Beijing Institute of Mathematical Sciences and Applications

In this talk, we consider bound preserving problems for multispecies and multireaction chemical reactive flows. In this problem, the density and pressure are nonnegative, and the mass fraction should be between 0 and 1. The mass fraction does not satisfy a maximum principle and hence it is not easy to preserve the upper bound 1. Also, most of the bound-preserving techniques available are based on Euler forward time integration. Therefore, for problems with stiff source, the time step will be significantly limited. Some previous ODE solvers for stiff problems cannot preserve the total mass and the positivity of the numerical approximations at the same time. In this work, we will construct third order conservative bound-preserving methods to overcome all these difficulties. Moreover, we will discuss how to control numerical oscillations.

Reconstructing cellular dynamics from sparsely sampled single-cell sequencing data is a major challenge in biology. Classical dynamical models, despite their superior interpretability and predictive power for perturbation analysis, meet with challenges due to the curse of dimensionality and insufficient observations. Can we revitalize models in the era of single-cell data science, by taking advantage of Artificial Intelligence?<br>In this talk, I will introduce our recent efforts to dynamically integrate sampled cell state distributions through generative AI, highlighting exciting opportunities in both algorithm development and theoretical innovation. I will begin by presenting a framework that employs flow-based generative models to uncover the underlying dynamics (i.e. PDEs) of scRNA-seq data, and demonstrate the development of a dimensionless solver capable of inferring continuous cell-state transitions, as well as proliferation and apoptosis, from real datasets.<br>For spatial transcriptomics, we have further extended this framework by developing stVCR, which addresses the critical challenge of aligning snapshots collected from (1) different biological replicates and (2) distinct temporal stages. stVCR enables interpretable reconstruction and simulation of cell differentiation, growth, and migration in physical space, aligning spatial coordinates from transcriptomic snapshots—effectively generating a "video" of tissue development from limited static "images." This approach will be illustrated through applications in axolotl brain regeneration and 3D Drosophila embryo development.<br>To further infer stochastic dynamics from static data, we explore a regularized unbalanced optimal transport (RUOT) formulation and its theoretical connections to the Schrödinger Bridge and diffusion models. I will also introduce a generative deep-learning solver designed for this problem, with applications in single-cell analysis.

We introduce an innovative research in the field of subtractive manufacturing, where we employ deep learning for real-time tool path planning. Our work commences with a comprehensive discussion on the evolution of tool path planning methodologies, setting the stage for our introduction of a novel approach. We present an adaptive iso-scallop height method for tool path generation, which we believe is particularly adept at integrating with learning algorithms. Additionally, we delve into the architecture of the B-spline surface reparameterization network and the design of corresponding loss functions. We conclude with a detailed exposition of our experimental results, which underscore the significant advantages of our method in CNC tool path planning.

In this talk, we explore various structures designed to achieve negative elastic metamaterials. First, we review the effective methods for constructing negative mass density and bulk modulus. Next, we focus on the strategies for achieving a negative shear modulus.

Learning a data representation for downstream supervised learning tasks under unlabeled scenario is both critical and challenging. In this talk, we propose a novel unsupervised transfer learning approach using adversarial contrastive training (ACT). Our experimental results demonstrate outstanding classification accuracy with both fine-tuned linear probe and K-NN protocol across various datasets, showing competitiveness with existing state-of-the-art self-supervised learning methods. Moreover, we provide an end-to-end theoretical guarantee for downstream classification tasks in a misspecified, over-parameterized setting, highlighting how a large amount of unlabeled data contributes to prediction accuracy in downstream task with a small sample size.

Projected policy gradient under the simplex parameterization, policy gradient and natural policy gradient under the softmax parameterization, are fundamental algorithms in reinforcement learning. There have been a flurry of recent activities in studying these algorithms from the theoretical aspect. Despite this, their convergence behavior is still not fully understood, even given the access to exact policy evaluations. In this talk, we give a systematic study of the aforementioned policy optimization methods. Several novel results are presented, including 1) Sublinear and finite iteration convergence of projected policy gradient for any constant step size, 2) sublinear convergence of softmax policy gradient for any constant step size, 3) policy convergence and exact asymptotic convergence rate of softmax natural policy gradient, 4) global linear convergence of entropy regularized softmax policy gradient for a wider range of constant step sizes than existing result, 5) tight local linear convergence rate of entropy regularized natural policy gradient, and 6) a new and concise local quadratic convergence rate of soft policy iteration without the assumption on the stationary distribution under the optimal policy. New and elementary analysis techniques have been developed to establish these results.

Incompressible Navier-stokes equations (NSEs) and their numerical approximations play an important role in many fields of science and engineering. In particular, The NSEs can be coupled with other nonlinear equations, such as Cahn-Hillard, Maxwell, Keller-Segel equations etc, to model various complex phenomenas in fluid mechanics and materials science.<br>However, how to construct stable higher-order fully decoupled schemes for the NSEs has been a long standing open problem. In this lecture, we shall discuss the main difficulties and review existing approaches in designing higher-order decoupled schemes for NSEs, and present our recent work on a new class of stiffly stable IMEX schemes and apply them to construct unconditionally stable higher-order fully decoupled schemes for the NSEs.<br>We shall also discuss how to extend our results for NSEs to construct higher-order fully decoupled schemes for complex nonlinear systems involving NSEs.

Non-Line-of-Sight (NLOS) imaging, a typical inverse problem, reconstructs hidden objects' shapes, positions, or motions by analyzing indirectly obtained optical or acoustic signals. Unlike traditional imaging, NLOS captures signals via complex reflections and scatterings, then uses inversion to recover obscured information. This process involves complex modeling and faces challenges like ill-posedness and noise interference. Using curvature regularization methods, we achieved high-quality, rapid NLOS reconstructions. Results from both synthetic and experimental data demonstrate that our approach accurately recovers hidden objects and outperforms state-of-the-art algorithms in quantitative metrics and visual quality.

Physical systems inherently exhibit critical properties such as energy conservation/dissipation relations, mass conservation, and positive densities. Designing numerical schemes that preserve fundamental characteristics is crucial for accurately modeling and simulating complex systems. In this talk, we present a natural framework for constructing energy-stable time discretization schemes for dissipative systems. By leveraging the Onsager principle, we demonstrate its efficacy in formulating partial differential equation models for diverse gradient flow systems. Furthermore, this principle provides a basis for developing numerical schemes that uphold some crucial physical properties. Within this framework, several widely used schemes emerge naturally, showing its versatility and applicability.

Machine learning methods for learning physical models, in particular (partial) differential equations, has been a popular topic nowadays. In this talk, we introduce a general framework involving such learning-informed models with neural networks as some nonlinear components in inverse and optimal control problems. Particularly, we present their numerical algorithms and approximation properties. We will provide both analytical and numerical aspects of such models, and show their applications in optimal control of partial differential equations and quantitative magnetic resonance imaging.

In this work, we present a novel approach for sediment concentration measurement in water flow, modeled as a multiscale inverse medium problem. To address the multiscale nature of the sediment distribution, we treat it as an inhomogeneous random field and use the homogenization theory in deriving the effective medium model. The inverse problem is formulated as the reconstruction of the effective medium model, specifically, the sediment concentration, from partial boundary measurements. Additionally, we develop numerical algorithms to improve the efficiency and accuracy of solving this inverse problem. Our numerical experiments demonstrate the effectiveness of the proposed model and methods in producing accurate sediment concentration estimates, offering new insights into sediment measurement in complex environments.

In this talk, we propose an RADI-type method for large-scale stochastic continuous-time algebraic Riccati equations with sparse and low-rank structures. The method is developed by using the Incorporation idea together with different Shifts to accelerate the convergence and Compressions to reduce the storage and complexity. Unlike many existing methods for large-scale problems such as Newton-type methods and homotopy method, it calculates the residual with low cost and does not need a stabilizing initial approximation difficult to find. Numerical experiments are given to show its efficiency.

Quasiperiodic systems, related to irrational numbers, are important space-filling structures without decay nor translational invariance. How to numerically compute these incommensurate systems poses challenges. In this talk, we will present some accurate and efficient methods for quasiperiodic systems based on the arithmetic property of irrational numbers, including periodic approximation method, projection method, and finite point recovery method. The corresponding approximation analysis is also given. Then we will briefly give some applications, such as quasicrystals and phase transitions, grain boundaries, quasiperiodic homogenization, quasiperiodic Schrödinger systems.

基于动态区域分解的模型杂交算法能大大提高多尺度动理学玻尔兹曼方程的模拟效率。但是目前基于矩矩阵特征值的指示子需要重构各阶导数且对人为参数较为敏感。我们基于全连接神经网络构造了一组新的指示子，从宏观量到区域分解端对端映射，且和我们的模型杂交算法适配。通过数值实验，模型杂交算法基于新的指示子有更好的表现。

In this talk, I will introduce a deep learning framework, which we proposed recently, for solving high-dimensional partial integro-differential equations (PIDEs) based on the temporal difference learning. We introduce a set of Levy processes and construct a corresponding reinforcement learning model. To simulate the entire process, we use deep neural networks to represent the solutions and non-local terms of the equations. Subsequently, we train the networks using the temporal difference error, termination condition, and properties of the non-local terms as the loss function. Our method demonstrates the advantages of low computational cost and robustness, making it well-suited for addressing problems with different forms and intensities of jumps. This talk is mainly based on the joint work with Liwei Lu (Tsinghua University), Hailong Guo (The University of Melbourne), and Xu Yang (University of California, Santa Barbara)

We consider two variants of restricted overlapping Schwarz methods for the non-trapping Helmholtz problems, which allow the optic-rays leaving a bounded domain in a uniform time. The first method, known as RAS-Imp, incorporates impedance boundary condition to formulate the local problems. The second method, RAS-PML, employs local perfectly matched layers (PML). These methods combine the local solutions additively with a partition of unity. We have shown that RAS-Imp has power contractivity for strip domain decompositions. More recently, we shown that RAS-PML has super-algebraic convergence with respective to wavenumber after a specified number of iterations. This is the first theoretical result for the non-trapping Helmholtz problems with variable wave speed. In this talk we review these results and illustrate how the error of the Schwarz methods propagates as optic-rays. We also investigate situations not covered by the theory. In particular, the theory needs the overlap of the domains or the PML widths to be independent of k. We present numerical experiments where this distances decrease with k.<br>Shihua Gong obtained his PhD degree in computational mathematics from Peking University in 2018. Before joining the Chinese University of Hong Kong (Shenzhen), he worked as a postdoctoral scholar at Pennsylvania State University and then as a research associate at the University of Bath. His research interests include scientific computing and numerical analysis, mainly focusing on finite element and preconditioning techniques for frequency-domain wave equations and coupled equations in multiphysics problems.

先验分布包含了待重建图像的统计信息和语义信息，在计算成像领域中至关重要。近期，生成式扩散模型由于其卓越的复杂图像先验建模能力，被越来越广泛地应用于解决成像反问题。然而，训练扩散模型严重依赖大规模的干净数据，而此类数据在成像领域中非常稀缺，因此限制了这种数据驱动先验分布的实际应用。针对这个问题，我们提出了EMDiffusion，一种基于期望最大化（EM）算法的方法，有效地从受损观测数据中训练扩散模型，最终获得干净的先验分布。具体来说，我们的方法通过在（a）使用已知扩散模型从受损数据中重建干净图像（E-step）和（b）基于这些重建结果优化扩散模型（M-step）之间交替进行，逐步使学习到的扩散模型收敛到真实的干净数据分布。我们在图像修补、去噪和去模糊等多种计算成像任务上广泛验证了这一方法，取得了最佳的表现。

In this talk, we'll discuss the inductive biases of deep convolutional neural networks (CNNs), which are believed to be vital drivers behind CNNs' exceptional performance on vision-like tasks. Specifically, we'll analyze the universality of CNNs and show that achieving it requires only a depth of $O(\log d)$, where $d$ is the input dimension. Additionally, we'll demonstrate that CNNs can efficiently capture long-range sparse correlations with only $O(\log^2d)$ samples. These are achieved through a novel combination of increased network depth and the utilization of multi-channeling and down-sampling.<br>We'll also explore the inductive biases of weight sharing and locality through the lens of symmetry group by introducing locally-connected networks (LCNs), which can be viewed as CNNs without weight sharing. We'll compare the performance of CNNs, LCNs, and fully-connected networks (FCNs) on a simple regression task and highlight the cruciality of weight sharing and the importance of locality. Our findings demonstrate that weight sharing and locality break different symmetries in the learning process, leading to provable separations between the two biases.

Liquid crystals polymer networks (LCN) deform spontaneously upon temperature or optical actuation. This property can be exploited in the design of materials to achieve non-trivial shapes. These materials combine the features of rubber and nematic liquid crystals. In this work, we first derive a 2D membrane model from 3D neo-Hookean elasticity via formal asymptotics, where the obtained limiting model and the derivation are inspired by Virga et al., Bhattacharya et al. and Cirak et al. The membrane model consists of a minimization problem of a non-convex energy functional for deformations of the materials, and the equilibrium shapes depend on the design of blueprinted nematic orders and pre-determined actuation parameters. We discuss properties of this energy functional and its global minimizers. We design a finite element discretization for this model, propose a novel iterative scheme to solve the non-convex discrete minimization problem, and prove stability of the scheme and a convergence of discrete minimizers. We present a wide range of numerical simulations to illustrate effectiveness and efficiency of our algorithm, as well as the fact that this model captures quite rich physical phenomenon. These simulations include some results of practical interests that matches lab experiments, the design of origami shapes and a quantitative study of our numerical method.