The Physics-Informed Neural Network (PINN) approach is a new and promising way to solve partial differential equations using deep learning. The $L^2$ Physics-Informed Loss is the de-facto standard in training Physics-Informed Neural Networks. In this paper, we challenge this common practice by investigating the relationship between the loss function and the approximation quality of the learned solution. In particular, we leverage the concept of stability in the literature of partial differential equation to study the asymptotic behavior of the learned solution as the loss approaches zero. With this concept, we study an important class of high-dimensional non-linear PDEs in optimal control, the Hamilton-Jacobi-Bellman(HJB) Equation, and prove that for general $L^p$ Physics-Informed Loss, a wide class of HJB equation is stable only if $p$ is sufficiently large. Therefore, the commonly used $L^2$ loss is not suitable for training PINN on those equations, while $L^{\infty}$ loss is a better choice. Based on the theoretical insight, we develop a novel PINN training algorithm to minimize the $L^{\infty}$ loss for HJB equations which is in a similar spirit to adversarial training. The effectiveness of the proposed algorithm is empirically demonstrated through experiments. Our code is released at https://github.com/LithiumDA/L_inf-PINN.
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…
The calibration of solid constitutive models with full-field experimental data is a long-standing challenge, especially in materials which undergo large deformation. In this paper, we propose a physics-informed deep-learning framework for the discovery of constitutive model parameterizations given full-field displacement data and global force-displacement data. Contrary to the majority of recent literature in this field, we work with the weak form of the governing equations rather than the strong form to impose physical constraints upon the neural network predictions. The approach presented in…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；NVIDIA（英伟达）
In this paper, we propose the Adaptive Physics-Informed Neural Networks (APINNs) for accurate and efficient simulation-free Bayesian parameter estimation via Markov-Chain Monte Carlo (MCMC). We specifically focus on a class of parameter estimation problems for which computing the likelihood function requires solving a PDE. The proposed method consists of: (1) constructing an offline PINN-UQ model as an approximation to the forward model; and (2) refining this approximate model on the fly using samples generated from the MCMC sampler. The proposed APINN method constantly refines this approximat…
Convex Physics Informed Neural Networks for the Monge-Ampère Optimal Transport Problem
Geneva School of Business Administration (HEG), University of Applied Sciences；Institute of Mathematics, Ecole Polytechnique F\'ed\'erale de Lausanne, 1015 Lausanne, Switzerland, Email
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…
Physics-informed neural network estimation of active material properties in time-dependent cardiac biomechanical models
Department of Mathematics and Scientific Computing, NAWI Graz, University of Graz, Austria（数学与科学计算系，格拉茨大学，奥地利）；BioTechMed-Graz, Austria（格拉茨生物技术医学研究所，奥地利）；MOX, Department of Mathematics, Politecnico di Milano, Milano, Italy（MOX，米兰理工学院数学系，意大利）；Gottfried Schatz Research Center: Division of Biophysics, Medical University of Graz, Austria（戈特弗里德·施茨研究中心：生物物理部，格拉茨医科大学，奥地利）
Understanding the thermal behavior of additive manufacturing (AM) processes is crucial for enhancing the quality control and enabling customized process design. Most purely physics-based computational models suffer from intensive computational costs and the need of calibrating unknown parameters, thus not suitable for online control and iterative design application. Data-driven models taking advantage of the latest developed computational tools can serve as a more efficient surrogate, but they are usually trained over a large amount of simulation data and often fail to effectively use small bu…
Is the neural tangent kernel of PINNs deep learning general partial differential equations always convergent ?
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）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
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…
Brown University（布朗大学）；Sandia National Laboratories（桑迪亚国家实验室）
Physics-informed neural networks (PINNs) are effective in solving inverse problems based on differential and integral equations with sparse, noisy, unstructured, and multi-fidelity data. PINNs incorporate all available information into a loss function, thus recasting the original problem into an optimization problem. In this paper, we extend PINNs to parameter and function inference for integral equations such as nonlocal Poisson and nonlocal turbulence models, and we refer to them as nonlocal PINNs (nPINNs). The contribution of the paper is three-fold. First, we propose a unified nonlocal ope…
Data-Driven Physics-Informed Neural Networks: A Digital Twin Perspective
Cho Chun Shik Graduate School of Mobility（赵春植移动研究生院）；Korea Advanced Institute of Science and Technology（韩国科学技术院）；Seoul National University（首尔大学）；Institute of Advanced Machines and Design（先进机械与设计研究所）；College of Information Sciences and Technology（信息科学与技术学院）；The Pennsylvania State University（宾夕法尼亚州立大学）；German Aerospace Center (DLR, Deutsches Zentrum für Luft- und Raumfahrt)（德国航空航天中心）
PIETRA: Physics-Informed Evidential Learning for Traversing Out-of-Distribution Terrain
Massachusetts Institute of Technology（麻省理工学院）；Mitsubishi Electric Research Laboratories (MERL)（三菱电机研究实验室）；George Mason University（乔治梅森大学）；DEVCOM Army Research Laboratory（DEVCOM陆军研究实验室）
Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources
Department of Mathematics, Spelman College（斯帕尔曼学院数学系）；Advanced Computing, Mathematics and Data Division, Pacific Northwest National Laboratory（太平洋西北国家实验室高级计算、数学与数据 division）
For many systems in science and engineering, the governing differential equation is either not known or known in an approximate sense. Analyses and design of such systems are governed by data collected from the field and/or laboratory experiments. This challenging scenario is further worsened when data-collection is expensive and time-consuming. To address this issue, this paper presents a novel multi-fidelity physics informed deep neural network (MF-PIDNN). The framework proposed is particularly suitable when the physics of the problem is known in an approximate sense (low-fidelity physics) a…
Physics-Informed Learning for Human Whole-Body Kinematics Prediction via Sparse IMUs
Artificial and Mechanical Intelligence, Italian Institute of Technology（人工智能与机械智能，意大利理工学院）；Department of Computer Science, The University of Manchester（计算机科学系，曼彻斯特大学）；Human-Robot Interfaces and Interaction, Italian Institute of Technology（人机接口与交互，意大利理工学院）
We introduce a computational efficient data-driven framework suitable for quantifying the uncertainty in physical parameters and model formulation of computer models, represented by differential equations. We construct physics-informed priors, which are multi-output GP priors that encode the model's structure in the covariance function. This is extended into a fully Bayesian framework that quantifies the uncertainty of physical parameters and model predictions. Since physical models often are imperfect descriptions of the real process, we allow the model to deviate from the observed data by co…
In typical machine learning tasks and applications, it is necessary to obtain or create large labeled datasets in order to to achieve high performance. Unfortunately, large labeled datasets are not always available and can be expensive to source, creating a bottleneck towards more widely applicable machine learning. The paradigm of weak supervision offers an alternative that allows for integration of domain-specific knowledge by enforcing constraints that a correct solution to the learning problem will obey over the output space. In this work, we explore the application of this paradigm to 2-D…
Because physics-based building models are difficult to obtain as each building is individual, there is an increasing interest in generating models suitable for building MPC directly from measurement data. Machine learning methods have been widely applied to this problem and validated mostly in simulation; there are, however, few studies on a direct comparison of different models or validation in real buildings to be found in the literature. Methods that are indeed validated in application often lead to computationally complex non-convex optimization problems. Here we compare physics-informed A…
FunDiff: Diffusion Models over Function Spaces for Physics-Informed Generative Modeling
Institute for Foundations of Data Science, Yale University（数据科学基础研究所，耶鲁大学）；Department of Statistics and Data Science, Yale University（统计与数据科学系，耶鲁大学）；Laboratoire Navier, École nationale des ponts et chaussées, Institut Polytechnique de Paris, University Gustave Eiffel, CNRS UMR 8205（纳维实验室，巴黎理工学院，格斯塔夫·埃菲尔大学，CNRS UMR 8205）；Department of Chemical and Environmental Engineering, Yale University（化学与环境工程系，耶鲁大学）
Physics-Informed Learning Using Hamiltonian Neural Networks with Output Error Noise Models
Eindhoven University of Technology（埃因霍温理工大学）；Eindhoven Artificial Intelligence Systems Institute（埃因霍温人工智能系统研究所）；Institute for Computer Science and Control（计算机科学与控制研究所）
Reservoir History Matching of the Norne field with generative exotic priors and a coupled Mixture of Experts -- Physics Informed Neural Operator Forward Model
Tuning arrays with rays: Physics-informed tuning of quantum dot charge states
National Institute of Standards and Technology（美国国家标准与技术研究院）；Intel Corporation（英特尔公司）；Joint Center for Quantum Information and Computer Science（量子信息与计算机科学联合中心）；University of Maryland（马里兰大学）
