Advances in physics-informed deep learning for imaging data: a review of methods and applications

<p>Deep learning models have been widely utilized in various scientific and engineering problems; however, their application still faces practical challenges, including high data volume requirements, limited physical consistency, and insufficient interpretability. Physics-informed deep learning (PIDL) has emerged as a promising paradigm to address these challenges by incorporating physical laws into the training process of deep learning models. By integrating data-driven approaches with physics-based constraints, PIDL enhances the accuracy and reliability of deep learning models, making it a powerful tool across diverse fields. Numerous variants of PIDL models have been developed to cater to different applications. This review provides a comprehensive examination of recent advancements in PIDL, with a particular focus on its applications in geoenergy development. We discuss key methodologies underlying PIDL, including weighting strategies in loss functions, network architectures, derivative calculations, and various forms of physical equations. Furthermore, we summarize the three most common application scenarios of PIDL models, including solving partial differential equations (PDEs), surrogate modeling, and inverse modeling. A series of case studies highlighting PIDL’s role in geoenergy development are also presented. Finally, current challenges and future directions of PIDL in the geoenergy field are summarized. This review aims to serve as a foundational and valuable resource for researchers and practitioners newly entering this field, while also highlighting the potential of PIDL in advancing geoenergy development.</p>

Physics-informed deep learning for traffic state estimation based on the traffic flow model and computational graph method

Water transport in the unsaturated zone is an important part of the hydrological cycle and is the link between the atmosphere‒soil‐groundwater for material and energy transport. The accurate prediction of soil moisture (SM) is essential for the rational exploitation of water resources. Data‐driven deep learning methods are widely used in many fields; however, the lack of physical mechanisms limits their application in hydrological fields, especially for SM prediction in unsaturated zones. To solve this problem, this study proposes a new deep learning method that introduces the water balance principle, Richard's equation, and SM boundary conditions as constraints to construct the new loss function that guides the training process of deep learning, called physics‐informed deep learning (PIDL). In tests consisting of a large number of data sets acquired from in situ observation sites in the field, PIDL exhibits higher accuracy than ordinary deep learning (long short‐term memory) and physical models, with 51.03% and 53.46% reduction in root mean square error of SM prediction, respectively. PIDL performance significantly improved in predicting scenarios that are difficult for ordinary deep learning to handle, such as sparse data sets, extreme values, and mutated values. In addition, PIDL maintains high accuracy over a longer prediction period. The addition of physical mechanisms allows deep learning to mine patterns not only from the data itself but also from a priori physical theoretical knowledge for guidance, and this hybrid modeling approach can also be generalized to prediction problems in other hydrological domains.

The reliability of rotating machinery is essential in industrial environments, where early fault detection can prevent significant losses. In this scenario, Condition-Based Maintenance (CBM) strategies benefit from combined signal processing and machine learning techniques. Although Deep Learning (DL) based models present good results in automatic fault classification, their exclusive dependence on data can generate inconsistent predictions and non-compliance with physics. To overcome this limitation, this work proposes an approach that uses physical principles based on Physics-Informed Deep Learning (PIDL) for fault classification in bearings, using vibration data obtained in an experimental bench. The dataset covers three operating conditions — healthy, light damage and heavy damage — with vibration signals captured by two sensors, one always healthy and the other with or without damage. The methodology involves the application of the Hilbert transform (HHT) for envelope analysis and defining amplitude thresholds that reflect the physical levels of degradation. These thresholds are incorporated into the learning process, guiding the classification and promoting greater interpretability and robustness. Initial results show that, in the multiclass classification problem, traditional DL outperformed PIDL, achieving a balanced accuracy of 97.3% compared to 84.78% for PIDL. However, in the binary classification scenario — distinguishing between healthy and unhealthy conditions — the PIDL model achieved performance comparable to DL's, with balanced accuracies of 95.17% and 95.71%, respectively. These findings highlight that incorporating physical constraints into DL models can enhance the robustness of predictions, particularly in simpler classification contexts.

Proton therapy requires accurate dose calculation for treatment planning to ensure the conformal doses are precisely delivered to the targets. The conversion of CT numbers to material properties is a significant source of uncertainty for dose calculation. The aim of this study is to develop a physics-informed deep learning (PIDL) framework to derive accurate mass density and relative stopping power maps from dual-energy computed tomography (DECT) images. The PIDL framework allows deep learning (DL) models to be trained with a physics loss function, which includes a physics model to constrain DL models. Five DL models were implemented including a fully connected neural network (FCNN), dual-FCNN (DFCNN), and three variants of residual networks (ResNet): ResNet-v1 (RN-v1), ResNet-v2 (RN-v2), and dual-ResNet-v2 (DRN-v2). An artificial neural network (ANN) and the five DL models trained with and without physics loss were explored to evaluate the PIDL framework. Two empirical DECT models were implemented to compare with the PIDL method. DL training data were from CIRS electron density phantom 062M (Computerized Imaging Reference Systems, Inc., Norfolk, VA). The performance of DL models was tested by CIRS adult male, adult female, and 5-year-old child anthropomorphic phantoms. For density map inference, the physics-informed RN-v2 was 3.3%, 2.9% and 1.9% more accurate than ANN for the adult male, adult female, and child phantoms. The physics-informed DRN-v2 was 0.7%, 0.6%, and 0.8% more accurate than DRN-v2 without physics training for the three phantoms, respectfully. The results indicated that physics-informed training could reduce uncertainty when ANN/DL models without physics training were insufficient to capture data structures or derived significant errors. DL models could also achieve better image noise control compared to the empirical DECT parametric mapping methods. The proposed PIDL framework can potentially improve proton range uncertainty by offering accurate material properties conversion from DECT.

We present a physics-informed deep learning (PIDL) approach to tackle the challenge of data sparsity and sensor noise in traffic state estimation (TSE). PIDL strengthens a deep learning (DL) neural network with the knowledge of traffic flow theory to accurately estimate traffic conditions. The ‘physics’—a priori information of the system—acts as a regularization agent during training. We illustrate the implementation of the proposed approach with two commonly used models representing traffic physics: Lighthill-Whitham-Richards (LWR) model and the cell transmission model (CTM). The LWR implementation is illustrated with Greenshields’ and inverse-lambda fundamental diagrams; whereas, CTM model implementation works with any fundamental diagram of choice. Two case studies validate the approach by reconstructing the velocity-field. Case study-I uses synthetic data generated to resemble the trajectory of connected and autonomous vehicles as captured by roadside units. Case study-II employs NGSIM data mimicking scant probe vehicle observations. We observe that the proposed PIDL approach is particularly better in state estimation with a lower amount of training data, illustrating the capability of PIDL in making precise and timely TSE even with sparse input. E.g., With 10% CAV penetration rate and a 15% added-noise, relative error for PIDL was at 22.9% compared to 30.8% for DL.

For its robust predictive power (compared to pure physics-based models) and sample-efficient training (compared to pure deep learning models), physics-informed deep learning (PIDL), a paradigm hybridizing physics-based models and deep neural networks (DNNs), has been booming in science and engineering fields. One key challenge of applying PIDL to various domains and problems lies in the design of a computational graph that integrates physics and DNNs. In other words, how the physics is encoded into DNNs and how the physics and data components are represented. In this paper, we offer an overview of a variety of architecture designs of PIDL computational graphs and how these structures are customized to traffic state estimation (TSE), a central problem in transportation engineering. When observation data, problem type, and goal vary, we demonstrate potential architectures of PIDL computational graphs and compare these variants using the same real-world dataset.

A robust and flexible satellite aerosol retrieval algorithm for multi-angle polarimetric measurements with physics-informed deep learning method

Toward improved lumped groundwater level predictions at catchment scale: Mutual integration of water balance mechanism and deep learning method

A physics-informed deep learning paradigm for car-following models

Data‐driven scientific discovery methods have been developed and applied to discover governing equations from data, involving the attempt to discover the unsaturated flow equation in soils from data. However, an important but unresolved problem is how to reconstruct the unsaturated flow equation from highly noisy and scarce discrete data. In this study, we present a new deep‐learning framework: DeepGS (deep‐learning‐based group sparsity framework), that leverages the synergy of group sparsity and physics‐informed deep learning (PIDL) to reconstruct the latent governing equation for unsaturated flow. In particular, we design a strategy that decomposes the identification of the unsaturated flow equation into two tasks: the determination of the partial differential equation structure and the reconstruction of the nonlinear coefficients. The tasks can be seamlessly handled by group sparse regression and the PIDL approach. Through the training, it realizes the simultaneous reconstruction of soil moisture dynamics and unsaturated flow governing equation. A series of comprehensive numerical experiments are conducted to determine the optimal architecture and test its performance. The results show the efficacy and robustness of DeepGS, which significantly outperform previous methods. We also conclude that accurately reconstructing soil moisture dynamics and spatiotemporal derivatives from noisy and scarce data play a critical role in governing equation discovery. This study further demonstrates the potential of discovering the governing equation for unsaturated flow from data in more complex scenarios, where rich and accurate soil moisture observations are generally intractable to access.

Since its introduction in 2017, physics-informed deep learning (PIDL) has garnered growing popularity in understanding the systems governed by physical laws in terms of partial differential equations (PDEs). However, empirical evidence points to the limitations of PIDL for learning certain types of PDEs. In this paper, we (a) present the challenges in training PIDL architecture, (b) contrast the performance of PIDL architecture in learning a first order scalar hyperbolic conservation law and its parabolic counterpart, (c) investigate the effect of training data sampling, which corresponds to various sensing scenarios in traffic networks, (d) comment on the implications of PIDL limitations for traffic flow estimation and prediction in practice. Case studies present the contrast in PIDL results between learning the traffic flow model (LWR PDE) and its diffusive variation. The outcome indicates that PIDL experiences significant challenges in learning the hyperbolic LWR equation due to the non-smoothness of its solution. Conversely, the architecture with parabolic PDE, augmented with the diffusion term, leads to the successful reassembly of the density data even with the shockwaves present. The paper concludes by providing a discussion on recent assessments of reasons behind the challenge PIDL encounters with hyperbolic PDEs and the corresponding mitigation strategies.

The increasing integration of renewable energy sources and the growing complexity of modern power grids necessitate the development of intelligent and adaptive control strategies for power electronic converters. Traditional control techniques often fail to provide the required flexibility and real-time adaptability to dynamic grid conditions. Physics-Informed Deep Learning (PIDL) has emerged as a transformative approach that combines deep learning with fundamental physical principles to enhance the efficiency, stability, and reliability of power converter control in IoT-integrated smart grids. By embedding domain knowledge into data-driven models, PIDL ensures physically consistent and generalizable solutions while optimizing real-time decision-making processes. This chapter explores the role of PIDL in adaptive power converter control, highlighting its advantages over conventional control methods. The integration of IoT and edge computing technologies enables real-time monitoring, decentralized decision-making, and predictive analytics, enhancing the resilience and efficiency of smart grids. The computational challenges associated with real-time implementation, including data scarcity, model complexity, and processing latency, are addressed through optimization techniques such as federated learning, hardware acceleration, and lightweight neural network architectures. Case studies on renewable energy grids demonstrate the effectiveness of PIDL-driven adaptive control strategies in managing fluctuations in generation and demand while improving grid reliability. The chapter further discusses future research directions in PIDL for smart grids, emphasizing the potential of quantum computing, neuromorphic processors, and hybrid AI models for enhancing computational efficiency and scalability. The convergence of physics-informed AI, IoT-driven adaptive control, and decentralized intelligence was expected to play a pivotal role in the evolution of next-generation power systems. This research contributes to the advancement of sustainable energy management by providing a framework for the deployment of intelligent, self-optimizing power converter control mechanisms in modern smart grids.

The multi‐spectral satellite sensors such as MODIS have a large swath, high spatial resolution, and well onboard calibration, enabling aerosol retrievals with daily global coverage. Despite numerous available bands, MODIS aerosol algorithms over land typically only utilize measurements from 2 to 3 spectral wavelengths to retrieve Aerosol Optical Depth (AOD) based on prescribed aerosol models. To make full use of multi‐spectral measurements and prior information, we developed an aerosol algorithm based on physics‐informed deep learning (PDL) approach. With physical constraint from radiative transfer simulation, PDL can construct model functions between the whole spectral measurements and each retrieved aerosol parameter separately. AERONET validations in eastern China show that MODIS PDL algorithm can accurately retrieve AOD and fine AOD (R = 0.936) at 1 km resolution and has reliable performance in coarse AOD as well as notable sensitivity to aerosol absorption. The flexible and efficient PDL method provides a generalized algorithm for common multi‐spectral satellite measurements.

Removing optical and atmospheric blur from galaxy images significantly improves galaxy shape measurements for weak gravitational lensing and galaxy evolution studies. This ill-posed linear inverse problem is usually solved with deconvolution algorithms enhanced by regularisation priors or deep learning. We introduce a so-called ’physics-informed deep learning’ approach to the Point Spread Function (PSF) deconvolution problem in galaxy surveys. We apply algorithm unrolling and the Plug-and-Play technique to the Alternating Direction Method of Multipliers (ADMM), in which a neural network learns appropriate hyperparameters and denoising priors from simulated galaxy images. We characterize the time-performance trade-off of several methods for galaxies of differing brightness levels, as well as our method’s robustness to systematic PSF errors and network ablations. We show an improvement in reduced shear ellipticity error of 38.6 per cent (SNR=20)/45.0 per cent (SNR=200) compared to classic methods and 7.4 per cent (SNR=20)/33.2 per cent (SNR = 200) compared to modern methods (https://github.com/Lukeli0425/Galaxy-Deconv).