Applied Mathematics and Data Science III

Title: An energy-stable mixed finite element method for Rosensweig ferrofluid flow model

Speaker: Xiaoping Xie，Sichuan University

Abstract: We develop a mixed finite element method for the Rosensweig's ferrofluid flow model. First, we establish some regularity results for the weak solution.  Next,  for  the  spatial semi-discretization of the model using mixed finite elements we  show that energy inequality of the continuous equation is preserved and  give   optimal error estimates in $L^\infty(L^2)$ and $L^2(H^1)$ norms. For  the full discretization using implicit Euler scheme we show the existence and uniqueness of solutions, the unconditional stability  and  optimal error estimates. Finally, we provide numerical experiments to verify the theoretical results.  

Title: Levy Score Function and Score-based Particle Method for Levy-Fokker-Planck Equation

Speaker: Xiang Zhou，City University of Hong Kong

Abstract: The score function, also known as the score or the gradient of the log-density, is a fundamental concept in the diffusion process with applications in generative modeling and density estimation. However, the corresponding score function of the Levy jump process is elusive. This talk addresses the challenges of mathematical derivation, numerical algorithm, and error analysis of the corresponding score function in non-Gaussian systems with jumps and discontinuity represented by the nonlinear Levy--Fokker--Planck equations.  We propose the Levy score function, which features an extra non-local double-integral term, and then design the score matching algorithm by minimizing the new loss function. Based on the probability flow equivalence between deterministic systems, we develop the particle method as an application of this generalized score function to sample the stochastic process at any discrete time point.  We proved the error analysis that the Kullback--Leibler divergence between the numerical and theoretical solutions is bounded by the score-matching loss function, as well as the discretization error of time step size. To show the usefulness and efficiency of our approach, numerical examples from applications in biology and finance are tested.  

Title: An AI-aided algorithm for multivariate polynomial reconstruction on Cartesian grids and its applications to numerical PDEs

Speaker: Qinghai Zhang，Zhejiang University

Abstract: Polynomial reconstruction on Cartesian grids is fundamental in many scientific and engineering applications, yet it is still an open problem how to construct for a rectangular grid a lattice T so that multivariate polynomial interpolation on this lattice is unisolvent. In this work, we solve this open problem of poised lattice generation (PLG) via an interdisciplinary research of approximation theory, abstract algebra, and artificial intelligence. Based on this algorithm, we further develop fourth- and higher-order finite-difference and finite-volume methods that retain the simplicity of Cartesian grids yet overcomes the disadvantage of legacy finite difference methods in handling irregular geometries. Test results of a variety of PDEs confirm the high-order convergence of the proposed methods.  

Title: Physics-informed Data-driven Cavitation Model for a Specific Mie–Grüneisen Equation of State

Speaker: Wenjun Ying，Shanghai Jiao Tong University

Abstract: We will present a novel one-fluid cavitation model of a specific Mie-Grüneisen equation of state(EOS), named polynomial EOS, based on an artificial neural network. Not only the physics-informed equation but also the experimental data are embedded into the proposed model by an optimization problem. The physics-informed data-driven model provides the concerned pressure within the cavitation region, where the pressure tends to zero, and the density has the same trend. The present model is then applied to computing the challenging compressible multi-phase flow simulation, such as nuclear and underwater explosions. Numerical simulations show that our model in application agrees well with the corresponding experimental data, ranging from one dimension to three dimensions with the h−adaptive mesh refinement algorithm and load balance techniques in the structured and unstructured grids.

Title: DeepParticle: learning invariant measure by a deep neural network minimizing Wasserstein distance on data generated from an interacting particle method

Speaker: Zhiwen Zhang，The University of Hong Kong

Abstract: High dimensional partial differential equations (PDE) are challenging to compute by traditional mesh-based methods especially when their solutions have large gradients or concentrations at unknown locations. Mesh-free methods are more appealing; however, they remain slow and expensive when a long time and resolved computation is necessary. In this talk, we present DeepParticle, an integrated deep learning (DL), optimal transport (OT), and interacting particle (IP) approach through a case study of Fisher-Kolmogorov-Petrovsky-Piskunov front speeds in incompressible flows. PDE analysis reduces the problem to the computation of the principal eigenvalue of an advection-diffusion operator. Stochastic representation via the Feynman-Kac formula makes possible a genetic interacting particle algorithm that evolves particle distribution to a large time-invariant measure from which the front speed is extracted. The invariant measure is parameterized by a physical parameter (the Peclet number). We learn this family of invariant measures by training a physically parameterized deep neural network on affordable data from IP computation at moderate Peclet numbers, then predict at a larger Peclet number when IP computation is expensive. Our methodology extends to a more general context of deep learning stochastic particle dynamics. For instance, we can learn and generate aggregation patterns in Keller-Segel chemotaxis systems.

Title: 结合物理方程的深度学习方法在海洋温盐重构和海洋内波中的应用初探

Speaker: Chunxin Yuan，Ocean University of China

Abstract: 人工智能方法近年来蓬勃发展，极大地促进了很多领域的研究。海洋中的动力学现象具有非常典型的四维时空特征，而且针对这些现象的观测数据也存在海洋表面数据多，而内部数据少的特点。这两个典型特点导致一些纯数据驱动的人工智能方法在物理海洋学的研究中取得的效果并不总是令人满意。因此，这里我们采用PINN和DeepONet深度网络学习方法将描述海洋动力过程的理论方程纳入到人工智能方法中，建立了由海洋表面的观测数据重构海洋内部数据的智能预报模型，其效果远优于纯数据驱动的人工智能模型和传统的动力学模型。

更进一步地，采用具有明确物理含义的海洋内波模态函数取代传统POD方法中的基函数，极大地提高了海洋内波模型的计算效率。该方法体现物理规律能够提升人工智能方法在海洋研究领域的泛化能力和计算精度。

Title: Transformed Model Reduction for Partial Differential Equations with Sharp Inner Layers

Speaker: Xianmin Xu，The State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences

Abstract: Small parameters in partial differential equations can give rise to solutions with sharp inner layers that evolve over time. However, the standard model reduction method becomes inefficient when applied to these problems due to the slowly decaying Kolmogorov N-width of the solution manifold. In this talk, we will present some recent efforts to deal with the difficulties. In particular, we show a new approach to transform the equation in such a way that the transformed solution manifold exhibits a fast decaying Kolmogorov N-width. We employ asymptotic analysis to identify slow variables and perform a transformation of the partial differential equations accordingly. Subsequently, we apply the Proper Orthogonal Decomposition (POD) method and a qDEIM technique to the transformed equation with the slow variables. Numerical experiments demonstrate that the new model reduction method yield significantly improved results compared to direct model reduction applied to the original equation. Furthermore, this approach can be applied to some well-known equations, such as the Allen-Cahn equation, the convection equation and the Burgers equation, etc.

Title: Beyond Unconstrained Features: Neural Collapse for Shallow Neural Networks with General Data

Speaker: Shuyang Ling，New York University Shanghai

Abstract: Neural collapse (NC) is a phenomenon that emerges at the terminal phase of the training (TPT) of deep neural networks (DNNs). The features of the data in the same class collapse to their respective sample means and the sample means exhibit a simplex equiangular tight frame (ETF). In the past few years, there has been a surge of works that focus on explaining why the NC occurs and how it affects generalization. Since the DNNs are notoriously difficult to analyze, most works mainly focus on the unconstrained feature model (UFM). In this work, we focus on shallow ReLU neural networks and try to understand how the width, depth, data dimension, and statistical property of the training dataset influence the neural collapse. We provide a complete characterization of when the NC occurs for two or three-layer neural networks. For two-layer ReLU neural networks, a sufficient condition on when the global minimizer of the regularized empirical risk function exhibits the NC configuration depends on the data dimension, sample size, and the signal-to-noise ratio in the data instead of the network width. For three-layer neural networks, we show that the NC occurs as long as the first layer is sufficiently wide. Regarding the connection between NC and generalization, we show the generalization heavily depends on the SNR (signal-to-noise ratio) in the data. Our results significantly extend the state-of-the-art theoretical analysis of the NC under the UFM by characterizing the emergence of the NC under shallow nonlinear networks and showing how it depends on data properties and network architecture.

Title: Exploring Effective Representations Using Hyperbolic Neural Networks

Speaker: Dongmian Zou，Duke Kunshan University

Abstract: Hyperbolic Neural Networks (HNNs) have recently found successful in representing hierarchical and complex data. However, unlike other domains, the exploration of hyperbolic neural operations and the development of effective hyperbolic representations remains limited. We delve into developing HNN architectures and addressing the stability and robustness issues. In this talk, we discuss some of our recent findings in the following aspects: first, we investigate representation learning through hyperbolic convolutions with provable properties; second, we enhance representations with Gromov-Wasserstein regularization; third, we improve stability and robustness by designing hyperbolic operations and regularization techniques. Finally, we demonstrate the practical applications of these advancements in tasks such as few-shot image classification, graph classification, and anomaly detection.

Title: A stabilized arbitrary Lagrangian-Eulerian sliding interface method for fluid-structure interaction with a rotating rigid structure

Speaker: Buyang Li，The Hong Kong Polytechnic University

Abstract: We introduce a novel sliding interface formulation for fluid-structure interaction (FSI) between a rotating rigid structure and incompressible fluid, improving existing methodologies with a skew-symmetric Nitsche's stabilization term applied on an artificial sliding interface, alongside a rotational arbitrary Lagrangian-Eulerian framework. This innovative approach not only preserves the energy-dissipating property at the continuous level but also provides a robust foundation for further advancements in FSI modeling. Our methodology includes a first-order full discretization that maintains these critical energy-dissipating properties at the discrete level, ensuring numerical stability and accuracy. While prior contributions such as the original sliding interface method introduced by Bazilevs & Hughes (Comput. Mech. 2008) have been significant, theoretical analyses such as the inf-sup condition on non-matching meshes have gone largely unaddressed. We fill this gap by proving the inf-sup condition within the context of the isoparametric finite element method (FEM), where meshes are not only non-matching but also overlapping, thus extending the applicability and robustness of our approach. Leveraging this inf-sup condition along with the inherent energy-dissipating properties, we establish the unique solvability of the fully discrete scheme. Through extensive numerical experiments, we illustrate the convergence, efficiency, and energy-dissipating property of the proposed method.

Title: 现象驱动的用深度学习求解微分方程

Speaker: Zhiqin Xu，Shanghai Jiao Tong University

Abstract: 本报告将从现象驱动的角度出发，挖掘深度学习在求解微分方程中遇到的困难，并提出合理的解决方法。第一个是例子，对于多尺度的问题，频率原则指出神经网络难以学习高频小尺度，因此，我们提出多尺度神经网络。第二个例子是低温的Fokker-Planck方程，由于低温使得该方程的算子存在一个小特征值，导致数值求解非常困难。我们观察到对于深度学习，该困难使得神经网络经常学到势能梯度为零的无意义近似解，并基于此观察，提出梯度退火的深度神经网络，有效求解低温的问题。

Title: DRM Revisited: A Complete Error Analysis

Speaker: Yuling Jiao，Wuhan University

Abstract: In this talk, we provide an answer to the question in deep Ritz Method (DRM) analysis: Given a desired precision, how can we determine the number of training samples, the parameters of neural networks, the step size of gradient descent, and the number of iterations such that the  output deep networks  of the gradient descent closely approximates the true solution of the PDEs with the specified precision?

Title: 可压缩颗粒两相流数值方法及应用

Speaker: Baolin Tian，Beihang University

Abstract: 可压缩颗粒多相流问题广泛见于国防科技、经济生活和自然界中，如何对其进行高效、高精度的数值模拟一直是计算流体力学领域的挑战性问题之一。针对可压缩气固颗粒两相流问题，本文提出了一种可统一模拟稀疏和稠密颗粒流问题的可压缩跨流态两相流质点网格法（CMP-PIC），集成了颗粒塑性力做功、颗粒传热和颗粒破碎等物理模型，并进一步发展了考虑颗粒和气体化学反应的计算功能。同时，在CMP-PIC基础上，课题组发展了一种浸没边界计算方法，实现了可解析颗粒外形的直接数值模拟。所发展的数值方法和程序已应用于颗粒两相流、两相爆轰、界面不稳定性和湍流混合等工程领域复杂流动问题的数值模拟。

Title: 等离子体物理模拟的能量守恒半拉格朗日格式

Speaker: Xiaofeng Cai，Beijing Normal University

Abstract: 动理学模型利用介观尺度来描述粒子的行为，这对于理解复杂的多尺度等离子体现象尤为有效。尽管显式动理学格式易于实现，但在处理等离子体振荡周期时，它们需要一个能够精细解析振荡周期的数值时间步长，并可能出现自我加热或自我冷却的问题。这些问题可以通过无条件稳定的隐式动理学格式来解决，但这需要非线性迭代求解器。在本次报告中，我将介绍一种高效的能量守恒半拉格朗日（ECSL）格式。ECSL 的新颖之处在于，它既保留了显式动理学格式的高效性和易于实现的特点，同时也保持了隐式动理学格式的能量守恒和无条件稳定的特性。数值实验验证了 ECSL 的准确性、高效性和守恒特性。

Title: Convergence rates of greedy algorithms for training shallow neural networks

Speaker: Yuwen Li，Zhejiang University

Abstract: In this talk, I will present new error bounds of orthogonal and Chebyshev greedy algorithms. When the dictionary is generated by activation functions such as ReLU, we obtain convergence rates of greedy training algorithms for shallow neural networks. The proposed convergence rates are all based on the metric entropy of underlying dictionary. This talk is partially based on joint work Jonathan Siegel.

Title: Thermodynamically consistent modeling and simulation of gas flow in porous media

Speaker: Huangxin Chen，Xiamen University

Abstract: Numerical simulation of gas flow in porous media is becoming increasingly attractive due to its importance in shale and natural gas production and carbon dioxide sequestration. In this talk, we will discuss a multicomponent Maxwell-Stefan (MS) model with rock compressibility. Benefiting from the definitions of gas and solid free energies, this MS formulation follows an energy dissipation law which is consistent with the second law of thermodynamics. An efficient energy-stable numerical scheme will be introduced for the solution of the multicomponent MS model. Furthermore, we will also introduce a thermodynamically consistent numerical method for gas flow in poroelasticity media, which is coupled with the single-phase compressible flow and poromechanics.

Title: When Water Waves Act Up: Hydrodynamic Quantum Analogs

Speaker: Zhan Wang，Institute of Mechanics ,Chinese Academy of Sciences

Abstract: Water waves can span scales of hundreds of kilometers to a few millimeters with fascinating and complex behaviors. At small scales, when gravity and surface tension are equally important, water waves exhibit a large variety of phenomena, one of which is the quantum-like behaviors when droplets walk on the surface of a vibrating bath, and almost all the classic quantum experiments, such as single and double slit experiments, tunneling and locking orbits, can be realized in the Faraday-bouncing droplet system. We describe the reasons for these similarities based on the combination of mathematical modeling and numerical simulations.

Title: A linear, mass-conserving, Gauss’s law preserving,charge-conserving, helicity-conserving finite element method for three dimensional MHD equations

Speaker: Shipeng Mao，Academy of Mathematics and Systems Science, Chinese Academy of Sciences

Abstract: In this paper, we propose a novel structure-preserving finite element scheme for the three-dimensional incompressible magnetohydrodynamic (MHD) equations. The contribution of our research is three fold. Firstly, the proposed scheme exactly preserves critical physical properties, including mass conservation, the magnetic Gauss's law, charge conservation, energy conservation and magnetic/fluid helicity conservation in their respective physical limits. To the best of our knowledge, this is the first numerical method that preserves all these properties simultaneously.

Secondly, it introduces the first linear scheme that upholds the helicity-preserving property for MHD, thus eliminating the necessity for fixed-point iterations as seen in existing literature. Last but not least, for the resulting large linear systems, we develop efficient block preconditioners that remain robust at high fluid and magnetic Reynolds numbers by incorporating techniques such as the augmented Lagrangian method and mass augmentation. Finally, a series of numerical experiments demonstrate that our method is accurate, stable, robust under extreme physical parameters and capable of preserving all the stated physical properties, including a benchmark problem of driven magnetic reconnection with fluid and magnetic Reynolds numbers at least $10^5$.

Title: SpecNet2: Orthogonalization-free Spectral Embedding by Neural Networks

Speaker: Yingzhou Li，Fudan University

Abstract: Spectral methods which represent data points by eigenvectors of kernel matrices or graph Laplacian matrices have been a primary tool in unsupervised data analysis. In many application scenarios, parametrizing the spectral embedding by a neural network that can be trained over batches of data samples gives a promising way to achieve automatic out-of-sample extension as well as computational scalability. Such an approach was taken in the original paper of SpectralNet (Shaham et al. 2018), which we call SpecNet1. The current work introduces a new neural network approach, named SpecNet2, to compute spectral embedding which optimizes an equivalent objective of the eigen-problem and removes the orthogonalization layer in SpecNet1. SpecNet2 also allows separating the sampling of rows and columns of the graph affinity matrix by tracking the neighbors of each data point through the gradient formula. Theoretically, we show that any local minimizer of the new orthogonalization-free objective reveals the leading eigenvectors. Furthermore, global convergence for this new orthogonalization-free objective using a batch-based gradient descent method is proved. Numerical experiments demonstrate the improved performance and computational efficiency of SpecNet2 on simulated data and image datasets.

Title: A hybrid method for multiscale unsteady PDEs

Speaker: Pingbing Ming，The State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences

Abstract: In this talk, we shall present a concurrent global-local numerical method for solving multiscale parabolic equations in divergence form. The proposed method employs hybrid coefficient to provide accurate macroscopic information while preserving essential microscopic details within a specified subdomain. Both the macroscopic and microscopic errors have been improved compared to existing results, eliminating the factor of δt^{-1/2}   when the diffusion coefficient is independent of t. Numerical experiments demonstrate that the proposed method effectively captures both global and local solution behaviors, aligning with theoretical predictions. This is a joint work with Yulei Liao (National University of Singapore) and Yang Liu (AMSS, Chinese Academy of Sciences).