-dimensional path planning problems in real-time.

architecture — We use SympNets to map the physical to a latent space where the potentially complex solution paths are stretched to straight lines

Our method can solve the high-dimensional path planning problem efficiently assuming that a physical model is completely known to us. We are exploring the possibility of combining our method with reinforcement learning models to handle uncertainty in the environments as a future work.

density — Path planning of 256 drones: minimizing energy cost while avoiding each other.

T. Meng, Z. Zhang(Co-first author), J. Darbon, G. Karniadakis. (Under review)

GENERIC formalism informed neural network

We propose the GENERIC formalism informed neural networks, which exactly satisfy the consistency conditions of thermodynamics, to facilitate prediction of the evolution of complex processes when the exact form of the governing equation is not readily available.
The general equation for non-equilibrium reversible-irreversible coupling (GENERIC) formalism provides a thermodynamically consistent way to characterize the evolutionary structure of molecular systems. It takes the form $\frac{dz}{dt} = L \frac{\delta E}{\delta z} + M \frac{\delta S}{\delta z}$, where $E$ and $S$ represent the energy and entropy of the system. $L$ and $M$ are the Poisson and dissipation matrices, which characterize the geometric and dissipation properties of the material in operator form. We design $L$, $M$, $E$, $S$ as neural networks with specific architectures to satisfy the structure and consistency condition of the formalism.

Our method can also be used to predict the stochastic dynamics of a diffusive particle and preserving the statistical properties better than purely data-driven methods, as can be seen from the figure below.

Z. Zhang, Y. Shin, G. Karniadakis. GFINNs: GENERIC Formalism Informed Neural Networks for Deterministic and Stochastic Dynamical Systems. Accepted by Philosophical Transactions of the Royal Society A.

Poisson Neural Networks

We propose the Poisson Neural Networks (PNNs) to learn Poisson system and perform long-time integration on such systems which can represent a broad class of dynamic motions.

Our new NN for physics, based on rigorous approximation theory can be used for video frame interpolation and long-time prediction of the motion of rigid body. The method is demonstrated successful on benchmark tasks including predicting the motion of a charged particle in a electromagnetic potential, solving the nonlinear Schrodinger equation and predicting two-body motion (image data) for long time.

P. Jin, Z. Zhang, Y. G. Kevrekidis, G. Karniadakis. Learning Poisson systems and trajectories of autonomous systems via Poisson neural networks. Accepted by IEEE TNNLS.

Symplectic Networks

Hamiltonian equations are the governing equations of classical mechanics as well as molecular dynamics. In this project, we proposed the so-called Symplectic Networks(SympNets) to learn the dynamics of arbitrary Hamiltonian equation.

The most interesting theoretical property of SympNets is that it can approximate arbitrary symplectic map:
Theorem If the activation function $\sigma$ is $r$-finite, then the set of SympNets $\Psi$ is $r$-uniformly dense on compacta in $\mathcal{SP}^r(U)$.
Practically, they are able to predict the motion of a pendulum, double pendulum and three body motion using limited amount of data. The algorithm is super efficient in the sense that one only need 14 parameters, to infer the motion of a pendulum using data from less than half period of the motion. SympNets could also have applications in generative modelings.

Traing data (left) and long-time prediction (right).

P. Jin, Z. Zhang(Co-first Author), A. Zhu, G. Karniadakis, Y. Tang. SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems. Neural Networks 132. 166-179 (2020)

Parallel version of a GIS algorithm(REU)

I spent a summer in the Joint Institute for Computational Science at UTK and ORNL in year 2016. I proposed a parallel version of the dasymetric mapping algorithm in GIS and implemented it in MPI. The new method effectively improved running efficiency. Check this website for my final report.