# Research

academic & industrial

# Projects

Currently I'm conducting research on three topics

- Kinetic Theory
- Uncertainty Quantification
- Neural Differential Equations

## KINETIC THEORY: FROM ATOM TO CONTINUUM

The world is multi-scale, multi-layered and multi-physics in nature. David Hilbert's sixth problem pointed out an intriguing beginning to link the behavior of interacting many-particle systems across different scales. The dramatically increasing computing power provides us the possibility to investigate the Hilbert’s passage from a numerical point of view. 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 focus is 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 named 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., differential equations. The goal of this project is to leverage deep neural networks as building blocks in a numerical method to solve unified mechanical-neural models. Basically a governing equation is a coarse-grained model of reality that possesses an intricate structure, which 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 and Martin Frank. RelaxNet: A structure-preserving neural network to approximate the Boltzmann collision operator. arXiv:2211.08149v1 [physics.comp-ph] 15 Nov. 2022. [PDF]

Jonas Kusch, Steffen Schotthöfer, Pia Stammer, Jannick Wolters, and Tianbai Xiao. KiT-RT: An extendable framework for radiative transfer and therapy. arXiv:2205.08417v1 [physics.med-ph] 12 May 2022. [PDF]

Tianbai Xiao, Steffen Schotthöfer, and Martin Frank. Predicting continuum breakdown with deep neural networks. arXiv:2203.02933v1 [physics.flu-dyn] 6 Mar. 2022. [PDF]

**2023**

Tianbai Xiao, Jonas Kusch, Julian Koellermeier, and Martin Frank. A Flux Reconstruction Stochastic Galerkin Scheme for Hyperbolic Conservation Laws. Journal of Scientific Computing, 95(1): 18, 2023. [PDF]

**2022**

Tianbai Xiao. A Well-Balanced Unified Gas-Kinetic Scheme for Multicomponent Flows under External Force Field. Entropy, 24(8): 1110, 2022. [PDF]

Tianbai Xiao. An Investigation of Uncertainty Propagation in Non-equilibrium Flows. International Journal of Computational Fluid Dynamics, 36(4): 294-318, 2022. [PDF]

Steffen Schotthöfer, Tianbai Xiao, Martin Frank, and Cory D. Hauck. Structure Preserving Neural Networks: A Case Study in the Entropy Closure of the Boltzmann Equation. Proceedings of the 39th International Conference on Machine Learning, PMLR: 19406-19433, 2022. [PDF][Video][Poster]

Steffen Schotthöfer, Tianbai Xiao, Martin Frank, and Cory D. Hauck. Neural network-based, structure-preserving entropy closures for the Boltzmann moment system. arXiv:2201.10364v1 [math.NA] 25 Jan. 2022. [PDF]

**2021**

Tianbai Xiao. A flux reconstruction kinetic scheme for the Boltzmann equation. Journal of Computational Physics, 447: 110689, 2021. [PDF]

Tianbai Xiao and Martin Frank. Using neural networks to accelerate the solution of the Boltzmann equation. Journal of Computational Physics, 443: 110521, 2021. [PDF]

Steffen Schotthöfer, Tianbai Xiao, Martin Frank, and Cory D. Hauck. A structure-preserving surrogate model for the closure of the moment system of the Boltzmann equation using convex deep neural networks. AIAA Aviation Forum, 2021. [PDF]

Tianbai Xiao. Kinetic.jl: A portable finite volume toolbox for scientific and neural computing. Journal of Open Source Software, 6(62): 3060, 2021. [PDF]

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

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: 109535, 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]