Topology optimization (TO) provides a principled mathematical approach for optimizing the performance of a structure by designing its material spatial distribution in a pre-defined domain and subject to a set of constraints. The majority of existing TO approaches leverage numerical solvers for design evaluations during the optimization and hence have a nested nature and rely on discretizing the design variables. Contrary to these approaches, herein we develop a new class of TO methods based on the framework of Gaussian processes (GPs) whose mean functions are parameterized via deep neural networks. Specifically, we place GP priors on all design and state variables to represent them via parameterized continuous functions. These GPs share a deep neural network as their mean function but have as many independent kernels as there are state and design variables. We estimate all the parameters of our model in a single for loop that optimizes a penalized version of the performance metric where the penalty terms correspond to the state equations and design constraints. Attractive features of our approach include $(1)$ having a built-in continuation nature since the performance metric is optimized at the same time that the state equations are solved, and $(2)$ being discretization-invariant and accommodating complex domains and topologies. To test our method against conventional TO approaches implemented in commercial software, we evaluate it on four problems involving the minimization of dissipated power in Stokes flow. The results indicate that our approach does not need filtering techniques, has consistent computational costs, and is highly robust against random initializations and problem setup.
Multi-material Multi-physics Topology Optimization with Physics-informed Gaussian Process Priors
Department of Mechanical and Aerospace Engineering, University of California, Irvine（机械与航空航天工程系，加州大学伊维特分校）；Department of Civil and Environmental Engineering, University of California, Irvine（土木与环境工程系，加州大学伊维特分校）
Gaussian Process Regression-based Knowledge Distillation Framework for Simultaneous Prediction of Physical and Mechanical Properties of Epoxy Polymers
Sindu B.S Special and Multifunctional Structures Laboratory, CSIR-Structural Engineering Research Centre, Taramani, Chennai, Tamil Nadu, India（Sindu B.S 特殊和多功能结构实验室，CSIR-结构工程研究中心，坦马尼，钦奈，泰米尔纳德邦，印度）；Fraunhofer Institute for Algorithms and Scientific Computing SCAI, Schloss Birlinghoven, 53757 Sankt Augustin , Germany（弗劳恩霍夫算法与科学计算研究所SCAI，伯灵霍芬城堡，53757圣奥古斯滕，德国）
Interval and fuzzy physics-informed neural networks for uncertain fields
Cornell University（康奈尔大学）；National Technical University of Athens（雅典国家技术大学）；Université Paris-Saclay（巴黎萨克雷大学）；ENS Paris-Saclay（巴黎萨克雷高等师范学校）；CNRS（法国国家科学研究中心）
A Coupled Physics-Informed Neural Network for Greenhouse Climate State Reconstruction and Parameter Identification under Sparse Sensor Measurements
Centro de Modelamiento Matemático (CMM), University of Chile（智利大学数学建模中心）；College of Science, King Khalid University（哈立德国王大学理学院）；Presidency University（总统大学）
Physics-informed active learning with simultaneous weak-form latent space dynamics identification
ANSYS Inc.（ANSYS公司）；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）
Rethinking Physics-Informed Regression Beyond Training Loops and Bespoke Architectures
Department of Electrical \& Electronic Engineering, Imperial College, SW7 2AZ London, UK (e-mail: ).；Department of Aeronautics, Imperial College, \ 7 2AZ London, UK (e-mail: )
HypNO: A Graph-Based Neural Operator with Physics-Informed Message Passing for Hyperbolic Conservation Laws
Laboratory of Computer Science (LIX), École Polytechnique（计算机科学实验室（LIX），巴黎综合理工学院）；Department of Electrical Engineering and Computer Sciences, University of California, Berkeley（电气工程与计算机科学系，加利福尼亚大学伯克利分校）
Identifying Memory Effects in Epidemics via a Fractional SEIRD Model and Physics-Informed Neural Networks
University of Wuppertal, Applied and Computational Mathematics（乌尔姆大学应用与计算数学系）；Moulay Ismail University of Meknes, FST Errachidia, Department of Mathematics, AMNEA Group（梅基斯穆拉伊大学，Errachidia 工程技术学院，数学系，AMNEA 团队）
Residual-loss Anomaly Analysis of Physics-Informed Neural Networks: An Inverse Method for Change-point Detection in Nonlinear Dynamical Systems with Regime Switching
School of Mathematics and Statistics（数学与统计学学院）；Huazhong University of Science and Technology（华中科技大学）；Northwestern Polytechnical University（西北工业大学）
Department of Mechanical Engineering, National University of Singapore（新加坡国立大学机械工程系）；Institute of High Performance Computing, Agency for Science Technology and Research（科技研究局高性能计算研究所）；Centre for Frontier AI Research, Agency for Science Technology and Research（科技研究局前沿人工智能研究中心）；College of Electronics and Information Engineering, Shenzhen University（深圳大学电子与信息工程学院）；Engineering Product Design Pillar, Singapore University of Technology and Design（新加坡科技设计大学工程产品设计支柱）；Technology Centre for Offshore and Marine, Singapore（新加坡海洋与近海技术中心）
Integrating Newton's Laws with deep learning for enhanced physics-informed compound flood modelling
Center for Complex Hydrosystems Research（复杂水系统研究中心）；Department of Civil, Construction and Environmental Engineering（土木、建设与环境工程系）；The University of Alabama（阿拉巴马大学）
Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks
Center for Mathematical Sciences, Huazhong University of Science and Technology（华中科技大学数学科学中心）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Applied Mathematics, College of Computing, Illinois Institute of Technology（伊利诺伊理工学院计算学院应用数学系）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
