We formulate a general framework for hp-variational physics-informed neural networks (hp-VPINNs) based on the nonlinear approximation of shallow and deep neural networks and hp-refinement via domain decomposition and projection onto space of high-order polynomials. The trial space is the space of neural network, which is defined globally over the whole computational domain, while the test space contains the piecewise polynomials. Specifically in this study, the hp-refinement corresponds to a global approximation with local learning algorithm that can efficiently localize the network parameter optimization. We demonstrate the advantages of hp-VPINNs in accuracy and training cost for several numerical examples of function approximation and solving differential equations.
INI-VPINN: A Variational Physics-Informed Neural Network with Implicit Neumann and Interface Handling for Multi-Material Domains with Geometric Singularities
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…
In this work we analyze how quadrature rules of different precisions and piecewise polynomial test functions of different degrees affect the convergence rate of Variational Physics Informed Neural Networks (VPINN) with respect to mesh refinement, while solving elliptic boundary-value problems. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions and highlight the i…
LRX-PINN: A Layer-Resolving XNet Physics-Informed Neural Network with Integrated Cauchy Activations for Convection-Dominated Problems
Institute for Advanced Research, Great Bay University（先进研究院，大湾大学）；College of Computer Science and Technology, Dongguan University of Technology（计算机科学与技术学院，东莞理工大学）；Department of Mathematics, Northwestern University（数学系，西北大学）
University of Naples Federico II（那不勒斯费德里科二世大学）；SISSA – International School for Advanced Studies（国际高等研究学院（SISSA））；University of Colorado Boulder（科罗拉多大学博尔德分校）
Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related adv…
Random test functions, $H^{-1}$ norm equivalence, and stochastic variational physics-informed neural networks
Mathematical Sciences Institute and France-Australia Mathematical Sciences and Interactions ANU-CNRS International Research Lab（数学科学研究所和法国-澳大利亚数学科学与互动 ANU-CNRS 国际研究实验室）
Defense Innovation Institute, Chinese Academy of Military Science（军事科学院国防创新研究院）
Physics Informed Neural Network (PINN) is a scientific computing framework used to solve both forward and inverse problems modeled by Partial Differential Equations (PDEs). This paper introduces IDRLnet, a Python toolbox for modeling and solving problems through PINN systematically. IDRLnet constructs the framework for a wide range of PINN algorithms and applications. It provides a structured way to incorporate geometric objects, data sources, artificial neural networks, loss metrics, and optimizers within Python. Furthermore, it provides functionality to solve noisy inverse problems, variatio…
CASUS - Center for Advanced System Understanding（先进系统理解中心）；Helmholtz-Zentrum Dresden-Rossendorf e.V. (HZDR)（亥姆霍兹德累斯顿罗森多夫研究中心）
We present a novel class of approximations for variational losses, being applicable for the training of physics-informed neural nets (PINNs). The loss formulation reflects classic Sobolev space theory for partial differential equations and their weak formulations. The loss computation rests on an extension of Gauss-Legendre cubatures, we term Sobolev cubatures, replacing automatic differentiation (A.D.). We prove the runtime complexity of training the resulting Soblev-PINNs (SC-PINNs) to be less than required by PINNs relying on A.D. On top of one-to-two order of magnitude speed-up the SC-PINN…
California Institute of Technology（加州理工学院）；Purdue University（普渡大学）
We introduce a scheme for probabilistic hypocenter inversion with Stein variational inference. Our approach uses a differentiable forward model in the form of a physics informed neural network, which we train to solve the Eikonal equation. This allows for rapid approximation of the posterior by iteratively optimizing a collection of particles against a kernelized Stein discrepancy. We show that the method is well-equipped to handle highly multimodal posterior distributions, which are common in hypocentral inverse problems. A suite of experiments is performed to examine the influence of the var…
Variational Quantum Physics-Informed Neural Networks for Hydrological PDE-Constrained Learning with Inherent Uncertainty Quantification
Faculty of Computer Science and Multimedia, Lincoln University College, Petaling Jaya, Selangor, Malaysia（计算机科学与多媒体学院，林肯大学学院，Petaling Jaya，Selangor，马来西亚）
Bauhaus Universität-Weimar（魏玛包豪斯大学）；University of Notre Dame（圣母大学）；Ton Duc Thang University（陈兴道大学）
We present a new physics informed neural network (PINN) algorithm for solving brittle fracture problems. While most of the PINN algorithms available in the literature minimize the residual of the governing partial differential equation, the proposed approach takes a different path by minimizing the variational energy of the system. Additionally, we modify the neural network output such that the boundary conditions associated with the problem are exactly satisfied. Compared to conventional residual based PINN, the proposed approach has two major advantages. First, the imposition of boundary con…
Bayesian Reasoning for Physics Informed Neural Networks
Institute for Theoretical Physics, University of Wroc aw（沃拉夫大学理论物理研究所）；Institute of Low Temperature and Structure Research（低温与结构研究所）；Polish Academy of Sciences（波兰科学院）
AQ-PINNs: Attention-Enhanced Quantum Physics-Informed Neural Networks for Carbon-Efficient Climate Modeling
SVKM’s Dwarkadas J. Sanghvi College of Engineering（SVKM德瓦卡达斯·J·桑格维工程学院）；New York University Abu Dhabi (NYUAD)（纽约大学阿布扎比分校）；NYUAD Research Institute（纽约大学阿布扎比分校研究院）；The Maersk Mc-Kinney Moller Institute（马士基麦克-金尼·莫勒研究所）；University of Southern Denmark（南丹麦大学）
PUNCH: Physics-informed Uncertainty-aware Network for Coronary Hemodynamics
AngioInsight Inc.（AngioInsight公司）；University of California, Los Angeles（加州大学洛杉矶分校）；University of Colorado Boulder（科罗拉多大学博尔德分校）；University of Michigan（密歇根大学）；Cardiovascular Research Foundation（心血管研究基金会）；University of California, Riverside（加州大学河滨分校）
Brown University（布朗大学）；Pacific Northwest National Laboratory（西北太平洋国家实验室）
We propose a Bayesian physics-informed neural network (B-PINN) to solve both forward and inverse nonlinear problems described by partial differential equations (PDEs) and noisy data. In this Bayesian framework, the Bayesian neural network (BNN) combined with a PINN for PDEs serves as the prior while the Hamiltonian Monte Carlo (HMC) or the variational inference (VI) could serve as an estimator of the posterior. B-PINNs make use of both physical laws and scattered noisy measurements to provide predictions and quantify the aleatoric uncertainty arising from the noisy data in the Bayesian framewo…
PiLocNet: Physics-informed neural network on 3D localization with rotating point spread function
Department of Mathematics, City University of Hong Kong（香港城市大学数学系）；Department of Statistics and Data Science, Southern University of Science and Technology（南方科技大学统计与数据科学系）；National Centre for Applied Mathematics Shenzhen（深圳应用数学中心）；School of Physics and Astronomy, University of Minnesota（明尼苏达大学物理与天文学学院）；Lingnan University（岭大大学）
Stochastic variational inference for Bayesian deep neural network (DNN) requires specifying priors and approximate posterior distributions over neural network weights. Specifying meaningful weight priors is a challenging problem, particularly for scaling variational inference to deeper architectures involving high dimensional weight space. We propose MOdel Priors with Empirical Bayes using DNN (MOPED) method to choose informed weight priors in Bayesian neural networks. We formulate a two-stage hierarchical modeling, first find the maximum likelihood estimates of weights with DNN, and then set …
