Multi-Fidelity Physics-Informed Neural Networks with Bayesian Uncertainty Quantification and Adaptive Residual Learning for Efficient Solution of Parametric Partial Differential Equations
Department of Applied Mathematics and Computer Science (DTU Compute)（应用数学与计算机科学系（DTU Compute））；Technical University of Denmark（丹麦技术大学）
Olaf Yunus Laitinen Imanov
备注：
The author has identified fundamental analytical errors in the formulation of the adaptive PDE residual loss and Bayesian uncertainty estimation pipeline that invalidate the numerical convergence results and core findings. The author is withdrawing the manuscript
Physics-informed neural networks (PINNs) have emerged as a powerful paradigm for solving partial differential equations (PDEs) by embedding physical laws directly into neural network training. However, solving high-fidelity PDEs remains computationally prohibitive, particularly for parametric systems requiring multiple evaluations across varying parameter configurations. This paper presents MF-BPINN, a novel multi-fidelity framework that synergistically combines physics-informed neural networks with Bayesian uncertainty quantification and adaptive residual learning. Our approach leverages abundant low-fidelity simulations alongside sparse high-fidelity data through a hierarchical neural architecture that learns nonlinear correlations across fidelity levels. We introduce an adaptive residual network with learnable gating mechanisms that dynamically balances linear and nonlinear fidelity discrepancies. Furthermore, we develop a rigorous Bayesian framework employing Hamiltonian Monte Carlo.
In this paper, we study a physics-informed algorithm for Wasserstein Generative Adversarial Networks (WGANs) for uncertainty quantification in solutions of partial differential equations. By using groupsort activation functions in adversarial network discriminators, network generators are utilized to learn the uncertainty in solutions of partial differential equations observed from the initial/boundary data. Under mild assumptions, we show that the generalization error of the computed generator converges to the approximation error of the network with high probability, when the number of sample…
Technical University of Munich（慕尼黑工业大学）；Pontificia Universidad Católica de Chile（智利天主教大学）；Harvard University（哈佛大学）
Physics-Informed Neural Networks (PINNs) are gaining popularity as a method for solving differential equations. While being more feasible in some contexts than the classical numerical techniques, PINNs still lack credibility. A remedy for that can be found in Uncertainty Quantification (UQ) which is just beginning to emerge in the context of PINNs. Assessing how well the trained PINN complies with imposed differential equation is the key to tackling uncertainty, yet there is lack of comprehensive methodology for this task. We propose a framework for UQ in Bayesian PINNs (B-PINNs) that incorpor…
Improved Uncertainty Quantification in Physics-Informed Neural Networks Using Error Bounds and Solution Bundles
Departamento de Ciencia de la Computación, Pontificia Universidad Católica de Chile（计算机科学系，天主教智利大学）；Department of Computer Science, University of Tübingen（计算机科学系，图宾根大学）；John A. Paulson School of Engineering and Applied Sciences, Harvard University（工程与应用科学学院，哈佛大学）
We present a deep learning framework for quantifying and propagating uncertainty in systems governed by non-linear differential equations using physics-informed neural networks. Specifically, we employ latent variable models to construct probabilistic representations for the system states, and put forth an adversarial inference procedure for training them on data, while constraining their predictions to satisfy given physical laws expressed by partial differential equations. Such physics-informed constraints provide a regularization mechanism for effectively training deep generative models as …
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…
We present a Parametrization of the Physics Informed Neural Network (P-PINN) approach to tackle the problem of uncertainty quantification in reservoir engineering problems. We demonstrate the approach with the immiscible two phase flow displacement (Buckley-Leverett problem) in heterogeneous porous medium. The reservoir properties (porosity, permeability) are treated as random variables. The distribution of these properties can affect dynamic properties such as the fluids saturation, front propagation speed or breakthrough time. We explore and use to our advantage the ability of networks to in…
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…
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）
Uncertainty quantification for noisy inputs-outputs in physics-informed neural networks and neural operators
Brown University（布朗大学）；Huazhong University of Science and Technology（华中科技大学）；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数学与应用科学交叉研究院）
Physics-informed neural networks have been widely applied to learn general parametric solutions of differential equations. Here, we propose a neural network to discover parametric eigenvalue and eigenfunction surfaces of quantum systems. We apply our method to solve the hydrogen molecular ion. This is an ab-initio deep learning method that solves the Schrodinger equation with the Coulomb potential yielding realistic wavefunctions that include a cusp at the ion positions. The neural solutions are continuous and differentiable functions of the interatomic distance and their derivatives are analy…
The placement of temperature sensitive and safety-critical components is crucial in the automotive industry. It is therefore inevitable, even at the design stage of new vehicles that these components are assessed for potential safety issues. However, with increasing number of design proposals, risk assessment quickly becomes expensive. We therefore present a parameterized surrogate model for the prediction of three-dimensional flow fields in aerothermal vehicle simulations. The proposed physics-informed neural network (PINN) design is aimed at learning families of flow solutions according to a…
Prediction of wind turbines power with physics-informed neural networks and evidential uncertainty quantification
University of Granada（格拉纳达大学）；Research Centre for Information and Communication Technologies (CITIC-UGR)（信息与通信技术研究中心（CITIC-UGR））；TECNALIA（泰克纳利亚研究中心）；Imperial College London（帝国理工学院）
Physics-informed neural operator for parametric phase-field modelling of interfacial degradation and microstructural evolution
Tongji University, College of Civil Engineering（同济大学，土木工程学院）；Tongji University, State Key Laboratory of Disaster Reduction in Civil Engineering（同济大学，灾害防治工程重点实验室）
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）
Brown University（布朗大学）；John Hopkins University（约翰斯·霍普金斯大学）；Air Force Research Laboratory（美国空军研究实验室）
We introduce an optimized physics-informed neural network (PINN) trained to solve the problem of identifying and characterizing a surface breaking crack in a metal plate. PINNs are neural networks that can combine data and physics in the learning process by adding the residuals of a system of Partial Differential Equations to the loss function. Our PINN is supervised with realistic ultrasonic surface acoustic wave data acquired at a frequency of 5 MHz. The ultrasonic surface wave data is represented as a surface deformation on the top surface of a metal plate, measured by using the method of l…
Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning
Oden Institute for Computational Sciences and Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算科学与工程研究所）；College of Computing, Georgia Institute of Technology（佐治亚理工学院计算学院）；Walker Department of Mechanical Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校沃克机械工程系）
LFR-PINO: A Layered Fourier Reduced Physics-Informed Neural Operator for Parametric PDEs
School of Aeronautics and Astronautics, Shanghai Jiao Tong University, Shanghai, China（上海交通大学航空宇航科学与技术学院）；School of Aeronautics and Astronautics, Zhejiang University, Hangzhou, China（浙江大学航空宇航科学与技术学院）；Key Laboratory for Satellite Digitalization Technology, Innovation Academy for Microsatellites of CAS, Shanghai, China（中国科学院卫星数字化技术重点实验室）
Physics-Informed Neural Networks for Methane Sorption: Cross-Gas Transfer Learning, Ensemble Collapse Under Physics Constraints, and Monte Carlo Dropout Uncertainty Quantification
Environmental Geosciences, Department of Geosciences, University of Oslo（奥斯陆大学环境地质学系）；National Key Laboratory of Petroleum Resources and Engineering, China University of Petroleum (Beijing)（中国石油大学（北京）石油资源与工程国家级重点实验室）；College of Geosciences, China University of Petroleum (Beijing)（中国石油大学（北京）地质学院）；State Key Laboratory of Coal Mine Disaster Prevention and Control, China University of Mining and Technology（中国矿业大学煤矿灾害防治控制国家重点实验室）；The Lyell Centre, Heriot-Watt University（赫瑞斯泰大学利尔中心）；Subsurface Energy Transition and Innovation Centre, Heriot-Watt University（赫瑞斯泰大学地下能源转型与创新中心）
Deterministic and statistical calibration of constitutive models from full-field data with parametric physics-informed neural networks
Institute of Applied Mechanics（应用力学研究所）；Technische Universität Braunschweig（Braunschweig技术大学）；Clausthal University of Technology（克莱斯特哈大学）；Institute for Acoustics and Dynamics（声学与动力学研究所）；Eidgenössische Technische Hochschule Zürich（苏黎世联邦理工学院）
Neural Spline Operators for Risk Quantification in Stochastic Systems
Department of Electrical and Computering Engineering, Carnegie Mellon University（电气与计算机工程系，卡内基梅隆大学）；School of Science and Engineering, Department of Mathematics and Computer Science, Duquesne University（科学与工程学院，数学与计算机科学系，杜克森大学）
本文提出 Neural Spline Operators（NeSO），用 B 样条增强的物理信息神经算子学习函数性动力学到风险概率的映射，并通过理论与两类实验验证其逼近能力和在线加速优势。
Gryphon Center for Artificial Intelligence and Theoretical Sciences（格里芬人工智能与理论科学中心）；Georgia Institute of Technology（佐治亚理工学院）；University of Pennsylvania（宾夕法尼亚大学）
We present a novel class of Physics-Informed Neural Networks that is formulated based on the principles of Evidential Deep Learning, where the model incorporates uncertainty quantification by learning parameters of a higher-order distribution. The dependent and trainable variables of the PDE residual loss and data-fitting loss terms are recast as functions of the hyperparameters of an evidential prior distribution. Our model is equipped with an information-theoretic regularizer that contains the Kullback-Leibler divergence between two inverse-gamma distributions characterizing predictive uncer…
Bayesian Physics Informed Neural Networks (B-PINNs) have gained significant attention for inferring physical parameters and learning the forward solutions for problems based on partial differential equations. However, the overparameterized nature of neural networks poses a computational challenge for high-dimensional posterior inference. Existing inference approaches, such as particle-based or variance inference methods, are either computationally expensive for high-dimensional posterior inference or provide unsatisfactory uncertainty estimates. In this paper, we present a new efficient infere…
King’s College London（伦敦国王学院）；St Thomas’ Hospital（圣托马斯医院）
The goal of this work is to train a neural network which approximates solutions to the Navier-Stokes equations across a region of parameter space, in which the parameters define physical properties such as domain shape and boundary conditions. The contributions of this work are threefold: 1) To demonstrate that neural networks can be efficient aggregators of whole families of parameteric solutions to physical problems, trained using data created with traditional, trusted numerical methods such as finite elements. Advantages include extremely fast evaluation of pressure and velocity at any poin…
Adaptive Physics-Informed Neural Networks with Multi-Category Feature Engineering for Hydrogen Sorption Prediction in Clays, Shales, and Coals
Faculty of Mathematics & Natural Sciences, University of Oslo（数学与自然科学学院，奥斯陆大学）；SINTEF Industry, Applied Geoscience Department（SINTEF工业公司，应用地球科学部门）；College of Geosciences, China University of Petroleum（地质科学学院，中国石油大学）
University of Tübingen（图宾根大学）；University of Genova（热那亚大学）；Istituto Italiano di Tecnologia（意大利技术研究院）；Massachusetts Institute of Technology（麻省理工学院）；Max-Planck-Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Neural operators are a type of deep architecture that learns to solve (i.e. learns the nonlinear solution operator of) partial differential equations (PDEs). The current state of the art for these models does not provide explicit uncertainty quantification. This is arguably even more of a problem for this kind of tasks than elsewhere in machine learning, because the dynamical systems typically described by PDEs often exhibit subtle, multiscale structure that makes errors hard to spot by humans. In this work, we first provide a mathematically detailed Bayesian formulation of the ''shallow'' (li…
Columbia University（哥伦比亚大学）；Data Science Institute, Columbia University（哥伦比亚大学数据科学研究所）
This paper proposes the TrafficFlowGAN, a physics-informed flow based generative adversarial network (GAN), for uncertainty quantification (UQ) of dynamical systems. TrafficFlowGAN adopts a normalizing flow model as the generator to explicitly estimate the data likelihood. This flow model is trained to maximize the data likelihood and to generate synthetic data that can fool a convolutional discriminator. We further regularize this training process using prior physics information, so-called physics-informed deep learning (PIDL). To the best of our knowledge, we are the first to propose an inte…
As applications of deep learning (DL) continue to seep into critical scientific use-cases, the importance of performing uncertainty quantification (UQ) with DL has become more pressing than ever before. In scientific applications, it is also important to inform the learning of DL models with knowledge of physics of the problem to produce physically consistent and generalized solutions. This is referred to as the emerging field of physics-informed deep learning (PIDL). We consider the problem of developing PIDL formulations that can also perform UQ. To this end, we propose a novel physics-infor…
Centre for Artificial Intelligence, University College London（人工智能研究中心，伦敦大学学院）；Heudiasyc Laboratory（Heudiasyc实验室）；Mila - Quebec AI Institute（魁北克人工智能研究所）
Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise, they typically require large training data-sets consisting of paired input-output observations which may be expensive to obtain, while their predictions may not be consistent with the underlying physical principles that generated the observed data. In this work, we propose a novel model class coined as physics-informed DeepONets, which introduces an effective r…
Motion-Robust T2* Quantification from Gradient Echo MRI with Physics-Informed Deep Learning
Institute of Machine Learning in Biomedical Imaging , Helmholtz Munich , Germany；School of Computation, Information \& Technology , Technical University of Munich , Germany；School of Medicine \& Health, Institute for Diagnostic；Interventional Neuroradiology , Technical University of Munich , Germany；School of Medicine \& Health, TUM-Neuroimaging Center , Technical University of Munich , Germany；Philips GmbH Market DACH , Germany；School of Medicine \& Health, Clinic of Neurology , Technical University of Munich , Germany；School of Biomedical Engineering \& Imaging Sciences , King’s College London , UK
Oden Institute for Computational Engineering & Sciences, The University of Texas at Austin（德克萨斯大学奥斯汀分校奥登计算工程与科学学院）；McKelvey School of Engineering, Washington University in St. Louis（圣路易斯华盛顿大学麦凯维工程学院）；Department of Mechanical Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校机械工程系）
Many-query problems, arising from uncertainty quantification, Bayesian inversion, Bayesian optimal experimental design, and optimization under uncertainty-require numerous evaluations of a parameter-to-output map. These evaluations become prohibitive if this parametric map is high-dimensional and involves expensive solution of partial differential equations (PDEs). To tackle this challenge, we propose to construct surrogates for high-dimensional PDE-governed parametric maps in the form of projected neural networks that parsimoniously capture the geometry and intrinsic low-dimensionality of the…
