This paper advances the use of physics-informed neural networks (PINNs) architectures to address moving interface problems via the level set method. Originally developed for other PDE-based problems, we particularly leverage PirateNet's features, including causal training, sequence-to-sequence learning, random weight factorization, and Fourier feature embeddings, and tailor them to handle problems with complex interface dynamics. Numerical experiments validate this framework on benchmark problems such as Zalesak's disk rotation and time-reversed vortex flow. We demonstrate that PINNs can efficiently solve level set problems exhibiting significant interface deformation without the need for upwind numerical stabilization, as generally required by classic discretization methods, or additional mass conservation schemes. However, incorporating an Eikonal regularization term in the loss function with an appropriate weight can further enhance results in specific scenarios. Our results indicate that PINNs with the PirateNet architecture surpass conventional PINNs in accuracy, achieving state-of-the-art error rates of $L^2=0.14\%$ for Zalesak's disk and $L^2=0.85 \%$ for the time-reversed vortex flow problem, as compared to reference solutions. Additionally, for a complex two-phase flow dam break problem coupling the level set with the Navier-Stokes equations, we propose a geometric reinitialization method embedded within the sequence-to-sequence training scheme to ensure long-term stability and accurate inference of the level set field.
In this paper, a novel immersed boundary method is developed, validated, and applied. Through devising a second-order three-step flow reconstruction scheme, the proposed method is able to enforce the Dirichlet, Neumann, Robin, and Cauchy boundary conditions in a straightforward and consistent manner. Equipped with a fluid-solid coupling framework that integrates high-order temporal and spatial discretization schemes, numerical experiments concerning flow involving stationary and moving objects, convex and concave geometries, no-slip and slip wall boundary conditions, as well as subsonic and su…
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…
Phonon Boltzmann transport equation (BTE) is a key tool for modeling multiscale phonon transport, which is critical to the thermal management of miniaturized integrated circuits, but assumptions about the system temperatures (i.e., small temperature gradients) are usually made to ensure that it is computationally tractable. To include the effects of large temperature non-equilibrium, we demonstrate a data-free deep learning scheme, physics-informed neural network (PINN), for solving stationary, mode-resolved phonon BTE with arbitrary temperature gradients. This scheme uses the temperature-depe…
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…
Boltzmann transport equation (BTE) is an ideal tool to describe the multiscale phonon transport phenomena, which are critical to applications like microelectronics cooling. Numerically solving phonon BTE is extremely computationally challenging due to the high dimensionality of such problems, especially when mode-resolved properties are considered. In this work, we demonstrate the use of physics-informed neural networks (PINNs) to efficiently solve phonon BTE for multiscale thermal transport problems with the consideration of phonon dispersion and polarization. In particular, a PINN framework …
Physics-Informed Machine Learning for Two-Phase Moving-Interface and Stefan Problems
Institute of Artificial Intelligence Innovation, National Yang Ming Chiao Tung University（人工智能创新研究所，阳明交通大学）；Department of Applied Mathematics, National Yang Ming Chiao Tung University（应用数学系，阳明交通大学）；National Center for Theoretical Sciences, National Taiwan University（理论科学研究中心，台湾大学）
This is a series of studies devoted to modeling and solving heat and mass transfer problems occurring in electric contacts where we employ and develop mathematical apparatus along with quantum algorithms for solving moving boundary value problems. In this particular study we utilize special functions and demonstrate the use of Harrow-Hassidim-Lloyd (HHL) quantum algorithm for finding exact and approximate solutions of Generalized Heat Equation with moving boundaries and as examples we consider plane and spherical cases. In spherical case the Generalized Heat Equation is reduced to linear movin…
A particle confined to an impassable box is a paradigmatic and exactly solvable one-dimensional quantum system modeled by an infinite square well potential. Here we explore some of its infinitely many generalizations to two dimensions, including particles confined to rectangle, elliptic, triangle, and cardioid-shaped boxes, using physics-informed neural networks. In particular, we generalize an unsupervised learning algorithm to find the particles' eigenvalues and eigenfunctions. During training, the neural network adjusts its weights and biases, one of which is the energy eigenvalue, so its o…
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…
The physics-informed neural network (PINN) is effective in solving the partial differential equation (PDE) by capturing the physics constraints as a part of the training loss function through the Automatic Differentiation (AD). This study proposes the hybrid finite difference with the physics-informed neural network (HFD-PINN) to fully use the domain knowledge. The main idea is to use the finite difference method (FDM) locally instead of AD in the framework of PINN. In particular, we use AD at complex boundaries and the FDM in other domains. The hybrid learning model shows promising results in…
We present a data-enabled physics-informed neural network (DEPINN) with comprehensive numerical study for solving industrial scale neutron diffusion eigenvalue problems (NDEPs). In order to achieve an engineering acceptable accuracy for complex engineering problems, a very small amount of prior data from physical experiments are suggested to be used, to improve the accuracy and efficiency of training. We design an adaptive optimization procedure with Adam and LBFGS to accelerate the convergence in the training stage. We discuss the effect of different physical parameters, sampling techniques, …
Scalable physics-informed deep generative model for solving forward and inverse stochastic differential equations
Institute of Interdisciplinary Research for Mathematics and Applied Science, School of Mathematics and Statistics, Huazhong University of Science and Technology（华中科技大学数学与统计学院交叉学科数学与应用科学研究所）；Department of Mathematics, Shanghai Normal University（上海师范大学数学系）
A hands-on introduction to Physics-Informed Neural Networks for solving partial differential equations with benchmark tests taken from astrophysics and plasma physics
Numerical solutions to the equation for advection are determined using different finite-difference approximations and physics-informed neural networks (PINNs) under conditions that allow an analytical solution. Their accuracy is examined by comparing them to the analytical solution. We used a machine learning framework like PyTorch to implement PINNs. PINNs approach allows training neural networks while respecting the PDEs as a strong constraint in the optimization as apposed to making them part of the loss function. In standard small-scale circulation simulations, it is shown that the convent…
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 …
We investigate the use of Physics-Informed Neural Networks (PINNs) for solving the wave equation. Whilst PINNs have been successfully applied across many physical systems, the wave equation presents unique challenges due to the multi-scale, propagating and oscillatory nature of its solutions, and it is unclear how well they perform in this setting. We use a deep neural network to learn solutions of the wave equation, using the wave equation and a boundary condition as direct constraints in the loss function when training the network. We test the approach by solving the 2D acoustic wave equatio…
The concept of physics-informed neural networks has become a useful tool for solving differential equations due to its flexibility. There are a few approaches using this concept to solve the eikonal equation which describes the first-arrival traveltimes of acoustic and elastic waves in smooth heterogeneous velocity models. However, the challenge of the eikonal is exacerbated by the velocity models producing caustics, resulting in instabilities and deterioration of accuracy due to the non-smooth solution behaviour. In this paper, we revisit the problem of solving the eikonal equation using neur…
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）
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…
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）
Solving the wave equation numerically constitutes the majority of the computational cost for applications like seismic imaging and full waveform inversion. An alternative approach is to solve the frequency domain Helmholtz equation, since it offers a reduction in dimensionality as it can be solved per frequency. However, computational challenges with the classical Helmholtz solvers such as the need to invert a large stiffness matrix can make these approaches computationally infeasible for large 3D models or for modeling high frequencies. Moreover, these methods do not have a mechanism to trans…
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…
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 …
RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations
Department of Civil and Environmental Engineering, Hong Kong Polytechnic University（香港理工大学土木与环境工程系）；Department of Statistics and Data Science, Yale University（耶鲁大学统计与数据科学系）
Physics-informed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In particular, in addition to the deep neural network (DNN) for the solution, a second DNN is considered that represents the residual of the PDE. The residual is then combined with the mismatch in the given data of the solution in order to formulate the loss function. This framework is effective but is lacking uncertainty quantification of the solution due to the inher…
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…
Quantum-Classical Physics-Informed Neural Networks for Solving Reservoir Seepage Equations
School of Petroleum Engineering, Yangtze University（长江大学石油工程学院）；College of Future Technology, Yangtze University（长江大学未来技术学院）；State Key Laboratory of Low Carbon Catalysis and Carbon Dioxide Utilization (Yangtze University)（长江大学低碳催化与二氧化碳利用国家重点实验室）；King Abdullah University of Technology, Physical Science and Engineering Division（卡布斯大学物理科学与工程学院）；Western Research Institute, Yangtze University（长江大学西部研究院）
BridgeNet: A Hybrid, Physics-Informed Machine Learning Framework for Solving High-Dimensional Fokker-Planck Equations
Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, Tehran, Iran（塔里亚特莫达雷斯大学应用数学系）；Department of Computer and Data Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran（谢赫·法赫鲁丁大学计算机与数据科学系）；Institute for Cognitive and Brain Sciences, Shahid Beheshti University,Tehran, Iran（谢赫·法赫鲁丁大学认知与脑科学研究所）
