北京雁栖湖应用数学研究院

扩散方程是一类含有二阶空间导数的偏微分方程，其解析解通常满足极大值原理（简称保界）。数值格式的解是否保界直接影响格式是否稳定以及解是否具有物理意义。局部间断有限元(LDG)方法是一类求解含高阶空间导数项偏微分方程的数值方法，具有高精度、模板紧致、网格灵活、并行度高等优点。本报告将介绍一类隐式求解扩散方程的高精度保界LDG方法。通过严格的矩阵分析，理论上给出了隐式 LDG 方法保界的充分条件。数值试验验证了我们证明结果的严格性及有效性。

原子簇展开（Atomic Cluster Expansion, ACE）方法提供了一种高效且可解释的对称多项式参数化形式，并已在多粒子体系模拟中取得成功。我们将 ACE 框架的实际适用范围扩展至多电子波函数的计算，发展了一套多水平变分蒙特卡洛算法，通过对一系列典型体系的模拟展示了该方法的精度和效率。

现代社会建立于数据之上，数据驱动着世界的发展。数据是离散的，连接这些离散数据的数学模型往往天然具有NP难的特征，例如复杂的组合优化问题，这迫使我们越来越重视离散数学工具的发展。另一方面，相较于离散的研究对象，连续的数学模型往往会呈现更多的结构信息，如凸性、对称性等，进而为产生更丰富的处理手段提高可能。于是一个合理的想法是将离散的基于数据的组合优化问题“嵌入”到连续问题来进行理论和算法的发展。沿着这条思路，传统的连续嵌入尝试往往遵循松弛-凑整的研究路径去讨论收敛性和估计近似比，但这样的分析极不平凡，多需要配合精心设计的凑整策略。即便如此，松弛和凑整这两个模块互相独立导致这种嵌入方式下重构的可行解依旧不够准确。为此，本报告将从多种NP难的图割问题出发来说说如何尝试构建一套离散到连续的准确嵌入方式，进而发展等价的非线性图谱理论和简单连续迭代算法，并通过启发式算法的高质量参考解来进行验证。

质子治疗是肿瘤精准放射治疗的先进技术，利用质子束的布拉格峰特性可精准释放能量于肿瘤区域，同时降低正常组织辐射剂量，具有很好的应用前景，但同时由于质子治疗与设备的特性对放疗计划的优化算法提出了更高要求。质子治疗具有很高精度，然而由于射程和设置的不确定性存在，因此在计划过程中需要考虑这些不确定性，需要构建具有鲁棒性的高效算法以避免剂量分布的明显退化。同时，质子治疗系统受到最小跳数(MU)的限制，如果在质子治疗计划优化中不考虑这一限制，将影响剂量分布的质量，因此需要设计适用问题的算法。

Artificial intelligence (AI) has shown unprecedented capability as a computational approach to solving scientific tasks. Nevertheless, scientific tasks hold their unique challenge of the enormous problem space, while data generation approaches are limited in accuracy, scale, and system coverage. This talk introduces recent advances that go beyond the data limitation by using physical laws, the blessing from the science side. We demonstrate a series of effective approaches for molecular science across scales, from improving the accuracy and generalizability of electronic structure prediction beyond data limitation by quantum-mechanical equations, which in turn improves energy and microscopic property prediction, to correcting data bias for molecular structure modeling and macroscopic property prediction by leveraging the connection to the energy function and measurable observables.

Deep operator networks (DeepONets) are a powerful data-driven framework for operator learning, capable of approximating a broad class of linear and nonlinear operators governing partial differential equations (PDEs). Despite their success in modeling complex systems such as diffusion–reaction dynamics and Burgers’ equations, their accuracy is often limited by high computational costs and optimization difficulties inherent in training deep neural networks. To overcome these challenges, we propose RaNN-DeepONets, which integrate DeepONets with randomized neural networks (RaNNs). In this hybrid architecture, hidden-layer parameters are randomly initialized and fixed, while output-layer weights are solved via least squares, reducing training time and mitigating optimization errors. Furthermore, we introduce physics-informed RaNN-DeepONets, which embed PDE constraints directly into the training process, alleviating the need for costly data generation. Evaluations on diffusion–reaction, Burgers’, and Darcy flow benchmarks demonstrate that RaNN-DeepONets achieve accuracy comparable to standard DeepONets while reducing computational cost by orders of magnitude. These results highlight their potential as an efficient and scalable alternative for operator learning in PDE-based modeling.

In this talk, we first present several high-order and physics-preserving numerical schemes for two-phase flow in porous media. The constructed schemes only need to solve one linear system and a nonlinear algebraic equation with negligible computational cost at each time step. We also prove that the proposed schemes are energy stable, global mass-conservative and bounds-preserving for each phase without any restrictions of time step size. Furthermore, we will present recent developments in ensuring local mass conservation and preserving original energy dissipation.

In this talk, I will introduce a new parametric finite element method for Willmore flow of hypersurfaces in a unified framework. The method is linear and employs a splitting of the normal and tangential velocity of the flow. The normal velocity is approximated via an evolution equation for the curvature, and follows the arbitrary Lagrangian-Eulerian approach. This enables an unconditional energy stability with respect to the discrete energy. We also incorporate the `BGN’tangential velocity through a curvature identity. This helps to preserve the mesh quality. We show various numerical examples to demonstrate the favorite properties of the method.

科学发现在塑造我们解决复杂挑战的能力方面发挥了重要作用，从推动医疗技术进步到应对可持续性问题。然而，传统研究方法往往难以应对现代科学问题所呈现的规模化和跨学科特性。为此，“科学基础模型”应运而生。本次报告将首先概述科学基础模型的最新进展，随后介绍重点介绍基于科学序列的基础模型，NatureLM。NatureLM解读了生物学、化学和物理学中自然的通用“语言”，为加速创新提供了一种全新的途径。通过在多样化的科学数据集上进行训练，例如蛋白质序列（FASTA）、分子结构（SMILES）以及交错的文本-科学数据对，我们引入了一种受 GPT 启发的基础模型，整合了多个领域的洞察。我将分享我们在将 NatureLM 应用于实际挑战（如药物设计和蛋白质工程）方面的最新进展，展示其在推动计算发现中的潜力。

This work proposes a new mathematical formulation for flow measurement based on the inverse source problem for wave equations with partial boundary measurement. Inspired by the design of acoustic Doppler current profilers (ADCPs), we formulate an inverse source problem that can recover the flow field from the observation data on boundary receivers. To our knowledge, this is the first mathematical model of flow measurement using partial differential equations. This model is proved well-posed, and the corresponding algorithm is derived to compute the velocity field efficiently. Extensive numerical simulations are performed to demonstrate the accuracy and robustness of our model. The comparison results demonstrate that our model is ten times more accurate than ADCP. Our formulation is capable of simulating a variety of practical measurement scenarios.

级联重映射方法最早由Nair等（2002）在大气与气候建模领域提出，用于构造高效、质量守恒的半拉格朗日（SL）输运算法。该方法通过在多维空间中采用连续的一维守恒重映射，实现了在保持SL方法大步长与高精度特性的同时，严格满足质量守恒。尽管该算法在地球物理流体动力学中取得了广泛应用，但在等离子体动理学领域的研究仍较为有限。鉴于级联守恒型半拉格朗日（CCSL）方法在质量守恒、计算效率与并行化方面的潜在优势，我们将其引入Vlasov方程及相关等离子体模型的数值求解中。理论分析表明，该格式在空间上具有二阶精度，其主要误差来源于回溯区域的几何近似。在数值试验中，我们进一步发现，自由流保持与保极值原理对等离子体数值模拟结果的准确性具有重要影响。针对这些问题，我们对原始CCSL格式进行了改进：引入自由流修正，通过几何调整保证在回溯过程中体积保持不变；构造极值原理限制器，在保持质量守恒与正性的同时，有效抑制高阶插值产生的非物理振荡。数值算例——包括线性输运方程、guiding-center模型以及相对论Vlasov–Maxwell系统——验证了改进CCSL方法的高精度与良好鲁棒性。与经典SL方法或守恒SL方法相比，本方法在保持无源场下的物理不变量方面表现更优。该研究为高保真等离子体动理学模拟提供了一种稳健且高效的数值框架，并具备良好的可扩展性，可进一步推广至高维和并行计算环境。

An integration of a high-fidelity Euler equations solver into reinforcement learning (RL) for aerodynamic shape optimization is presented, which provides a potential way to overcome the limitation of gradient-free methods. A task-specific reward function targets continuous optimization, enabling efficient exploration. Using Bézier curves for airfoil geometry and a Newton-geometric multigrid solver, the method optimizes a 132-variable airfoil within around 5,000 simulations, which is competitive in the typical gradient-free capabilities. Laplacian smoothing and Bézier fitting prevent mesh tangling and ensure precise geometry control. The framework represents the potential towards quality applications in the design of airfoils.

We propose a supervised learning scheme for the first order Hamilton–Jacobi PDEs in high dimensions. The scheme is designed by using the geometric structure of Wasserstein Hamiltonian flows via a density coupling strategy. It is equivalently posed as a regression problem using the Bregman divergence, which provides the loss function in learning while the data is generated through the particle formulation of Wasserstein Hamiltonian flow. We prove a posterior estimate on L1 residual of the proposed scheme based on the coupling density. Furthermore, the proposed scheme can be used to describe the behaviors of Hamilton–Jacobi PDEs beyond the singularity formations on the support of coupling density. Several numerical examples with different Hamiltonians are provided to support our findings.

In this talk, I will present the mathematical theory of resonances in subwavelength structures. By combining potential theory with Fourier-mode expansion techniques, I will show that these resonances are governed by finite-dimensional linear systems, so that their explicit formulas can be obtained asymptotically. These formulas, in turn, provide guidance for designing subwavelength structures with closely spaced resonances. Such structures can be used to realize super-resolution imaging, which I will illustrate by some numerical experiments.

An integration of a high-fidelity Euler equations solver into reinforcement learning (RL) for aerodynamic shape optimization is presented, which provides a potential way to overcome the limitation of gradient-free methods. A task-specific reward function targets continuous optimization, enabling efficient exploration. Using Bézier curves for airfoil geometry and a Newton-geometric multigrid solver, the method optimizes a 132-variable airfoil within around 5,000 simulations, which is competitive in the typical gradient-free capabilities. Laplacian smoothing and Bézier fitting prevent mesh tangling and ensure precise geometry control. The framework represents the potential towards quality applications in the design of airfoils.

Lagrangian data assimilation (DA) aims to reconstruct flow fields from particle trajectories, but its nonlinear and high-dimensional nature often limits efficiency and accuracy. In this talk, we will introduce LEMDA, a Lagrangian–Eulerian multiscale data assimilation framework that bridges particle-based and continuum-based perspectives. We begin with a brief overview of our recent work on the multiscale modeling of sea-ice floe particles, developed within the conceptual framework of Hilbert’s sixth problem. Starting from a Boltzmann kinetic formulation of particle dynamics, we employ the BBGKY hierarchy to derive Eulerian continuum equations that capture the statistical behavior and large-scale evolution of the particle system. Building upon this particle–continuum multiscale model, we introduce the LEMDA (Lagrangian–Eulerian Multiscale Data Assimilation) framework, which integrates both Lagrangian and Eulerian observational data. By incorporating a stochastic surrogate model for the dynamics of both particles and continuum fields, the posterior distribution of the system can be expressed in closed analytic form, thereby avoiding ensemble-based approximations and empirical tuning. The resulting framework consists of two complementary components: a Lagrangian DA module suited for sparse particle data and an Eulerian DA module designed for the large-scale behavior of dense, colliding, or multiscale particle systems. We will show numerical experiments to illustrate LEMDA’s accuracy, robustness, and computational efficiency.

量子化学是材料设计与药物研发的核心工具，能够在原子尺度上揭示结构、反应与性质之间的内在联系。然而，传统量子化学方法依赖于求解薛定谔方程，计算复杂度极高，难以在大体系和长时间尺度下应用；而经验或半经验方法虽计算快速，却难以兼顾精度与可迁移性。人工智能（AI）的崛起正深刻改变这一局面。通过学习大量量子化学数据，AI模型能够以接近力场的计算速度逼近从头算精度，大幅拓展了可研究体系的规模与复杂性。AI不再仅仅是加速工具，更成为构建高精度势能面、预测分子性质和探索反应路径的全新范式。本次报告将介绍AI在量子化学中，尤其是分子动力学模拟和哈密顿量预测中的关键进展，探讨其在推动科学计算智能化方面的潜力与挑战。

This talk presents a novel stabilized nonconforming finite element method for solving the surface biharmonic problem. The method extends the New-Zienkiewicz-type (NZT) element to polyhedral (approximated) surfaces by employing the Piola transform to define the connection of vertex gradients between adjacent elements. Key features of the surface NZT finite element space include its $H^1$-relative conformity and the weak H(div) conformity of the surface gradient, allowing for stabilization without the need for artificial parameters. Assuming that the exact solution and the dual problem possess only $H3$ regularity, we establish optimal error estimates in the energy norm and provide, for the first time, a detailed analysis yielding optimal second-order convergence in the broken $H^1$ norm. Numerical experiments are provided to support the theoretical results, and they suggest that the stabilization term may not be necessary.

Lagrangian data assimilation (DA) aims to reconstruct flow fields from particle trajectories, but its nonlinear and high-dimensional nature often limits efficiency and accuracy. In this talk, we will introduce LEMDA, a Lagrangian–Eulerian multiscale data assimilation framework that bridges particle-based and continuum-based perspectives. We begin with a brief overview of our recent work on the multiscale modeling of sea-ice floe particles, developed within the conceptual framework of Hilbert’s sixth problem. Starting from a Boltzmann kinetic formulation of particle dynamics, we employ the BBGKY hierarchy to derive Eulerian continuum equations that capture the statistical behavior and large-scale evolution of the particle system. Building upon this particle–continuum multiscale model, we introduce the LEMDA (Lagrangian–Eulerian Multiscale Data Assimilation) framework, which integrates both Lagrangian and Eulerian observational data. By incorporating a stochastic surrogate model for the dynamics of both particles and continuum fields, the posterior distribution of the system can be expressed in closed analytic form, thereby avoiding ensemble-based approximations and empirical tuning. The resulting framework consists of two complementary components: a Lagrangian DA module suited for sparse particle data and an Eulerian DA module designed for the large-scale behavior of dense, colliding, or multiscale particle systems. We will show numerical experiments to illustrate LEMDA’s accuracy, robustness, and computational efficiency.

In this paper, we study the Vlasov-Poisson system with massless electrons (VPME) near quasineutrality and with uncertainties. Based on the idea of reformulation on the Poisson equation by [P. Degond et.al., JCP,2010], we first consider the deterministic problem and develop an efficient asymptotic preserving particle-in-cell (AP-PIC) method to capture the quasineutral limit numerically, without resolving the discretizations subject to the small Debye length in plasma. The main challenge and difference compared to previous related works is that we consider the nonlinear Poisson in the VPME system, which contains nonlinear electric potential and provide an explicit scheme. In the second part, we extend to study the uncertainty quantification (UQ) problem and develop an efficient bi-fidelity method for solving the VPME system with multidimensional random parameters, by choosing the Euler-Poisson equation as the low-fidelity model. Several numerical experiments are shown to demonstrate the asymptotic-preserving property of our deterministic solver and the effectiveness of our bi-fidelity method for solving the model with random uncertainties.

Numerical resolution of moderately high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality for the classical numerical methods including finite difference, finite element and spectral methods. In this talk, we discuss a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation, 7D Hamiltonian-Jacobi-Bellman equation and 1000000-D linear fractional diffusion equation demonstrate the potential of SPM in solving moderately high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy.

In this talk, we will present recent results on applying multigrid structures to neural operators for problems in numerical PDEs. First, we will discuss some basic background on operator learning, including the problem setup, a uniform abstract framework, and a general universal approximation result. Motivated by the general definition of neural operators, we propose MgNO, which utilizes multigrid structures to parameterize these linear operators between neurons, offering a new and concise architecture in operator learning. For the implementation issue of MgNO, we will illustrate MgNet as a unified framework for convolutional neural networks and multigrid methods. This approach provides both mathematical rigor and practical expressivity, with many interesting numerical properties and observations.

自然科学系统中的功能源于一维序列与其对应的三维结构。然而，现有科学任务中生成式模型普遍存在三方面特点：模型建模很少以功能优化为目标；离散序列与连续坐标处理割裂；对构象集合的建模不足，这严重限制了新材料、药物分子、以及功能蛋白的科学发现。因此，我们提出 UniGenX，一种统一不同模态、不同表征的生成式模型，在蛋白质、小分子与晶体材料三大领域，以功能为目标，联合生成所需分子的序列与结构。UniGenX 将科学表达的异质输入统一为“符号–数值”混合令牌序列，利用解码器式自回归 Transformer 提供全局上下文，条件扩散头在特定令牌的引导下准确生成数值。UniGenX除了在结构预测各项任务上取得新的基准（SOTA）外，在跨领域目标功能驱动的设计中更刷新多项基准：在材料设计中，实现多个“冲突”属性的条件生成（如同时要求高硬度、低密度、高导热），获得了 11 个具有新颖组分、4 个经 DFT 证实稳定的晶体材料；在化学分子设计任务中，在五个性质目标与构象集生成上刷新了科学基准，并定义了新的统一生成指标；在生物学中，使诱导契合（induced fit）蛋白建模的成功率（RMSD < 2 Å）提升了 23 倍，在指定功能的酶的设计任务上，大大提高了根据 EC分类的酶的设计科学基准。消融实验与跨域迁移结果证明了UniGenX离散–连续联合训练的优势，表明了 UniGenX 实现了从“预测”到“可控、功能驱动生成”的范式跃迁。在这个报告中，我们将从扩散模型、自回归模型技术以及科学任务驱动出发，详细介绍统一离散和连续的必要性、方式方法、目前取得的结果，以及未来技术应用于科学发现的讨论。