Inverse problems arise across scientific and engineering domains, where the goal is to infer hidden parameters or physical fields from indirect and noisy observations. Classical approaches, such as variational regularization and Bayesian inference, provide well established theoretical foundations for handling ill posedness. However, these methods often become computationally restrictive in high dimensional settings or when the forward model is governed by complex physics. Physics Informed Neural Networks (PINNs) have recently emerged as a promising framework for solving inverse problems by embedding physical laws directly into the training process of neural networks. In this paper, we introduce a new perspective on the Bayesian Physics Informed Neural Network (BPINN) framework, extending classical PINNs by explicitly incorporating training data generation, modeling and measurement uncertainties through Bayesian prior modeling and doing inference with the posterior laws. Also, as we focus on the inverse problems, we call this method BPINN-IP, and we show that the standard PINN formulation naturally appears as its special case corresponding to the Maximum A Posteriori (MAP) estimate. This unified formulation allows simultaneous exploitation of physical constraints, prior knowledge, and data-driven inference, while enabling uncertainty quantification through posterior distributions. To demonstrate the effectiveness of the proposed framework, we consider inverse problems arising in infrared image processing, including deconvolution and super-resolution, and present results on both simulated and real industrial data.
Bayesian PINNs for uncertainty-aware inverse problems (BPINN-IP)
ISCT, Bures-sur-Yvette, France（法国ISCT）；Institute of Digital Twin (IDT), EIT, Ningbo, China（中国宁波EIT数字孪生研究所）；Dept. of Statistics, Central South University, Changcha, China（中国长沙中央南大学统计系）
Enhanced BPINN Training Convergence in Solving General and Multi-scale Elliptic PDEs with Noise
University of California at Berkeley, CA, USA（加州大学伯克利分校）；Ceyear Technologies Co., Ltd, Qingdao 266555, China（青岛凯年科技有限公司）；Shandong University, Qingdao 266237, China（山东大学）；The University of Queensland, St Lucia 4072, Australia（昆士兰大学）；Guangdong University of Finance and Economics, Guangzhou 510320, China（广东财经大学）
Drug Release Modeling using Physics-Informed Neural Networks
Division of Applied Mathematics, Brown University（应用数学系，布朗大学）；School of Engineering, Brown University（工程学院，布朗大学）；Institute for Biology, Engineering and Medicine, Brown University（生物、工程与医学研究所，布朗大学）
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…
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）
Source separation problems are ubiquitous in the physical sciences; any situation where signals are superimposed calls for source separation to estimate the original signals. In this tutorial I will discuss the Bayesian approach to the source separation problem. This approach has a specific advantage in that it requires the designer to explicitly describe the signal model in addition to any other information or assumptions that go into the problem description. This leads naturally to the idea of informed source separation, where the algorithm design incorporates relevant information about the …
Idaho National Laboratory（爱达荷国家实验室）；Johns Hopkins University（约翰斯·霍普金斯大学）
Although the no-u-turn sampler (NUTS) is a widely adopted method for performing Bayesian inference, it requires numerous posterior gradients which can be expensive to compute in practice. Recently, there has been a significant interest in physics-based machine learning of dynamical (or Hamiltonian) systems and Hamiltonian neural networks (HNNs) is a noteworthy architecture. But these types of architectures have not been applied to solve Bayesian inference problems efficiently. We propose the use of HNNs for performing Bayesian inference efficiently without requiring numerous posterior gradient…
A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning
Université Jean Monnet Saint-Étienne, CNRS, Institut d’Optique Graduate School, Laboratoire Hubert Curien UMR 5516（里昂蒙特大学圣埃蒂安分校、法国国家科学研究中心、光学研究生院、Hubert Curien实验室 UMR 5516）；Inria and University College London, France and United Kingdom（Inria 和 英国伦敦大学学院，法国和英国）
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…
Lund University（隆德大学）；University of Freiburg（弗赖堡大学）；Bosch Center for Artificial Intelligence（博世人工智能中心）；Stanford University（斯坦福大学）；DBtune（DBtune公司）
Information-theoretic Bayesian optimization techniques have become popular for optimizing expensive-to-evaluate black-box functions due to their non-myopic qualities. Entropy Search and Predictive Entropy Search both consider the entropy over the optimum in the input space, while the recent Max-value Entropy Search considers the entropy over the optimal value in the output space. We propose Joint Entropy Search (JES), a novel information-theoretic acquisition function that considers an entirely new quantity, namely the entropy over the joint optimal probability density over both input and outp…
Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）；Microsoft Research Cambridge（微软研究院剑桥实验室）
Computer vision is hard because of a large variability in lighting, shape, and texture; in addition the image signal is non-additive due to occlusion. Generative models promised to account for this variability by accurately modelling the image formation process as a function of latent variables with prior beliefs. Bayesian posterior inference could then, in principle, explain the observation. While intuitively appealing, generative models for computer vision have largely failed to deliver on that promise due to the difficulty of posterior inference. As a result the community has favoured effic…
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…
Proxy-informed Bayesian transfer learning with unknown sources
Department of Computer Science University of Manchester（曼彻斯特大学计算机科学系）；Inserm Bordeaux Population Health, Vaccine Research Institute, Université de Bordeaux, Inria Bordeaux Sud-ouest（法国波尔多大学疫苗研究学院）；Department of Computer Science Aalto University（埃斯波芬兰阿尔托大学计算机科学系）；University of Manchester（曼彻斯特大学）；Aalto University（阿尔托大学）；ELLIS Institute Finland（芬兰ELLIS研究所）
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）
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…
University of Potsdam（波茨坦大学）；Institute of Mathematics（数学研究所）；Institute of Pharmacy（药剂学研究所）
Model-informed precision dosing (MIPD) using therapeutic drug/biomarker monitoring offers the opportunity to significantly improve the efficacy and safety of drug therapies. Current strategies comprise model-informed dosing tables or are based on maximum a-posteriori estimates. These approaches, however, lack a quantification of uncertainty and/or consider only part of the available patient-specific information. We propose three novel approaches for MIPD employing Bayesian data assimilation (DA) and/or reinforcement learning (RL) to control neutropenia, the major dose-limiting side effect in a…
Uncertainty-Aware Prediction of Lung Tumor Growth from Sparse Longitudinal CT Data via Bayesian Physics-Informed Neural Networks
Department of Mathematics, Vanderbilt University（范德比大学数学系）；John A. Paulson School of Engineering and Applied Science, Harvard University（哈佛大学约翰·A·保罗森工程与应用科学学院）
Existing Bayesian spatial priors for functional magnetic resonance imaging (fMRI) data correspond to stationary isotropic smoothing filters that may oversmooth at anatomical boundaries. We propose two anatomically informed Bayesian spatial models for fMRI data with local smoothing in each voxel based on a tensor field estimated from a T1-weighted anatomical image. We show that our anatomically informed Bayesian spatial models results in posterior probability maps that follow the anatomical structure.
Coupling-Informed Transport Maps for Bayesian Filtering in Nonlinear Dynamical Systems
School of Mathematical Sciences, Tongji University（同济大学数学科学学院）；Key Laboratory of Intelligent Computing and Applications (Ministry of Education), Tongji University（智能计算与应用重点实验室（教育部））
Harvard University（哈佛大学）；Vector Institute for Artificial Intelligence（矢量人工智能研究所）；University of Toronto（多伦多大学）；Atinary Technologies Sàrl（Atinary技术有限公司）
Designing functional molecules and advanced materials requires complex design choices: tuning continuous process parameters such as temperatures or flow rates, while simultaneously selecting catalysts or solvents. To date, the development of data-driven experiment planning strategies for autonomous experimentation has largely focused on continuous process parameters despite the urge to devise efficient strategies for the selection of categorical variables. Here, we introduce Gryffin, a general purpose optimization framework for the autonomous selection of categorical variables driven by expert…
Across the empirical sciences, few statistical procedures rival the popularity of the frequentist t-test. In contrast, the Bayesian versions of the t-test have languished in obscurity. In recent years, however, the theoretical and practical advantages of the Bayesian t-test have become increasingly apparent and various Bayesian t-tests have been proposed, both objective ones (based on general desiderata) and subjective ones (based on expert knowledge). Here we propose a flexible t-prior for standardized effect size that allows computation of the Bayes factor by evaluating a single numerical in…
Hamburg University of Technology（汉堡工业大学）；Stanford University（斯坦福大学）；Brown University（布朗大学）
Understanding real-world dynamical phenomena remains a challenging task. Across various scientific disciplines, machine learning has advanced as the go-to technology to analyze nonlinear dynamical systems, identify patterns in big data, and make decision around them. Neural networks are now consistently used as universal function approximators for data with underlying mechanisms that are incompletely understood or exceedingly complex. However, neural networks alone ignore the fundamental laws of physics and often fail to make plausible predictions. Here we integrate data, physics, and uncertai…
Identifying a low-dimensional informed parameter subspace offers a viable path to alleviating the dimensionality challenge in the sampled-based solution to large-scale Bayesian inverse problems. This paper introduces a novel gradient-based dimension reduction method in which the informed subspace does not depend on the data. This permits an online-offline computational strategy where the expensive low-dimensional structure of the problem is detected in an offline phase, meaning before observing the data. This strategy is particularly relevant for multiple inversion problems as the same informe…
We introduce a framework for efficient Markov Chain Monte Carlo (MCMC) algorithms targeting discrete-valued high-dimensional distributions, such as posterior distributions in Bayesian variable selection (BVS) problems. We show that many recently introduced algorithms, such as the locally informed sampler and the Adaptively Scaled Individual adaptation sampler (ASI), can be viewed as particular cases within the framework. We then describe a novel algorithm, the Adaptive Random Neighbourhood Informed sampler (ARNI), by combining ideas from both of these existing approaches. We show using several…
Bayesian Physics Informed Neural Networks for Reliable Transformer Prognostics
Department of Electronics & Computing, Mondragon University（蒙龙大学电子与计算系）；Department of Physical Chemistry, University of the Basque Country (UPV/EHU)（巴斯克国家大学物理化学系）；Department of Mathematics, University of the Basque Country (UPV/EHU)（巴斯克国家大学数学系）；Department of Computer Science and Artificial Intelligence, University of the Basque Country (UPV/EHU)（巴斯克国家大学计算机科学与人工智能系）；Basque Centre for Applied Mathematics (BCAM)（巴斯克应用数学中心）；Ikerbasque, Basque Foundation for Science（巴斯克科学基金会）
Physics-Informed Neural Networks with Unknown Partial Differential Equations: an Application in Multivariate Time Series
School of Automotive and Traffic Engineering, Jiangsu University（江苏大学汽车与交通工程学院）；International Science Consulting and Training (ISCT)（国际科学咨询与培训中心）；Shanfeng Company（杉峰公司）
Microorganisms play a critical role in host health. The advancement of high-throughput sequencing technology provides opportunities for a deeper understanding of microbial interactions. However, due to the limitations of 16S ribosomal RNA sequencing, microbiome data are zero-inflated, and a quantitative comparison of microbial abundances cannot be made across subjects. By leveraging a recent microbiome profiling technique that quantifies 16S ribosomal RNA microbial counts, we propose a novel Bayesian graphical model that incorporates microorganisms' evolutionary history through a phylogenetic …
Arterial Spin Labelling (ASL) functional Magnetic Resonance Imaging (fMRI) data provides a quantitative measure of blood perfusion, that can be correlated to neuronal activation. In contrast to BOLD measure, it is a direct measure of cerebral blood flow. However, ASL data has a lower SNR and resolution so that the recovery of the perfusion response of interest suffers from the contamination by a stronger hemodynamic component in the ASL signal. In this work we consider a model of both hemodynamic and perfusion components within the ASL signal. A physiological link between these two components …
Disentangling Aleatoric and Epistemic Uncertainty in Physics-Informed Neural Networks. Application to Insulation Material Degradation Prognostics
Department of Electronics and Computer Science（电子与计算机科学系）；Mondragon University（蒙得龙大学）；Research & Development（研发）；CARTIF Technology Centre（CARTIF技术中心）；Department of Physical Chemistry（物理化学系）；University of the Basque Country (EHU)（巴斯克国家大学（EHU））；Basque Centre for Applied Mathematics (BCAM)（巴斯克应用数学中心（BCAM））；Department of Computer Science and Artificial Intelligence（计算机科学与人工智能系）
Damper-B-PINN: Damper Characteristics-Based Bayesian Physics-Informed Neural Network for Vehicle State Estimation
School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai 200240, China（上海交通大学自动化与智能感知学院）；Department of Civil, Architectural, and Environmental Engineering, The University of Texas at Austin（德克萨斯大学奥斯汀分校土木、建筑与环境工程系）；College of Electrical Engineering, Zhejiang University（浙江大学电气工程学院）；School of Automotive Studies, Tongji University（同济大学汽车学院）；Key Laboratory of Road and Traffic Engineering of Ministry of Education, Tongji University（同济大学交通工程教育部长三角交通工程重点实验室）；School of Architecture, The University of Texas at Austin（德克萨斯大学奥斯汀分校建筑系）
