Haomin Zhou
Georgia Institute of Technology, GA·Mathematics & Statistics
Partial differential equations (PDEs) describe how quantities such as heat, fluids, probabilities and prices change across space and time. Most PDEs cannot be solved with pen and paper, so they are simulated on computers.
- Fields
Mathematics & Statistics
Computer Science
- Topics
partial differential equations
high-dimensional PDEs
curse of dimensionality
- Grants
- 2 known grants, through Aug 2028
What they work on
- Establish parameterized Wasserstein sub-manifolds, which represent evolving probability distributions through a neural network's parameters via a push-forward map and pullback metric, and develop their mathematical foundations.
- Develop efficient formulations of parameterized Wasserstein gradient and Hamiltonian flows, prove convergence and error estimates, and apply them to classical PDEs such as the Fokker-Planck and Schrödinger equations.
- Create a supervised learning method using density coupling to solve high-dimensional Hamilton-Jacobi equations, which describe optimal control problems.
- Build a parameter control framework and a deep tangent bundle method for rapid simulation of high-dimensional evolution PDEs across varying conditions, with analysis of computational complexity and error bounds.
More about the research
When a problem has many variables, the cost of simulation grows explosively, a barrier called the curse of dimensionality. Deep neural networks scale better as dimensions grow, but there is little theory explaining when they give reliable answers. These two collaborative projects build mathematical theory and algorithms for solving high-dimensional PDEs with neural networks. Both draw on optimal transport, the mathematics of moving one probability distribution into another at the least cost.
Why it matters
PDEs model physical, chemical and biological processes, financial systems and engineering designs, but high-dimensional versions are often too costly to simulate. Reliable neural-network solvers with proven error guarantees could make such simulations practical and faster to repeat under changing conditions. The work also strengthens the mathematical foundations behind related tools such as generative models and mean-field control.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Collaborative Research: Theory, computation and applications of parameterized solution operator approximation for high-dimensional PDEs
NSF · Sep 2025 – Aug 2028
Collaborative Research: Theory, computation and applications of parameterized Wasserstein gradient and Hamiltonian flows
NSF · Jul 2023 – Jun 2027
Closest labs
Yiran Wang
University of Alabama, AL·Mathematics & Statistics
Develop hybrid reduced order models in which neural networks add basis functions to precomputed multiscale ones, guided by stable Petrov-Galerkin formulations and residual-based error indicators.
Hung Tran
University of Wisconsin-Madison, WI·Mathematics & Statistics
Establish regularity and large-time behavior results for Hamilton-Jacobi equations, a class of nonlinear PDE from optimal control, and use them to prove that mass-conserving solutions of critical coagulation-fragmentation equations exist.
Tamer Zaki
Johns Hopkins University, MD·Engineering
Develop evolutional deep neural network (EDNN) algorithms that solve high-dimensional partial differential equations by evolving network parameters according to the governing physics, without relying on training data.
The school
Georgia Institute of Technology: 50 labs on PI Pages, $1.5B a year on research, 572 research doctorates a year.
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