我们提出AROMA（Attentive Reduced Order Model with Attention，带注意力的注意力降阶模型），一个旨在利用局部神经场增强偏微分方程（PDE）建模的框架。我们灵活的编码器-解码器架构能够从多种数据类型（包括不规则网格输入和点云）中获得空间物理场的平滑潜在表示。这种通用性消除了分块处理的需要，并允许高效处理多样化的几何结构。我们的潜在表示具有顺序性，可以在空间上进行解释，并允许使用条件transformer对PDE的时间动态进行建模。通过采用基于扩散的公式，与传统的MSE训练相比，我们实现了更高的稳定性并支持更长的 rollout。AROMA在模拟1D和2D方程方面的优越性能凸显了我们方法在捕捉复杂动力学行为方面的有效性。
英文摘要：
We present AROMA (Attentive Reduced Order Model with Attention), a framework designed to enhance the modeling of partial differential equations (PDEs) using local neural fields. Our flexible encoder-decoder architecture can obtain smooth latent representations of spatial physical fields from a variety of data types, including irregular-grid inputs and point clouds. This versatility eliminates the need for patching and allows efficient processing of diverse geometries. The sequential nature of our latent representation can be interpreted spatially and permits the use of a conditional transformer for modeling the temporal dynamics of PDEs. By employing a diffusion-based formulation, we achieve greater stability and enable longer rollouts compared to conventional MSE training. AROMA's superior performance in simulating 1D and 2D equations underscores the efficacy of our approach in capturing complex dynamical behaviors.
OCL-PDE: A Generative Framework for PDE Inverse Problems with Observation-Complementary Latents
The Hong Kong University of Science and Technology（香港科技大学）；University of Michigan（密歇根大学）；HKUST Shenzhen-Hong Kong Collaborative Innovation Research Institute（香港科技大学深圳-香港协同创新研究院）
Latent Neural Operator Pretraining for Solving Time-Dependent PDEs
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）
Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations
School of Science, Harbin Institute of Technology, Shenzhen, China（哈尔滨工业大学深圳校区科学学院）；Department of Theoretical Physics（理论物理系）；Applied Mathematics, University of Cambridge, Cambridge, United Kingdom（应用数学系，剑桥大学，英国）
Time-Dependent PDE-Constrained Optimization via Weak-Form Latent Dynamics
Department of Applied Mathematics, University of Colorado（应用数学系，科罗拉多大学）；Center for Applied Scientific Computing, Lawrence Livermore National Laboratory（应用科学计算中心，劳伦斯利弗莫尔国家实验室）
Latent Mamba Operator for Partial Differential Equations
Department of Electrical Communication Engineering, Indian Institute of Science, Bangalore, India（电子通信工程系，印度科学研究院，班加罗尔，印度）；Yardi School of AI, Indian Institute of Technology, New Delhi, India（AI学院，印度理工学院，新德里，印度）；Department of Civil Engineering, Indian Institute of Technology, New Delhi, India（土木工程系，印度理工学院，新德里，印度）；Department of Computer Science and Automation, Indian Institute of Science, Bangalore, India（计算机科学与自动化系，印度科学研究院，班加罗尔，印度）
针对神经算子求解偏微分方程时面临的高维扩展性差、计算成本高及难以捕捉长程依赖等问题，提出 Latent Mamba Operator (LaMO)，结合潜空间状态空间模型与核积分公式，并在多种 PDE 基准上取得 SOTA 性能。
CALM-PDE: Continuous and Adaptive Convolutions for Latent Space Modeling of Time-dependent PDEs
Machine Learning and Simulation Lab（机器学习与仿真实验室）；Institute for Artificial Intelligence（人工智能研究所）；University of Stuttgart（斯图加特大学）；International Max Planck Research School for Intelligent Systems (IMPRS-IS)（智能系统国际马克斯·普朗克研究学校）；Stuttgart Center for Simulation Science (SimTech)（斯图加特模拟科学中心）
HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries
School of Mathematics, Nanjing University of Aeronautics and Astronautics（南京航空航天大学数学学院）；Key Laboratory of Mathematical Modelling and High Performance Computing of Air Vehicles (NUAA), MIIT（工业和信息化部南京航空航天大学数学建模与高性能计算重点实验室（NUAA））；School of Mathematics and Information Sciences, Yantai University（烟台大学数学与信息科学学院）
Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling
Department of Mechanical Engineering, University of Utah, Salt Lake City, UT, USA（犹他大学机械工程系）；Imaging Institute, University of Utah, Salt Lake City, UT, USA（犹他大学成像研究所）
Latent Neural Operator for Solving Forward and Inverse PDE Problems
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；School of Artificial Intelligence, University of Chinese Academy of Sciences（中国科学院大学人工智能学院）
GenPANIS: A Latent-Variable Generative Framework for Forward and Inverse PDE Problems in Multiphase Media
Technical University of Munich, Professorship of Data-driven Materials Modeling, School of Engineering and Design（慕尼黑技术大学，数据驱动材料建模教授职位，工程与设计学院）；Munich Data Science Institute (MDSI - Core member)（慕尼黑数据科学研究所（MDSI—核心成员））
LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process
Faculty of Science and Technology, Beijing Normal-Hong Kong Baptist University（科学技术学院，北京师范大学-香港 Baptist大学）；Department of Mathematics, Shanghai Normal University（上海师范大学数学系）；Department of Mathematics and Center for Computation and Technology, Louisiana State University（数学系和计算技术中心，路易斯安那州立大学）；School of Mathematical Sciences, Institute of Natural Sciences, and MOE-LSC, Shanghai Jiaotong University（数学科学学院，自然科学研究院，MOE-LSC，上海交通大学）；Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（计算数学与科学/工程计算研究所，数学与系统科学学院，中国科学院）
Generalizing Dynamics Modeling More Easily from Representation Perspective
College of Software and Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, China（软件学院，教育部符号计算与知识工程重点实验室，吉林大学，中国）；School of Computing, Belfast, Northern Ireland, UK（computing 学院，贝尔法斯特，北爱尔兰，英国）
Department of Mathematics, Emory University（埃默里大学数学系）；School of Mathematical and Statistical Sciences, Clemson University（克林斯敦大学数学与统计科学学院）；Argonne National Laboratory（阿贡国家实验室）
