Modeling Glucocorticoid-Induced Renin Regulation from Sparse Data Using Physics-Informed Neural Networks

Long-distance open-channel water diversion projects, such as China’s South-to-North Water Diversion Project, have significantly mitigated regional water supply-demand imbalances. However, the hydraulic behavior of open channels during water conveyance is highly complex, particularly under abnormal conditions like extreme weather or equipment failures, which can cause abrupt hydraulic changes, rapid water level rises, and even local overtopping or other safety hazards. Therefore, global, real-time, and accurate monitoring of open-channel hydraulics is essential to ensure the project's safe and efficient operation. Hydraulic characteristics of open channels are typically obtained through hydrological monitoring systems and numerical simulations. The reasonable placement and number of monitoring sections in a hydrological system are crucial for balancing monitoring accuracy and construction costs across the entire open channel. Numerical simulation accuracy and reliability depend on clear boundary conditions, precise Manning roughness coefficients, and other key parameters. However, these parameters can vary over time and are often difficult to determine in practical applications. Physics-Informed Neural Networks (PINNs) provide an effective solution to these challenges. This study develops a PINN model to predict the hydraulic characteristics of unsteady flow in open channels by integrating sparse hydrological data with physical laws. The study also examines how the number and placement of monitoring sections affect the accuracy of hydraulic predictions for the entire channel. Results demonstrate that PINNs can achieve high-precision hydraulic predictions along the channel using data from only three optimally placed monitoring sections, with average relative L2 errors below 0.5%. PINNs exhibit strong generalization across diverse boundary conditions, accurately predicting complex flow scenarios and demonstrating significantly higher noise resistance compared to traditional methods. Even with Gaussian noise levels of 10%, PINN predictions maintain relative L2 errors within 3%. Furthermore, PINNs show substantial potential for inverting key parameters such as the Manning roughness coefficient. PINNs offer an efficient and rapid approach to hydraulic predictions for long-distance water conveyance projects, aiding in the design and optimization of monitoring systems while minimizing the number of sensors, equipment, and costs.

The accurate modeling of the Earth gravity field and geoid is critical for geodesy, yet traditional methods face limitations in handling the growing complexity and heterogeneity of modern geodetic data. To address these challenges, this study proposes a physics-informed neural network (PINN) framework for high-precision geoid modeling. The PINN employs convolutional neural networks (CNNs) to extract multi-scale features from terrestrial and airborne gravity data, which are then processed by a multilayer perceptron (MLP) to establish an accurate mapping between these features and the disturbing potential. Physical constraints, including Laplace’s equation and differential equations governing gravity anomaly and gravity disturbance, are embedded into the loss function to enhance both accuracy and interpretability. The proposed method is applied to the Colorado 1 cm geoid experiment. Compared to GNSS/leveling data of the Geoid Slope Validation Survey 2017 (GSVS17), the PINN-derived geoid model achieves a standard deviation (STD) of 2.1 cm. This represents a 12.5%–27.6% improvement over traditional methods and purely data-driven networks (DDNs). The PINN exhibits strong generalization under sparse data conditions, achieving 28.5% higher accuracy than the DDN with only 500 samples. Furthermore, analysis of geoid slopes and physical constraint contributions demonstrates that PINN’s dual physical constraints effectively balance global characteristics and localized fidelity of the geoid. This study establishes the PINN as a robust, physically interpretable machine learning approach for geoid modeling, outperforming classical methods and offering a promising pathway for gravity field estimation in regions with sparse or heterogeneous data. By bridging purely data-driven machine learning with fundamental geodetic principles, this work paves the way for future advancements in physics-informed machine learning-based geodetic modeling.

1-D coupled surface flow and transport equations revisited via the physics-informed neural network approach

Thermal deformation is a critical factor affecting the precision of machine tools, requiring accurate thermal modeling to predict temperature fields and thermal parameters. Traditional approaches, such as the Finite Element Method (FEM), require well-defined boundary conditions, which are often unknown or difficult to measure in real machining environments. This paper explores the use of Physics-informed neural networks (PINNs) as an alternative method for solving steady-state heat conduction problems in two dimensions. PINNs integrate sparse sensor data with physical laws, enabling temperature field prediction and convection coefficient estimation without the need for fully specified boundary conditions. We evaluate five different PINN models, varying the balance between data-driven and physics-informed constraints. Results show that enforcing the heat equation alone yields high accuracy in temperature prediction, but accurate convection coefficient estimation requires explicit enforcement of convection conditions. While PINNs successfully infer missing parameters, their sensitivity to temperature gradients can impact accuracy. Additionally, the need for retraining PINNs when conditions change limits real-time applicability, making them more suitable for offline thermal analysis rather than adaptive modeling in dynamic environments.

Adaptive transfer learning for PINN

Assimilating field data is essential to calibrate rock properties in hydrogeological applications. While such workflows are challenging for physics‐based reservoir simulators, data‐driven Artificial Neural Networks (ANN) offer a potential solution by effectively managing noisy data to infer fluid pressure diffusion p ( x , t ). Nevertheless, the sparse observation wells in hydrogeology projects make training ANN models difficult. Therefore, we employ Physics‐Informed Neural Networks (PINN) to bridge this gap between sparse spatial data. We train both ANN and PINN on a synthetic p ( x , t ) data set within a low‐permeability porous medium containing a highly permeable material. Having an identical architecture, both models integrate direct observations of the target variable, while the PINN further optimizes the diffusion equation residual. By comparing the model output as a function of the number of observation wells, we find that the PINN outperforms the ANN when spatial observations are limited to a small number of wells. Adding noise to the training data further underscores the robustness of PINN against random measurement errors or ambient noise. Furthermore, we evaluate the effectiveness of PINN in representing a high‐permeability zone without observations inside the domain. The findings suggest that accuracy is maintained even with reduced thickness of the equivalent material, although this leads to an increase in computation time. Given the sparsity of observation wells in hydrogeological applications, PINNs potentially provide a more suitable machine learning workflow than purely data‐driven ANN for reservoir modeling.

Physics-informed neural networks in water and wastewater systems: a critical review.

Accurate and rapid prediction of structural responses to seismic loading is critical for ensuring structural safety. Recently, there has been active research focusing on the application of deep learning techniques, including Physics-Informed Neural Networks (PINNs) and Long Short-Term Memory (LSTM) networks, to predict the dynamic behavior of structures. While these methods have shown promise, each comes with distinct limitations. PINNs offer physical consistency but struggle with capturing long-term temporal dependencies in nonlinear systems, while LSTMs excel in learning sequential data but lack physical interpretability. To address these complementary limitations, this study proposes a hybrid LSTM-PINN model, combining the temporal learning ability of LSTMs with the physics-based constraints of PINNs. This hybrid approach allows the model to capture both nonlinear, time-dependent behaviors and maintain physical consistency. The proposed model is evaluated on both single-degree-of-freedom (SDOF) and multi-degree-of-freedom (MDOF) structural systems subjected to the El-Centro ground motion. For validation, the 1940 El-Centro NS earthquake record was used, and the ground acceleration data were normalized and discretized for numerical simulation. The proposed LSTM-PINN is trained under the same conditions as the conventional PINN models (e.g., same optimizer, learning rate, and loss structure), but with fewer training epochs, to evaluate learning efficiency. Prediction accuracy is quantitatively assessed using mean error and mean squared error (MSE) for displacement, velocity, and acceleration, and results are compared with PINN-only models (PINN-1, PINN-2). The results show that LSTM-PINN consistently achieves the most stable and precise predictions across the entire time domain. Notably, it outperforms the baseline PINNs even with fewer training epochs. Specifically, it achieved up to 50% lower MSE with only 10,000 epochs, compared to the PINN's 50,000 epochs, demonstrating improved generalization through temporal sequence learning. This study empirically validates the potential of physics-guided time-series AI models for dynamic structural response prediction. The proposed approach is expected to contribute to future applications such as real-time response estimation, structural health monitoring, and seismic performance evaluation.

Surrogate modeling of parameterized multi-dimensional premixed combustion with physics-informed neural networks for rapid exploration of design space

In this paper, we develop a deep learning approach for the accurate solution of challenging problems of near-field microscopy that leverages the powerful framework of physics-informed neural networks (PINNs) for the inversion of the complex optical parameters of nanostructured environments. Specifically, we show that PINNs can be flexibly designed based on full-vector Maxwell’s equations to inversely retrieve the spatial distributions of the complex electric permittivity and magnetic permeability of unknown scattering objects in the resonance regime from near-field data. Moreover, we demonstrate that PINNs achieve excellent convergence to the true material parameters under both plane wave and point source (localized) excitations, enabling parameter retrieval in scanning near-field optical microscopy. Our method is computationally efficient compared to traditional data-driven deep learning approaches as it requires only a single dataset for training. Furthermore, we develop and successfully demonstrate adaptive PINNs with trainable loss weights that largely improve the accuracy of the inverse reconstruction for high-index materials compared to standard PINNs. Finally, we demonstrate the full potential of our approach by retrieving the space-dependent permittivity of a three-dimensional unknown object from near-field data. The presented framework paves the way to the development of a computationally driven, accurate, and non-invasive platform for the simultaneous retrieval of the electric and magnetic parameters of resonant nanostructures from measured optical images, with applications to biomedical imaging, optical remote sensing, and characterization of metamaterial devices.

Traffic state estimation (TSE) is a critical component of the efficient intelligent transportation systems (ITS) operations. In the literature, TSE methods are divided into model-driven methods and data-driven methods. Each approach has its limitations. The physics information-based neural network (PINN) framework emerges to mitigate the limitations of the traditional TSE methods, while the state-of-art of such a framework has focused on single road segments but can hardly deal with traffic networks. This paper introduces a PINN framework that can effectively make use of a small amount of observational speed data to obtain high-quality TSEs for a traffic network. Both model-driven and data-driven components are incorporated into PINNs to combine the advantages of both approaches and to overcome their disadvantages. Simulation data of simple traffic networks are used for studying the highway network TSE. This paper demonstrates how to solve the popular LWR physical traffic flow model with a PINN for a traffic network. Experimental results confirm that the proposed approach is promising for estimating network traffic accurately.

A self-adaptive loss physics-informed neural network (SAL-PINN) method is proposed to address the challenges of limited generalization capability exhibited by purely data-driven neural networks under sparse data conditions and the insufficient accuracy of standard physics-informed neural networks in capturing complex flow details. This method enables high-fidelity reconstruction of the plane turbine cascade flow field with limited measurement points. By incorporating a dynamic weight balancing mechanism, SAL-PINN adaptively adjusts the loss function weights for regions characterized by different numerical scales within the flow field, thereby facilitating targeted reinforcement learning for critical regions. This approach yields excellent performance in predicting both global flow structures and local flow details. The results demonstrate that under sparse data conditions, SAL-PINN achieves a further reduction in prediction errors for key physical quantities across the entire field by approximately 1.5%–2.0% compared to the standard physics-informed neural networks. In boundary regions with complex flows, the error reduction is even more pronounced, reaching 12.0%–25.4%. This study also systematically evaluates the models' sensitivity to data volume, revealing that the incorporation of physical information significantly reduces their dependence on data, while the adaptive weighting mechanism further diminishes this sensitivity. This work provides an effective solution for achieving high-accuracy flow field reconstruction with minimal data, offering significant potential for engineering applications, particularly in scenarios where experimental measurements are high-cost.

Integrated Finite Element Neural Network (I-FENN) for non-local continuum damage mechanics

A nonlocal physics-informed deep learning framework using the peridynamic differential operator

Physics-Informed Neural Networks (PINNs) have emerged as a promising paradigm for solving partial differential equations (PDEs) by embedding physical laws into the learning process. However, standard PINNs often suffer from training instabilities and unbalanced optimization when handling multi-term loss functions, especially in problems involving singular perturbations, fractional operators, or multi-scale behaviors. To address these limitations, we propose a novel gradient variance weighting physics-informed neural network (GVW-PINN), which adaptively adjusts the loss weights based on the variance of gradient magnitudes during training. This mechanism balances the optimization dynamics across different loss terms, thereby enhancing both convergence stability and solution accuracy. We evaluate GVW-PINN on three representative PDE models and numerical experiments demonstrate that GVW-PINN consistently outperforms the conventional PINN in terms of training efficiency, loss convergence, and predictive accuracy. In particular, GVW-PINN achieves smoother and faster loss reduction, reduces relative errors by one to two orders of magnitude, and exhibits superior generalization to unseen domains. The proposed framework provides a robust and flexible strategy for applying PINNs to a wide range of integer- and fractional-order PDEs, highlighting its potential for advancing data-driven scientific computing in complex physical systems.