Deep learning and the collocation method are merged and used to solve partial differential equations describing structures' deformation. We have considered different types of materials: linear elasticity, hyperelasticity (neo-Hookean) with large deformation, and von Mises plasticity with isotropic and kinematic hardening. The performance of this deep collocation method (DCM) depends on the architecture of the neural network and the corresponding hyperparameters. The presented DCM is meshfree and avoids any spatial discretization, which is usually needed for the finite element method (FEM). We show that the DCM can capture the response qualitatively and quantitatively, without the need for any data generation using other numerical methods such as the FEM. Data generation usually is the main bottleneck in most data-driven models. The deep learning model is trained to learn the model's parameters yielding accurate approximate solutions. Once the model is properly trained, solutions can be obtained almost instantly at any point in the domain, given its spatial coordinates. Therefore, the deep collocation method is potentially a promising standalone technique to solve partial differential equations involved in the deformation of materials and structural systems as well as other physical phenomena.
SRM Institute of Science and Technology（SRM科学技术研究所）；University of Utah（犹他大学）
We present a new technique for the accelerated training of physics-informed neural networks (PINNs): discretely-trained PINNs (DT-PINNs). The repeated computation of partial derivative terms in the PINN loss functions via automatic differentiation during training is known to be computationally expensive, especially for higher-order derivatives. DT-PINNs are trained by replacing these exact spatial derivatives with high-order accurate numerical discretizations computed using meshless radial basis function-finite differences (RBF-FD) and applied via sparse-matrix vector multiplication. The use o…
Sharif University of Technology（谢里夫理工大学）；Massachusetts Institute of Technology（麻省理工学院）
Physics-Informed Neural Networks (PINNs) have received increased interest for forward, inverse, and surrogate modeling of problems described by partial differential equations (PDE). However, their application to multiphysics problem, governed by several coupled PDEs, present unique challenges that have hindered the robustness and widespread applicability of this approach. Here we investigate the application of PINNs to the forward solution of problems involving thermo-hydro-mechanical (THM) processes in porous media, which exhibit disparate spatial and temporal scales in thermal conductivity, …
Physics Informed Neural Networks for design optimisation of diamond particle detectors for charged particle fast-tracking at high luminosity hadron colliders
Istituto Nazionale di Fisica Nucleare, Sezione di Firenze（意大利佛罗伦萨核物理研究所）；ICSC - Centro Nazionale di Ricerca in High Performance Computing, Big Data & Quantum Computing（高性能计算、大数据与量子计算国家研究中心）
Brown University（布朗大学）；Xiamen University（厦门大学）；Dalian University of Technology（大连理工大学）
Despite the significant progress over the last 50 years in simulating flow problems using numerical discretization of the Navier-Stokes equations (NSE), we still cannot incorporate seamlessly noisy data into existing algorithms, mesh-generation is complex, and we cannot tackle high-dimensional problems governed by parametrized NSE. Moreover, solving inverse flow problems is often prohibitively expensive and requires complex and expensive formulations and new computer codes. Here, we review flow physics-informed learning, integrating seamlessly data and mathematical models, and implementing the…
SPINONet: Scalable Spiking Physics-informed Neural Operator for Computational Mechanics Applications
Department of Applied Mechanics（应用力学系）；Indian Institute of Technology Delhi（德里印度理工学院）；Institute of Structural Mechanics and Dynamics in Aerospace Engineering（航空航天工程中的结构力学与动力学研究所）；Faculty of Aerospace Engineering and Geodesy（航空航天工程与大地测量学系）；University of Stuttgart（斯图加特大学）；Experimental Hepatobiliary Surgery Group（肝胆外科实验小组）；Department of Hepatobiliary Surgery and Visceral Transplantation（肝胆外科与内脏移植部）；University of Leipzig Medical Center（莱比锡大学医学中心）；Department of Civil and Systems Engineering（土木与系统工程系）；Johns Hopkins University（约翰霍普金斯大学）；Data Science and AI Institute（数据科学与人工智能研究所）；Yardi School of Artificial Intelligence (YScAI)（Yardi人工智能学院（YScAI））
We present hidden fluid mechanics (HFM), a physics informed deep learning framework capable of encoding an important class of physical laws governing fluid motions, namely the Navier-Stokes equations. In particular, we seek to leverage the underlying conservation laws (i.e., for mass, momentum, and energy) to infer hidden quantities of interest such as velocity and pressure fields merely from spatio-temporal visualizations of a passive scaler (e.g., dye or smoke), transported in arbitrarily complex domains (e.g., in human arteries or brain aneurysms). Our approach towards solving the aforement…
Physics-Informed Neural Networks (PINNs) have gained much attention in various fields of engineering thanks to their capability of incorporating physical laws into the models. However, the assessment of PINNs in industrial applications involving coupling between mechanical and thermal fields is still an active research topic. In this work, we present an application of PINNs to a non-Newtonian fluid thermo-mechanical problem which is often considered in the rubber calendering process. We demonstrate the effectiveness of PINNs when dealing with inverse and ill-posed problems, which are impractic…
Siemens AG, Corporate Technology（西门子股份公司，企业技术部）；Technical University of Munich（慕尼黑工业大学）
Identifying accurate dynamic models is required for the simulation and control of various technical systems. In many important real-world applications, however, the two main modeling approaches often fail to meet requirements: first principles methods suffer from high bias, whereas data-driven modeling tends to have high variance. Additionally, purely data-based models often require large amounts of data and are often difficult to interpret. In this paper, we present physics-informed neural ordinary differential equations (PINODE), a hybrid model that combines the two modeling techniques to ov…
Traditional approaches based on finite element analyses have been successfully used to predict the macro-scale behavior of heterogeneous materials (composites, multicomponent alloys, and polycrystals) widely used in industrial applications. However, this necessitates the mesh size to be smaller than the characteristic length scale of the microstructural heterogeneities in the material leading to computationally expensive and time-consuming calculations. The recent advances in deep learning based image super-resolution (SR) algorithms open up a promising avenue to tackle this computational chal…
Long-term simulation of physical and mechanical behaviors using curriculum-transfer-learning based physics-informed neural networks
College of Mechanics and Engineering Science, Hohai University（河海大学力学与工程科学学院）；China Three Gorges Corporation（中国长江三峡集团有限公司）；Cairo University（开罗大学）；Leibniz Universität Hannover（汉诺威莱布尼茨大学）
SPARC-Net: A Spectral, Causality-Aware, and Hard-Constrained Physics-Informed Architecture for Stiff and Shock-Dominated Partial Differential Equations
Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks
Institute for Mathematics and Computer-Based Simulation (IMCS), University of the Bundeswehr Munich（慕尼黑联邦国防军大学数学与计算机模拟研究所）；Institute for the Protection of Terrestrial Infrastructures, German Aerospace Center (DLR)（德国航空航天中心陆地基础设施保护研究所）
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）
We apply Physics-Informed Neural Networks (PINNs) for solving identification problems of nonhomogeneous materials. We focus on the problem with a background in elasticity imaging, where one seeks to identify the nonhomogeneous mechanical properties of soft tissue based on the full-field displacement measurements under quasi-static loading. In our model, we apply two independent neural networks, one for approximating the solution of the corresponding forward problem, and the other for approximating the unknown material parameter field. As a proof of concept, we validate our model on a prototypi…
Shape-informed cardiac mechanics surrogates in data-scarce regimes via geometric encoding and generative augmentation
MOX Laboratory, Department of Mathematics, Politecnico di Milano（米兰Politecnico大学数学系MOX实验室）；Division 2.2 Process Simulation, Bundesanstalt für Materialforschung und -prüfung (BAM)（材料研究所（BAM）第2.2过程模拟部）；Department of Mathematics, University of Trento（特伦托大学数学系）
LT-PINN: Lagrangian Topology-conscious Physics-informed Neural Network for Boundary-focused Engineering Optimization
Department of Civil and Environmental Engineering, The Hong Kong Polytechnic University, Hong Kong, China（土木与环境工程系，香港理工大学，香港，中国）；Research Institute for Sustainable Urban Development (RISUD), The Hong Kong Polytechnic University, Hong Kong, China（可持续城市发展研究院（RISUD），香港理工大学，香港，中国）；Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hong Kong, China（机械工程系，香港理工大学，香港，中国）；School of Fashion and Textiles, The Hong Kong Polytechnic University, Hong Kong, China（时尚与纺织学院，香港理工大学，香港，中国）；Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China（机械工程系，香港大学，香港，中国）
Standard neural networks can approximate general nonlinear operators, represented either explicitly by a combination of mathematical operators, e.g., in an advection-diffusion-reaction partial differential equation, or simply as a black box, e.g., a system-of-systems. The first neural operator was the Deep Operator Network (DeepONet), proposed in 2019 based on rigorous approximation theory. Since then, a few other less general operators have been published, e.g., based on graph neural networks or Fourier transforms. For black box systems, training of neural operators is data-driven only but if…
Aragon Institute of Engineering Research (I3A)（阿拉贡工程研究所）；University of Zaragoza（萨拉戈萨大学）；Higher Technical School of Industrial Engineering, Polytechnic University of Madrid（马德里理工大学工业工程高等技术学院）；Polytechnic University of Madrid（马德里理工大学）；ENSAM Institute of Technology（法国国立高等工程技术学院）
In this paper we present a deep learning method to predict the temporal evolution of dissipative dynamic systems. We propose using both geometric and thermodynamic inductive biases to improve accuracy and generalization of the resulting integration scheme. The first is achieved with Graph Neural Networks, which induces a non-Euclidean geometrical prior with permutation invariant node and edge update functions. The second bias is forced by learning the GENERIC structure of the problem, an extension of the Hamiltonian formalism, to model more general non-conservative dynamics. Several examples a…
Numerical methods such as finite element have been flourishing in the past decades for modeling solid mechanics problems via solving governing partial differential equations (PDEs). A salient aspect that distinguishes these numerical methods is how they approximate the physical fields of interest. Physics-informed deep learning is a novel approach recently developed for modeling PDE solutions and shows promise to solve computational mechanics problems without using any labeled data. The philosophy behind it is to approximate the quantity of interest (e.g., PDE solution variables) by a deep neu…
Multitask-Informed Prior for In-Context Learning on Tabular Data: Application to Steel Property Prediction
Dep. of Electrical and Computer Engineering, McGill University（麦吉尔大学电气与计算机工程系）；Mila – Québec AI Institute（魁北克AI研究所）；Dep. of Mining and Materials Engineering, McGill University（麦吉尔大学采矿与材料工程系）；Algoma Steel Inc.（阿尔戈马钢铁公司）
Differential equations are used to model and predict the behaviour of complex systems in a wide range of fields, and the ability to solve them is an important asset for understanding and predicting the behaviour of these systems. Complicated physics mostly involves difficult differential equations, which are hard to solve analytically. In recent years, physics-informed neural networks have been shown to perform very well in solving systems with various differential equations. The main ways to approximate differential equations are through penalty function and reparameterization. Most researche…
Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials
Universite Paris-Saclay, ENS Paris-Saclay, CNRS, Centre Borelli（巴黎-萨克莱大学、巴黎-萨克莱高等学院、国家科学研究中心、Borelli中心）；Michelin, Centre de Recherche de Ladoux（米其林公司、Ladoux研究中心）；CEA, DAM, DIF（法国原子能委员会、DAM、DIF）；ENSIIE, Évry-Courcouronnes（巴黎-萨克莱高等工程师学院、Évry-Courcouronnes）
University of British Columbia（不列颠哥伦比亚大学）；University of Arizona（亚利桑那大学）；Massachusetts Institute of Technology（麻省理工学院）
Deep learning has been the most popular machine learning method in the last few years. In this chapter, we present the application of deep learning and physics-informed neural networks concerning structural mechanics and vibration problems. Demonstration problems involve de-noising data, solution to time-dependent ordinary and partial differential equations, and characterizing the system's response for a given data.
Texas State University（德克萨斯州立大学）；University of Minho（米尼奥大学）；University of Colorado Boulder（科罗拉多大学博尔德分校）；Texas A&M University（德克萨斯农工大学）
Recent breakthroughs in computing power have made it feasible to use machine learning and deep learning to advance scientific computing in many fields, including fluid mechanics, solid mechanics, materials science, etc. Neural networks, in particular, play a central role in this hybridization. Due to their intrinsic architecture, conventional neural networks cannot be successfully trained and scoped when data is sparse, which is the case in many scientific and engineering domains. Nonetheless, neural networks provide a solid foundation to respect physics-driven or knowledge-based constraints d…
Sandia National Laboratories（桑迪亚国家实验室）；Arizona State University（亚利桑那州立大学）
We introduce physics-informed multimodal autoencoders (PIMA) - a variational inference framework for discovering shared information in multimodal scientific datasets representative of high-throughput testing. Individual modalities are embedded into a shared latent space and fused through a product of experts formulation, enabling a Gaussian mixture prior to identify shared features. Sampling from clusters allows cross-modal generative modeling, with a mixture of expert decoder imposing inductive biases encoding prior scientific knowledge and imparting structured disentanglement of the latent s…
Pacific Northwest National Laboratory（太平洋西北国家实验室）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Real-time state estimation and forecasting is critical for efficient operation of power grids. In this paper, a physics-informed Gaussian process regression (PhI-GPR) method is presented and used for probabilistic forecasting and estimating the phase angle, angular speed, and wind mechanical power of a three-generator power grid system using sparse measurements. In standard data-driven Gaussian process regression (GPR), parameterized models for the prior statistics are fit by maximizing the marginal likelihood of observed data, whereas in PhI-GPR, we compute the prior statistics by solving sto…
Separated-Variable Spectral Neural Networks: A Physics-Informed Learning Approach for High-Frequency PDEs
School of Mathematics and Statistics（数学与统计学学院）；Department of Engineering Mechanics（工程力学系）；Department of Aeronautical Engineering（航空工程系）；MIIT Key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University（复杂系统动力学与控制国家重点实验室，西北工业大学）；College of Computer Science and Technology（计算机科学与技术学院）；National University of Defense Technology（国防科技大学）；Department of Aerospace Engineering（航空航天工程系）
Karlsruhe Institute of Technology (KIT)（卡尔斯鲁厄理工学院（KIT））；RWTH Aachen University（亚琛工业大学）
(Artificial) neural networks have become increasingly popular in mechanics to accelerate computations with model order reduction techniques and as universal models for a wide variety of materials. However, the major disadvantage of neural networks remains: their numerous parameters are challenging to interpret and explain. Thus, neural networks are often labeled as black boxes, and their results often elude human interpretation. In mechanics, the new and active field of physics-informed neural networks attempts to mitigate this disadvantage by designing deep neural networks on the basis of mec…
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）
Real-time distortion prediction in metallic additive manufacturing via a physics-informed neural operator approach
School of Mechanical and Power Engineering, Nanjing Tech University（南京工业大学机械与动力工程学院）；Institute of Reliability centered Manufacturing, Nanjing Tech University（南京工业大学可靠性制造研究所）；Jiangsu Provincial Key Laboratory of Energy Power Manufacturing Equipment and Reliability Technology, Nanjing Tech University（南京工业大学江苏省能源动力制造装备与可靠性技术重点实验室）
University of Oxford（牛津大学）；Harvard University（哈佛大学）
Solving differential equations efficiently and accurately sits at the heart of progress in many areas of scientific research, from classical dynamical systems to quantum mechanics. There is a surge of interest in using Physics-Informed Neural Networks (PINNs) to tackle such problems as they provide numerous benefits over traditional numerical approaches. Despite their potential benefits for solving differential equations, transfer learning has been under explored. In this study, we present a general framework for transfer learning PINNs that results in one-shot inference for linear systems of …
