We investigate the use of discrete and continuous versions of physics-informed neural network methods for learning unknown dynamics or constitutive relations of a dynamical system. For the case of unknown dynamics, we represent all the dynamics with a deep neural network (DNN). When the dynamics of the system are known up to the specification of constitutive relations (that can depend on the state of the system), we represent these constitutive relations with a DNN. The discrete versions combine classical multistep discretization methods for dynamical systems with neural network based machine learning methods. On the other hand, the continuous versions utilize deep neural networks to minimize the residual function for the continuous governing equations. We use the case of a fedbatch bioreactor system to study the effectiveness of these approaches and discuss conditions for their applicability. Our results indicate that the accuracy of the trained neural network models is much higher for the cases where we only have to learn a constitutive relation instead of the whole dynamics. This finding corroborates the well-known fact from scientific computing that building as much structural information is available into an algorithm can enhance its efficiency and/or accuracy.
Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks
Institute of Applied Mechanics（应用力学研究所）；Technische Universität Braunschweig（Braunschweig技术大学）；Clausthal University of Technology（克莱斯特哈大学）；Institute for Acoustics and Dynamics（声学与动力学研究所）；Eidgenössische Technische Hochschule Zürich（苏黎世联邦理工学院）
Neuroevolution of Physics-Informed Neural Nets: Benchmark Problems and Comparative Results
Center for Frontier AI Research（前沿人工智能研究中心）；Agency for Science, Technology, and Research（新加坡科技研究局）；Institute for High Performance Computing（高性能计算研究所）；Nanyang Technological University（南洋理工大学）；Singapore Institute of Manufacturing Technology（新加坡制造技术研究所）
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）
PINN-Obs: Physics-Informed Neural Network-Based Observer for Nonlinear Dynamical Systems
TICLab, International University of Rabat, Rabat 11100, Morocco（拉巴特国际大学TIC实验室）；CNRS–CRAN–7039, University of Lorraine, France（法国洛林大学CNRS–CRAN–7039机构）；Faculty of Engineering, University of Leeds, LS2 9JT, United Kingdom（利兹大学工程学院）；INRIA Nancy—LARSEN, France（法国INRIA南锡—LARSEN机构）
Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfaces
Department of Mechanical Engineering, IIT Kharagpur（印度克里希纳加尔帕大学机械工程系）；Department of Computer Science & Engineering, IIT Kharagpur（印度克里希纳加尔帕大学计算机科学与工程系）
Physics-Informed Gaussian Process Regression for the Constitutive Modeling of Concrete: A Data-Driven Improvement to Phenomenological Models
Hopkins Extreme Materials Institute（霍普金斯极端材料研究所）；Johns Hopkins University（约翰霍普金斯大学）；Dept. of Civil and Systems Engineering（土木与系统工程系）；Karagozian and Case, Inc.（卡罗金和凯斯公司）；Dept. of Mechanical Engineering（机械工程系）
Residual Attention Physics-Informed Neural Networks for Robust Multiphysics Simulation of Steady-State Electrothermal Energy Systems
Nanophotonics and Biophotonics Key Laboratory of Jilin Province, School of Physics, Changchun University of Science and Technology（吉林省纳米光子与生物光子重点实验室，物理学院，长春理工大学）
Physics-informed neural networks in the recreation of hydrodynamic simulations from dark matter
University of Houston（休斯顿大学）；University of Edinburgh（爱丁堡大学）；Carnegie Mellon University（卡内基梅隆大学）；Pittsburgh Supercomputing Center（匹兹堡超级计算中心）；University of the Western Cape（西开普大学）；South African Astronomical Observatories（南非天文台）
A thermodynamically consistent physics-informed deep learning material model for short fiber/polymer nanocomposites
Institute of Structural Analysis, Leibniz Universität Hannover（莱布尼茨汉诺威大学结构分析研究所）；Department of Mechanical, Electrical and Chemical Engineering, Oslo Metropolitan University（奥斯陆城市大学机械、电气与化学工程系）；Institute of Lightweight Systems, Multifunctional Materials, DLR (German Aerospace Center)（德国航空航天中心轻量化系统与多功能材料研究所）
A Comparative Evaluation of Additive Separability Tests for Physics-Informed Machine Learning
National University of Singapore（新加坡国立大学）；Singapore Institute of Manufacturing Technology, Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局（A*STAR）新加坡制造技术研究院）
Non-linear mechanical field reconstruction coupling recurrent neural networks with physics-informed graph neural networks
Univ. Bordeaux, CNRS, Bordeaux INP, I2M, UMR 5295（波尔多大学、国家科学研究中心、波尔多工业学院、I2M、UMR 5295）；Univ. Bordeaux, CNRS, Bordeaux INP, LaBRI, UMR 5800（波尔多大学、国家科学研究中心、波尔多工业学院、LaBRI、UMR 5800）；Institute of Mechanics and Fluid Dynamics, TU Bergakademie Freiberg（力学与流体动力学研究所、弗赖堡技术大学）；Univ. Grenoble Alpes, CNRS, UMR 5525（格勒诺布尔阿尔卑斯大学、国家科学研究中心、UMR 5525）；Arts et Metiers Institute of Technology, CNRS, Bordeaux INP, I2M, UMR 5295（艺术与工艺技术学院、国家科学研究中心、波尔多工业学院、I2M、UMR 5295）
Constitutive parameterized deep energy method for solid mechanics problems with random material parameters
National Center for Applied Mathematics, Tianjin University（天津大学国家应用数学中心）；School of Mechanical and Aerospace Engineering, Jilin University（吉林大学机械与航空航天工程学院）
A recursive Bayesian neural network for constitutive modeling of sands under monotonic and cyclic loading
Department of Civil Engineering（土木工程系）；Indian Institute of Technology Delhi（德里印度理工学院）；Department of Transportation Systems Engineering（交通运输系统工程系）；Technical University of Munich（慕尼黑技术大学）；Department of Applied Mechanics（应用力学系）
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: )
