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Scientific Interests

As the core of scientific computing—the third scientific methodology—computational mathematics has grown rapidly with the advent of electronic computers, evolving into a major branch of mathematics and an indispensable theoretical tool for solving scientific and engineering problems using mathematics and computers. Among its subfields, numerical solutions of differential equations (including integro-differential equations and integral equations, collectively referred to as numerical solutions of differential equations for convenience) constitute a remarkably rich and active area. Since the CFL condition first appeared in 1928, numerical methods for differential equations have achieved tremendous success over nearly a century, culminating in three fundamental classes of numerical methods: finite differences in the 1950s, finite elements in the 1960s, and spectral methods in the 1970s. For regular types of differential equations, ready-made numerical schemes and even simulation software are directly available. The convergence of all three methods relies on the regularity of solutions, and achieving optimal convergence orders demands careful mesh refinement—with the prerequisite that a differential equation exists in the first place.

With the deepening of scientific research and the demands of societal development, some naturally arising common and important differential equations exhibit very high dimensionality. Solving high-dimensional differential equations using these mesh-based numerical methods encounters the so-called "curse of dimensionality": computational costs grow exponentially with dimension. To date, no universal method has been found to completely overcome this curse. Moreover, with the advent of data science and intelligence (brain) science, the situation worsens, as there may no longer be a differential equation to solve; instead, one must confront data or discrete networks directly, where regularity may be out of the question, and high dimensionality is inherent. Today's technologies can already produce high-resolution whole-brain images of organisms, such as the neural network of a fruit fly consisting of approximately 100,000 neurons, and a complete three-dimensional atlas of a mouse brain involving hundreds of millions of neurons. Exploring the mechanisms underlying intelligence inevitably requires tackling these ultra-large-scale neural networks and the mathematical models defined upon them. Although these scenarios were not the primary focus when numerical methods were developed a century ago, they are precisely the hotspots, key challenges, and difficulties that we must address as we look toward the next hundred years of computational mathematics. To this end, grounded in foundational frontier mathematical theories and the design of efficient practical algorithms, our group pursues interdisciplinary research integrating intelligence, quantum science, and computation from the following two perspectives.

>>> I. Differential Modeling and Numerical Methods

Our core research philosophy is to introduce stochastic particles and fully leverage their inherent adaptive properties to alleviate (or overcome) the curse of dimensionality: particles are randomly generated, randomly annihilated, and perform random walks in the corresponding space. To ensure computational accuracy and reliable results, the "birth," "death," and "movement" of particles are governed by a precise stochastic decision-making process. Generating enough particles ensures accuracy, annihilating redundant particles ensures efficiency, and particles moving with the solution—the three processes complement and reinforce each other—form an efficient high-dimensional adaptive numerical method. Borrowing the language of mesh-based adaptive methods: where the mesh needs refinement, more particles are generated to enhance resolution; where the mesh needs coarsening, more particles are annihilated to reduce computational cost; and the random walks of particles can be likened to mesh movement, capturing the dynamical behavior of the solution. In other words, the stochastic particle methods derived from this philosophy can simultaneously incorporate the three classical adaptive strategies: h-, p-, and r-refinement.

Specific research directions include: Numerical Methods for High-dimensional Problems, Computational Quantum Mechanics, Numerical Methods for the Wigner Equation, Differential Modeling and Numerical Methods for Low-Regularity/Singular/Unbounded/Long-Time Problems, and Mathematical Theory and Numerical Methods for the Dirac Equation.

=== Selected Publications === 

  • Jingyang Huang, Zhengyang Lei, Sihong Shao, Stochastic particle method with birth-death dynamics, Advances in Applied Mathematics and Mechanics 18 (2026) 1860.
  • Zhengyang Lei, Sihong Shao, Yunfeng Xiong, An efficient stochastic particle method for moderately high-dimensional nonlinear PDEs, Journal of Computational Physics 527 (2025) 113818.
  • Yunfeng Xiong, Sihong Shao, Overcoming the numerical sign problem in the Wigner dynamics via adaptive particle annihilation, SIAM Journal on Scientific Computing 46 (2024) B107.
  • Yunfeng Xiong, Yong Zhang, Sihong Shao, A characteristic-spectral-mixed scheme for six-dimensional Wigner-Coulomb dynamics, SIAM Journal on Scientific Computing 45 (2023) B906.
  • Sihong Shao, Yunfeng Xiong, Branching random walk solutions to the Wigner equation, SIAM Journal on Numerical Analysis 58 (2020) 2589.
  • Sihong Shao, Yunfeng Xiong, A branching random walk method for many-body Wigner quantum dynamics, Numerical Mathematics: Theory, Methods and Applications 12 (2019) 21.
  • Yunfeng Xiong, Zhenzhu Chen, Sihong Shao, An advective-spectral-mixed method for time-dependent many-body Wigner simulations, SIAM Journal on Scientific Computing 38 (2016) B491.
  • Sihong Shao, Jean Michel Sellier*, Comparison of deterministic and stochastic methods for time-dependent Wigner simulations, Journal of Computational Physics 300 (2015) 167.
  • Sihong Shao, Niurka R. Quintero, Franz G. Mertens, Fred Cooper, Avinash Khare, Avadh Saxena, Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity, Physical Review E 90 (2014) 032915.
  • Jian Xu, Sihong Shao, Huazhong Tang, Numerical methods for nonlinear Dirac equation, Journal of Computational Physics 245 (2013) 131.
  • Zhendong Li,Sihong Shao, Wenjian Liu, Relativistic explicit correlation: Coalescence conditions and practical suggestions, Journal of Chemical Physics 136 (2012) 144117.
  • Sihong Shao, Tiao Lu, Wei Cai, Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport, Communications in Computational Physics 9 (2011) 711.

 >>>> II. Discrete Modeling and Combinatorial Optimization

The core mathematical problem is the P vs NP problem. This direction shifts from the predominantly DE (Differential Equation)-based mathematical modeling paradigm to a DM (Discrete Model)-based one, emphasizing the design, analysis, and application of discrete structures.

Specific research directions include: Computational Complexity Theory, Combinatorial Optimization, Spectral Graph Theory and Algorithms, Efficient Algorithms for NPC Problems, and Continuous Algorithms for Graph Cut Problems.

==== Selected Publications ====

  • Sihong Shao, Chuan Yang, Xinyang Ye, Conductance estimation in digraphs: submodular transformation, Lovász extension, and Dinkelbach iteration, Accepted by SIAM Journal on Matrix Analysis and Applications on 2026/3/31.
  • Sihong Shao, Dong Zhang, Weixi Zhang, A simple iterative algorithm for maxcut, Journal of Computational Mathematics 42 (2024) 1277.
  • Kung-Ching Chang, Sihong Shao, Dong Zhang, Weixi Zhang, Lovász extension and graph cut, Communications in Mathematical Sciences 19 (2021) 761.
  • Kung-Ching Chang, Sihong Shao, Dong Zhang, Weixi Zhang, Nonsmooth critical point theory and applications to the spectral graph theory, SCIENCE CHINA Mathematics 64 (2021) 1.
  • Kung-Ching Chang, Sihong Shao, Dong Zhang, Nodal domains of eigenvectors for 1-Laplacian on graphs, Advances in Mathematics 308 (2017) 529.
  • Kung-Ching Chang, Sihong Shao, Dong Zhang, The 1-Laplacian Cheeger cut: Theory and algorithms, Journal of Computational Mathematics 33 (2015) 443.
  • Sihong Shao, Yishan Wu, An ODE approach to multiple choice polynomial programming, East Asian Journal on Applied Mathematics 15 (2025) 1.
  • Kung-Ching Chang, Sihong Shao, Dong Zhang, Cheeger's cut, maxcut and the spectral theory of 1-Laplacian on graphs, SCIENCE CHINA Mathematics 60 (2017) 1963.