This paper firstly introduces the development of quantum Monte Carlo and several types of typical quantum Monte Carlo methods in detail, and then summarizes the recent researches of condensed and warm dense matter systems, including transport, magnetism, superconductivity and thermodynamic properties. Finally, the development prospect of quantum Monte Carlo methods is discussed.

Inverse problems are of wide and important applications in many areas such as radar and sonar, medical imaging, nondestructive testing and geophysical prospection. Inverse problems are ill-posed problems, so it is challenging to construct stable and highly effective inversion algorithms for them. One of the important methods to tackle this challenging issue is to devise an appropriate regularization strategy based on the a priori information of the unknown solution. The success of traditional regularization methods heavily depends on correctly encoding the a priori information of the unknown solution into the inversion algorithms, but this is in general very difficult in practical computation. With the development of deep learning techniques in recent years, it becomes possible to directly learn the a priori information of the unknown solutions of the inverse problems from data, which is helpful in developing highly effective and stable inversion algorithms. In this paper, we review some recent progres on inversion algorithms based on deep learning, focusing mainly on those based on learnable regularization framework. In addition, we also summarize the advantages and shortcomings of the inversion algorithms based on deep learning, and discuss their future research directions.

Turbulent mixing induced by interfacial instabilities widely exists in various natural phenomena and engineering problems. Specifically, in typical engineering flows, involved turbulent mixing problems are usually triggered by intricate initial perturbations. This type of mixing problems often has various complexities such as coexistence of multiple mixing mechanisms, multi-stage continuous evolution, and wide and drastic changes in main control parameters. Accurately predicting the spatiotemporal evolution of mixing zone under the aforementioned complexity is crucial for engineering applications. In the current and foreseeable future, Reynolds-Averaged Navier-Stokes (RANS) and large eddy simulation (LES) modeling are the most feasible numerical methods. However, existing RANS and LES models face key challenges which must overcome, such as difficulty in uniformly predicting various mixing problems. This review will focus on these issues, standing from the perspective of engineering application, and provide a detailed overview of the latest understanding and existing problems of RANS and LES modeling among academic peers, as well as ours' recent progress.

Nanoscale heat conduction has attracted considerable attention due to the unique transport behaviors exhibited by its intrinsic heat-carrying phonons, as well as its significant potential in addressing thermal management issues in electronic devices and enhancing thermoelectric conversion efficiency. This paper reviews the research progress in nanoscale heat conduction theory and simulations over the past three decades. It particularly introduces and discusses methods based on both wave and particle perspectives, focusing on nanoscale coherent thermal conduction mechanisms. Finally, the paper analyzes the challenges facing the field of nanoscale heat conduction research and outlines potential future directions, aiming to deepen the understanding of the wave-particle duality of heat-carrying phonons and promote their application in critical areas.

In recent years, the use of machine learning methods to solve differential equations has attracted increasing attention from researchers in different fields. However, with the deepening of the research, researchers have begun to identify numerous challenges associated with the use of machine learning methods for solving time development equations. This paper presents a summary of the machine learning methods for solving the evolution equation. First, we present a summary of data-driven methods and deep learning methods based on equation learning. Then we introduce targeted algorithms for solving the problem under different neural network architectures. Finally, this paper presents a summary of the training features and recent work on the use of PINN method for solving the evolution equation. It also provides an outlook for future work.

With the development of artificial intelligence and scientific computing, deep learning plays a significant role in mathematical modeling. In this work we develop an active learning algorithm that uses microscopic data to establish a macroscopic model for collective behavior. Specifically, we take the Cucker-Smale model in this work and develop the corresponding active learning algorithm that integrates neural operator networks and Bayesian neural networks by utilizing microscopic particle data and partial physics. This algorithm is used to efficiently establishes the corresponding macroscopic Euler model through microscopic data. Finally, the effectiveness of the active learning algorithm is validated through one-dimensional and two-dimensional numerical simulations.

The radioactive particles in the cloud of near-earth nuclear explosion diffuse and settle under the driving of wind at different heights, causing a wide range of radioactive contamination. Traditional fallout prediction models are based on ideal terrain and meteorological conditions, which can not solve the problem of prediction of radioactive contamination under complex conditions. This paper introduces the research progress of the authors' team on the numerical prediction of radioactive contamination under complex conditions in the past 10 years. Firstly, the numerical simulation method of atmospheric transport and settlement of radioactive particles is established based on the gas-solid two-phase flow model. Then the raindrop collision model and cascade downscaling technique are used to realize the prediction of radioactive contamination in rainfall weather and complex terrain conditions respectively. Finally, the numerical prediction ability of radioactive contamination under complex conditions is used to calculate the distribution law of contaminated radiation environment, and some significant results are obtained.

In order to meet the demand for flow field prediction, this paper proposes KAN coupling model (KADS) combining Kolmogorov-Arnold network (KAN) and dynamic upsample (DySample: Upsampling by Dynamic Sampling), and uses two-dimensional diamond-shaped airfoil data to carry out flow field data prediction applications. In this paper, the activation function of the original KAN B-Spline is changed, and the KAN structures such as FourierKAN, GRBFKAN, RBFKAN, ChebyKAN are constructed, and their performance after coupling with DySample is evaluated. By comparing with the traditional MLP, it is found that ChebyKAN with Chebyshev polynomial as the activation function can achieve high accuracy with less training time and times, and there will be no overfitting during the test. The results show that the KADS model proposed in this paper can be applied to the task of flow field prediction and analysis, and can provide new modeling methods and ideas for the deep learning fluid intelligence modeling task.