# Research

# Projects

Currently I'm conducting research on three topics

- Kinetic Theory
- Uncertainty Quantification
- Neural Differential Equations

## KINETIC THEORY: FROM ATOM TO CONTINUUM

We live in a world that is multi-scale, multi-layered and multi-physics in nature. David Hilbert's sixth problem pointed out an intriguing beginning to describe the behavior of interacting many-particle systems across different scales. The dramatically increasing computing power provides us the possibility to investigate the discrete Hilbert’s passage. This project is dedicated to developing numerical algorithms for the study of multi-scale transports of molecule, plasma, electrons, photons, phonons, skyrmion, etc. Specifically, the attention is paid on:

- Direct solvers for the Boltzmann and related models
- Moment models with physics-oriented closures
- Particle-based simulation frameworks

## UNCERTAINTY QUANTIFICATION: FROM DATA TO KNOWLEDGE

How will the climate develop, how secure is our energy supply, and what chances does molecular medicine offer? The rapidly increasing amount of data offers radically new opportunities to address today’s most pressing questions but also requires novel mathematical and statistical methods to handle them. The uncertainties coming from stochastic data or immature prior knowledge are often considered as an unavoidable burden in real-world applications. By employing probabilistic data science techniques, uncertainty can also be turned into a valuable source of information and a powerful enrichment of black-box approaches from artificial intelligence. To harness this source of information, in this project we identify common challenges between several common use cases and foster translational research at the interface of disciplinary and mathematical research. The goal is to enable more reliable knowledge sourcing from data by developing tools and methods within the field of uncertainty quantification (UQ).

## NEURAL DIFFERENTIAL EQUATIONS: FROM MACHINE TO REALITY

The thriving deep neural networks provide the possibility for solving problems that seemed challenging in the past, e.g., computer vision and natural language processing. The same momentum is beginning to build in computational sciences, leading to a field which has been called scientific machine learning (SciML). While typical classification and regression tasks in classical machine learning applications mostly handle discrete and localized data, in scientific machine learning, information at different locations is expected to be connected by mathematical modeling and physical constraints, e.g. ordinary and partial differential equations. The goal of this project is to use deep neural networks as building blocks in a numerical method to solve the unified mechanical-neural models. Basically the governing equation is a coarse-grained model of reality that possesses an intricate structure that a numerical method needs to preserve. Thus, artificial neural networks are likely to be beneficial, but cannot be used out-of-the-box. The project therefore touches upon the applications point of view of deep learning with a focus on interpretability and robustness, and on the mathematical methodologies point of view to solve differential and integral equations.

# OUTCOME

I can be found on Google Scholar and ResearchGate.

**Preprints**

Tianbai Xiao. A flux reconstruction kinetic scheme for the Boltzmann equation. arXiv:2103.10371v1 [physics.comp-ph] 18 Mar 2021. [PDF]

Tianbai Xiao and Martin Frank. Using neural networks to accelerate the solution of the Boltzmann equation. arXiv:2010.13649v1 [physics.comp-ph] 26 Oct 2020. [PDF]

Tianbai Xiao. Modeling and Simulation of Non-equilibrium Flows with Uncertainty Quantification. arXiv:2008.02503v1 [physics.comp-ph] 6 Aug 2020. [PDF]

**2021**

Tianbai Xiao and Martin Frank. A stochastic kinetic scheme for multi-scale flow transport with uncertainty quantification. Journal of Computational Physics, 437: 110337, 2021. [PDF]

Tianbai Xiao and Martin Frank. A stochastic kinetic scheme for multi-scale plasma transport with uncertainty quantification. Journal of Computational Physics, 432: 110139, 2021. [PDF]

**2020**

Tianbai Xiao, Chang Liu, Kun Xu, and Qingdong Cai. A velocity-space adaptive unified gas kinetic scheme for continuum and rarefied flows. Journal of Computational Physics, 415: 109539, 2020. [PDF]

**2019**

Tianbai Xiao, Kun Xu, and Qingdong Cai. A unified gas-kinetic scheme for multiscale and multicomponent flow transport. Applied Mathematics and Mechanics, 40(3), 355-372, 2019. [PDF]

**2018**

Tianbai Xiao, Kun Xu, Qingdong Cai, and Tiezheng Qian. An investigation of non-equilibrium heat transport in a gas system under external force field. International Journal of Heat and Mass Transfer, 126: 362-379, 2018. [PDF]

**2017**

Tianbai Xiao, Qingdong Cai, and Kun Xu. A well-balanced unified gas-kinetic scheme for multiscale flow transport under gravitational field. Journal of Computational Physics, 332: 475-491, 2017. [PDF]

**2016**

Tianbai Xiao and Kun Xu. Investigation of multiscale non-equilibrium flow dynamics under external force field. arXiv:1610.05544 [physics.flu-dyn] 19 Oct 2016. [PDF]