Generative models such as denoising diffusion models are quickly advancing their ability to approximate highly complex data distributions. They are also increasingly leveraged in scientific machine learning, where samples from the implied data distribution are expected to adhere to specific governing equations. We present a framework that unifies generative modeling and partial differential equation fulfillment by introducing a first-principle-based loss term that enforces generated samples to fulfill the underlying physical constraints. Our approach reduces the residual error by up to two orders of magnitude compared to previous work in a fluid flow case study and outperforms task-specific frameworks in relevant metrics for structural topology optimization. We also present numerical evidence that our extended training objective acts as a natural regularization mechanism against overfitting. Our framework is simple to implement and versatile in its applicability for imposing equality and inequality constraints as well as auxiliary optimization objectives.
STeP-Diff: Spatio-Temporal Physics-Informed Diffusion Models for Mobile Fine-Grained Pollution Forecasting
Shenzhen International Graduate School, Tsinghua University, China（清华大学深圳国际研究生院）；Department of Electronic Engineering, Beijing National Research Center for Information Science and Tech- nology (BNRist), Tsinghua University, China（电子工程系，北京信息科学与技术国家研究中心（BNRist），清华大学）；Shenzhen Smartcity Communication, Shenzhen, China（深圳智慧城市通信）；College of Soil and Water Conservation, Southwest Forestry University, China（水土保持学院，西南林业大学）；College of Civil Engineering, Southwest Forestry University, China（土木工程学院，西南林业大学）
FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling
Institute for Foundations of Data Science, Yale University（数据科学基础研究所，耶鲁大学）；Department of Statistics and Data Science, Yale University（统计与数据科学系，耶鲁大学）；Laboratoire Navier, École nationale des ponts et chaussées, Institut Polytechnique de Paris, University Gustave Eiffel, CNRS UMR 8205（纳维实验室，巴黎理工学院，格斯塔夫·埃菲尔大学，CNRS UMR 8205）；Department of Chemical and Environmental Engineering, Yale University（化学与环境工程系，耶鲁大学）
iRadioDiff: Physics-Informed Diffusion Model for Indoor Radio Map Construction and Localization
State Key Laboratory of ISN（信息与通信工程国家重点实验室）；Xidian University（西安电子科技大学）；School of Telecommunications Engineering（电信工程学院）；School of Artificial Intelligence（人工智能学院）；School of Information Science and Technology（信息科学与技术学院）；Southwest Jiaotong University（西南交通大学）；Department of Electrical and Computer Engineering（电气与计算机工程系）
Physics-informed Diffusion Generation for Geomagnetic Map Interpolation
State Key Laboratory of Blockchain and Data Security（区块链与数据安全国家重点实验室）；High-Performance Intelligent Computing Research Center for Ultra-Large Scale Graph Data（超大规模图数据高性能智能计算研究中心）；School of Computer and Computing Science（计算机与计算科学学院）；Hangzhou City University（杭州市大学）；Zhejiang Lab（浙江实验室）
Synthetic Power Flow Data Generation Using Physics-Informed Denoising Diffusion Probabilistic Models
Department of Electrical Engineering and Computer Science, York University（约克大学电气工程与计算机科学系）；Faculty of Computer Science, Dalhousie University（达尔豪斯大学计算机科学学院）；Cistel Technology（Cistel技术公司）
Residual resampling-based physics-informed neural network for neutron diffusion equations
Chongqing University of Posts and Telecommunications（重庆邮电大学）；Nuclear Power Institute of China（中国核动力研究设计院）；CNNC Engineering Research Center of Nuclear Energy Software and Digital Reactor（中核集团核能软件与数字反应堆工程研究中心）
SiDGen: Structure-informed Diffusion for Generative modeling of Ligands for Proteins
Department of Computer Science, Indian Institute of Technology, Delhi, India（计算机科学系，印度理工学院，德里，印度）；Department of Chemistry, Indian Institute of Technology, Delhi, India（化学系，印度理工学院，德里，印度）
Joint Initialization of Flux Networks and Effective Multiplication Factor for Physics-Informed Neural Networks Solving Neutron Diffusion Problems
School of Computer Science and Technology, Chongqing University of Posts and Telecommunications（重庆邮电大学计算机科学与技术学院）；Center for Scientific Intelligence Innovation, University of Science and Technology of China（中国科学技术大学科学智能创新中心）；Institute of Advanced Technology, University of Science and Technology of China（中国科学技术大学先进技术研究院）
Efficient Diversity-Preserving Diffusion Alignment via Gradient-Informed GFlowNets
Mila, Université de Montréal（蒙特利尔大学Mila实验室）；Max Planck Institute for Intelligent Systems - Tübingen（智能系统马克斯·普朗克研究所（图宾根））；The Chinese University of Hong Kong (Shenzhen)（香港中文大学（深圳））；University of Tübingen（图宾根大学）；University of Cambridge（剑桥大学）；Microsoft Research（微软研究院）
A scaled TW-PINN: A physics-informed neural network for traveling wave solutions of reaction-diffusion equations with general coefficients
Department of Mathematics \& POSTECH MINDS (Mathematical Institute for Data Science), Pohang University of Science；Graduate School of Artificial Intelligence \& POSTECH MINDS (Mathematical Institute for Data Science), Pohang University of Science
Physics-Informed Neural Networks for Biological $2\mathrm{D}{+}t$ Reaction-Diffusion Systems
Dept. of Information Technology（信息科技系）；Uppsala University（乌普萨拉大学）；Dept. of Medical Genetics（医学遗传学系）；Oslo University Hospital（奥斯陆大学医院）；Dept. of Mathematics and Statistics（数学与统计学系）；The College of New Jersey（新泽西学院）
PIDM-DP: Physics-Informed Diffusion with Dormand-Prince Integration for Chaotic System Identification and State Reconstruction across Multiple Dynamical Regimes
fPINN-DeepONet: A Physics-Informed Operator Learning Framework for Multi-term Time-fractional Mixed Diffusion-wave Equations
Department of Mathematics, Purdue University West Lafayette, IN 47907, USA（普渡大学数学系）；Department of Mathematics, Purdue University School of Mechanical Engineering, Purdue University West Lafayette, IN 47907, USA（普渡大学机械工程学院数学系）
Partition of Unity Physics-Informed Neural Networks (POU-PINNs): An Unsupervised Framework for Physics-Informed Domain Decomposition and Mixtures of Experts
The University of Texas at El Paso（德克萨斯大学埃尔帕索分校）；University of California, Santa Cruz（加州大学圣克鲁兹分校）；Clarkson University（克拉克森大学）；Texas A&M University in Kingsville（德克萨斯农工大学金斯维尔分校）
