本文提出物理信息深度均衡模型（PIDEQs），融合深度均衡模型的隐式输出表示与物理信息训练技术，用于高效求解常微分方程初值问题，并以 Van der Pol 振荡器验证其有效性。
摘要：
本文介绍了用于求解常微分方程（ODEs）初值问题（IVPs）的物理信息深度均衡模型（PIDEQs）。PIDEQs 利用深度均衡模型（DEQs）和物理信息神经网络（PINNs）的最新进展，将 DEQs 的隐式输出表示与物理信息训练技术相结合。我们以 Van der Pol 振荡器作为基准问题验证 PIDEQs，展示其在求解 IVPs 方面的效率和有效性。我们的分析包括优化 PIDEQ 性能的关键超参数考虑因素。通过连接深度学习与基于物理的建模，这项工作推进了求解 IVPs 的计算技术，对科学计算和工程应用具有重要意义。
英文摘要：
This paper introduces Physics-Informed Deep Equilibrium Models (PIDEQs) for solving initial value problems (IVPs) of ordinary differential equations (ODEs). Leveraging recent advancements in deep equilibrium models (DEQs) and physics-informed neural networks (PINNs), PIDEQs combine the implicit output representation of DEQs with physics-informed training techniques. We validate PIDEQs using the Van der Pol oscillator as a benchmark problem, demonstrating their efficiency and effectiveness in solving IVPs. Our analysis includes key hyperparameter considerations for optimizing PIDEQ performance. By bridging deep learning and physics-based modeling, this work advances computational techniques for solving IVPs, with implications for scientific computing and engineering applications.
The objective of this paper is to investigate the ability of physics-informed neural networks to learn the magnetic field response as a function of design parameters in the context of a two-dimensional (2-D) magnetostatic problem. Our approach is as follows. First, we present a functional whose minimization is equivalent to solving parametric magnetostatic problems. Subsequently, we use a deep neural network (DNN) to represent the magnetic field as a function of space and parameters that describe geometric features and operating points. We train the DNN by minimizing the physics-informed funct…
Technical University of Denmark（丹麦技术大学）；The Danish Hydrocarbon Research and Technology Centre（丹麦碳氢化合物研究与技术中心）
This paper presents the potential of applying physics-informed neural networks for solving nonlinear multiphysics problems, which are essential to many fields such as biomedical engineering, earthquake prediction, and underground energy harvesting. Specifically, we investigate how to extend the methodology of physics-informed neural networks to solve both the forward and inverse problems in relation to the nonlinear diffusivity and Biot's equations. We explore the accuracy of the physics-informed neural networks with different training example sizes and choices of hyperparameters. The impacts …
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…
Solving a class of stochastic optimal control problems by physics-informed neural networks
School of Mathematics and Statistics（数学与统计学学院）；MOE Key Laboratory for Complexity Science in Aerospace（复杂科学在航空航天中的关键实验室）；State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics（流体动力与机械系统国家重点实验室，力学系）
University of Tehran（德黑兰大学）；KTH Royal Institute of Technology（瑞典皇家理工学院）
Physics-informed neural networks (PINNs) are successful machine-learning methods for the solution and identification of partial differential equations (PDEs). We employ PINNs for solving the Reynolds-averaged Navier$\unicode{x2013}$Stokes (RANS) equations for incompressible turbulent flows without any specific model or assumption for turbulence, and by taking only the data on the domain boundaries. We first show the applicability of PINNs for solving the Navier$\unicode{x2013}$Stokes equations for laminar flows by solving the Falkner$\unicode{x2013}$Skan boundary layer. We then apply PINNs for…
Solving Euler equations with Multiple Discontinuities via Separation-Transfer Physics-Informed Neural Networks
Graduate School, China Academy of Engineering Physics（中国工程物理研究院研究生院）；China Academy of Engineering Physics（中国工程物理研究院）；Zhejiang Institute of Modern Physics（浙江现代物理研究所）；School of Physics, Zhejiang University（浙江大学物理学院）；Beijing Computational Science Research Center（北京计算科学研究院）
Adaptively trained Physics-informed Radial Basis Function Neural Networks for Solving Multi-asset Option Pricing Problems
College of Mathematics and Statistics, Northwest Normal University（西北师范大学数学与统计学院）；Department of Mathematics, City University of Hong Kong（香港城市大学数学系）；Department of Information Technology, Uppsala University（乌普萨拉大学信息科技系）
Kolmogorov Arnold Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov Arnold Networks
Tsinghua University（清华大学）；Bauhaus-Universität Weimar（魏玛包豪斯大学）；CNPC Engineering Technology RD Company Limited（中国石油集团工程技术研究院有限公司）；Queensland University of Technology（昆士兰科技大学）；Leibniz University Hannover（汉诺威莱布尼茨大学）
University of Washington（华盛顿大学）；DuoTech LLC.（DuoTech有限责任公司）
A physics-informed neural network is developed to solve conductive heat transfer partial differential equation (PDE), along with convective heat transfer PDEs as boundary conditions (BCs), in manufacturing and engineering applications where parts are heated in ovens. Since convective coefficients are typically unknown, current analysis approaches based on trial and error finite element (FE) simulations are slow. The loss function is defined based on errors to satisfy PDE, BCs and initial condition. An adaptive normalizing scheme is developed to reduce loss terms simultaneously. In addition, th…
Minzu University of China（中央民族大学）；North China University of Technology（北方工业大学）
As a typical application of deep learning, physics-informed neural network (PINN) {has been} successfully used to find numerical solutions of partial differential equations (PDEs), but how to improve the limited accuracy is still a great challenge for PINN. In this work, we introduce a new method, symmetry-enhanced physics informed neural network (SPINN) where the invariant surface conditions induced by the Lie symmetries or non-classical symmetries of PDEs are embedded into the loss function in PINN, to improve the accuracy of PINN for solving the forward and inverse problems of PDEs. We test…
The University of Arizona（亚利桑那大学）；Massachusetts Institute of Technology（麻省理工学院）
This study presents a novel unsupervised convolutional Neural Network (NN) architecture with nonlocal interactions for solving Partial Differential Equations (PDEs). The nonlocal Peridynamic Differential Operator (PDDO) is employed as a convolutional filter for evaluating derivatives the field variable. The NN captures the time-dynamics in smaller latent space through encoder-decoder layers with a Convolutional Long-short Term Memory (ConvLSTM) layer between them. The ConvLSTM architecture is modified by employing a novel activation function to improve the predictive capability of the learning…
In practical engineering experiments, the data obtained through detectors are inevitably noisy. For the already proposed data-enabled physics-informed neural network (DEPINN) \citep{DEPINN}, we investigate the performance of DEPINN in calculating the neutron diffusion eigenvalue problem from several perspectives when the prior data contain different scales of noise. Further, in order to reduce the effect of noise and improve the utilization of the noisy prior data, we propose innovative interval loss functions and give some rigorous mathematical proofs. The robustness of DEPINN is examined on …
QCPINN: Quantum-Classical Physics-Informed Neural Networks for Solving PDEs
Computational Science and Engineering Department, Informatics Institute Istanbul Technical University（信息学院理工学院计算科学与工程系）；Scientific Computing, Rutherford Appleton Laboratory Science and Technology Facilities Council (STFC)（科学与技术设施委员会（STFC）科学计算部）
Northeastern University（东北大学）；University of Notre Dame（圣母大学）；MIT（麻省理工学院）
Partial differential equations (PDEs) play a fundamental role in modeling and simulating problems across a wide range of disciplines. Recent advances in deep learning have shown the great potential of physics-informed neural networks (PINNs) to solve PDEs as a basis for data-driven modeling and inverse analysis. However, the majority of existing PINN methods, based on fully-connected NNs, pose intrinsic limitations to low-dimensional spatiotemporal parameterizations. Moreover, since the initial/boundary conditions (I/BCs) are softly imposed via penalty, the solution quality heavily relies on h…
Compared with conventional numerical approaches to solving partial differential equations (PDEs), physics-informed neural networks (PINN) have manifested the capability to save development effort and computational cost, especially in scenarios of reconstructing the physics field and solving the inverse problem. Considering the advantages of parameter sharing, spatial feature extraction and low inference cost, convolutional neural networks (CNN) are increasingly used in PINN. However, some challenges still remain as follows. To adapt convolutional PINN to solve different PDEs, considerable effo…
CI-PINN: Causal Integral Physics-Informed Neural Network for Solving Evolution Equations
Beijing Normal-Hong Kong Baptist University（北京师范大学-香港浸会大学联合国际学院）；Beijing Normal University（北京师范大学）；Guangzhou Nanfang College（广州南方学院）；Louisiana State University（路易斯安那州立大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
Physics-informed neural networks (PINNs) are demonstrating remarkable promise in integrating physical models with gappy and noisy observational data, but they still struggle in cases where the target functions to be approximated exhibit high-frequency or multi-scale features. In this work we investigate this limitation through the lens of Neural Tangent Kernel (NTK) theory and elucidate how PINNs are biased towards learning functions along the dominant eigen-directions of their limiting NTK. Using this observation, we construct novel architectures that employ spatio-temporal and multi-scale ra…
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 …
A Theory-guided Weighted $L^2$ Loss for solving the BGK model via Physics-informed neural networks
Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Center for Mathematical Machine Learning and its Applications (CM2LA), Department of Mathematics, Pohang University of Science and Technology（浦项科技大学数学系数学机器学习及其应用中心(CM2LA)）；Department of Mathematics, Gyeongsang National University（庆尚国立大学数学系）；Research Institute of Mathematics, Seoul National University（首尔大学数学研究所）
Solving stiff dark matter equations via Jacobian Normalization with Physics-Informed Neural Networks
Departamento de Física and CFTP Instituto Superior Técnico, Universidade de Lisboa（物理系和CFTP理工学院，里斯本大学）；Institute of Experimental and Applied Physics, Czech Technical University in Prague（实验与应用物理研究所，布拉格捷克技术大学）
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）
University of Science and Technology of China（中国科学技术大学）；Huawei Technologies Co. Ltd（华为技术有限公司）
In recent years, deep learning technology has been used to solve partial differential equations (PDEs), among which the physics-informed neural networks (PINNs) emerges to be a promising method for solving both forward and inverse PDE problems. PDEs with a point source that is expressed as a Dirac delta function in the governing equations are mathematical models of many physical processes. However, they cannot be solved directly by conventional PINNs method due to the singularity brought by the Dirac delta function. We propose a universal solution to tackle this problem with three novel techni…
Universidad Politecnica de Madrid（马德里理工大学）；Hess Corporation（赫斯公司）
Physics-Informed Neural Networks (PINNs) have enabled significant improvements in modelling physical processes described by partial differential equations (PDEs). PINNs are based on simple architectures, and learn the behavior of complex physical systems by optimizing the network parameters to minimize the residual of the underlying PDE. Current network architectures share some of the limitations of classical numerical discretization schemes when applied to non-linear differential equations in continuum mechanics. A paradigmatic example is the solution of hyperbolic conservation laws that deve…
As an emerging technology in deep learning, physics-informed neural networks (PINNs) have been widely used to solve various partial differential equations (PDEs) in engineering. However, PDEs based on practical considerations contain multiple physical quantities and complex initial boundary conditions, thus PINNs often returns incorrect results. Here we take heat transfer problem in multilayer fabrics as a typical example. It is coupled by multiple temperature fields with strong correlation, and the values of variables are extremely unbalanced among different dimensions. We clarify the potenti…
Brown University（布朗大学）；Worcester Polytechnic Institute（伍斯特理工学院）；Pacific Northwest National Laboratory（太平洋西北国家实验室）
Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a Petrov-Galerkin version of PINNs based on the nonlinear approximation of deep neural networks (DNNs) by selecting the {\em trial space} to be the space of neural networks and the {\em test space} to be the space of Legendre polynomials. We formulate the \textit{variational residual} of the PDE using the DNN approximation by incorporating the variational form o…
On the training of physics-informed neural operators for solving parametric partial differential equations
College of Civil Engineering, Tongji University（同济大学土木工程学院）；Department of Engineering Science, University of Oxford（牛津大学工程科学系）；Institute for Foundations of Data Science, Yale University（耶鲁大学数据科学基础研究所）
Technical University of Denmark（丹麦技术大学）；The Danish Hydrocarbon Research and Technology Centre（丹麦油气研究与技术中心）
In biomedical engineering, earthquake prediction, and underground energy harvesting, it is crucial to indirectly estimate the physical properties of porous media since the direct measurement of those are usually impractical/prohibitive. Here we apply the physics-informed neural networks to solve the inverse problem with regard to the nonlinear Biot's equations. Specifically, we consider batch training and explore the effect of different batch sizes. The results show that training with small batch sizes, i.e., a few examples per batch, provides better approximations (lower percentage error) of …
Advanced Physics-Informed Neural Network with Residuals for Solving Complex Integral Equations
Department of Computer and Data Sciences, Shahid Beheshti University（计算机与数据科学系，谢赫·贝赫什提大学）；Department of Cognitive Modeling, Shahid Beheshti University（认知建模系，谢赫·贝赫什提大学）
In this work we present a novel, accurate, and robust physics-informed method for solving problems involving parametric differential equations (DEs) called the Extreme Theory of Functional Connections, or X-TFC. The proposed method is a synergy of two recently developed frameworks for solving problems involving parametric DEs, 1) the Theory of Functional Connections, TFC, and the Physics-Informed Neural Networks, PINN. Although this paper focuses on the solution of exact problems involving parametric DEs (i.e. problems where the modeling error is negligible) with known parameters, X-TFC can al…
A shallow physics-informed neural network for solving partial differential equations on surfaces
National Central University（中央大学）；National Yang Ming Chiao Tung University（国立阳明交通大学）；National Center for Theoretical Sciences（理论科学研究中心）；National Taiwan University（台湾大学）
Osaka University（大阪大学）；The Institute of Scientific and Industrial Research, Osaka University（大阪大学科学与工业研究所）
Recently, researchers have utilized neural networks to accurately solve partial differential equations (PDEs), enabling the mesh-free method for scientific computation. Unfortunately, the network performance drops when encountering a high nonlinearity domain. To improve the generalizability, we introduce the novel approach of employing multi-task learning techniques, the uncertainty-weighting loss and the gradients surgery, in the context of learning PDE solutions. The multi-task scheme exploits the benefits of learning shared representations, controlled by cross-stitch modules, between multip…
GRINN: A Physics-Informed Neural Network for solving hydrodynamic systems in the presence of self-gravity
Jet Propulsion Laboratory, California Institute of Technology（加州理工学院喷气推进实验室）；University of Western Ontario（西安大略大学）；Perimeter Institute For Theoretical Physics（圆周理论物理研究所）
