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

The exploration of deep learning methodologies has recently generated significant interest in the use of Physics-Informed Neural Networks (PINNs) to address complex physical problems governed by partial differential equations (PDEs). The PINN is trained using information from physical laws, including governing PDEs, boundary conditions, and initial conditions. However, achieving a well-trained PINN typically necessitates an appropriate balance between the weights of each loss function, which can considerably increase manual effort. This paper introduces a novel training approach that integrates non-dimensionalization with a multi-task learning technique, termed Automatic Weighted Loss (AWL), to autonomously achieve an optimal balance for each loss function. In the baseline PINN training for solving time-dependent PDEs, multiple weights (usually more than six) must be manually tuned for the model, whereas this method can reduce the number of scaling weights to only one. The proposed approach, referred to as the Non-dimensionalization Automatic Weighted Loss (NDAWL), is evaluated through six free-surface flow problems modelled by the Shallow Water Equations (SWEs). Furthermore, a comparative analysis is conducted between the solutions obtained using NDAWL-PINN and those from the original PINN, which relies on manual fine-tuning of loss functions. The numerical results indicate that the NDAWL-PINN method achieves comparable or superior accuracy to the original PINN, demonstrating its effectiveness in automating the balancing of loss functions.

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 real-time forecasting of flood dynamics is a long-standing challenge traditionally addressed through numerical solutions of the Shallow Water Equations (SWEs). Numerical solutions of realistic flow problems using numerical schemes are often hindered by high computational costs, particularly due to the need for fine spatial and temporal discretization, complex boundary conditions, and the resolution of non-linearities inherent in the governing equations. In this study, we investigate the use of Physics-Informed Neural Networks (PINNs) to solve 1D and 2D SWEs in dam-break scenarios. The proposed PINN framework incorporates the governing partial differential equations along with the initial and boundary conditions directly within the training process of the network, ensuring physically consistent solutions. We conduct a systematic comparison of the solutions of SWE using the classical numerical scheme (Lax-Wendroff) with estimates of physics informed neural networks. For 1D SWE, a neural network is trained and validated on a dam-break problem, revealing that physics-informed models produce smoother but still acceptable estimates of wave propagation compared to standard numerical results. For 2D SWE, we consider various configurations of dam geometries along with varying initial profiles for water heights. Across all scenarios, reproduce the numerical baselines, albeit with limited accuracy, while avoiding spurious oscillations and numerical artifacts. Further tuning, achieved by incorporating numerical solutions into the PINN training, improved accuracy. This proof of concept demonstrates the potential of hybridized PINNs as a mesh-free, scalable, and generalizable framework for approximating solutions to nonlinear hyperbolic systems. Our results indicate that pre-trained, physics-informed models could serve as a viable alternative for real-time flood forecasting in complex domains.

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.

A Physics-Informed Neural Network (PINN) was developed to solve the complex Navier-Stokes (N-S) equations for multiphase flow measurement including flowrates of individual phases in the multiphase mixture, and phase ratios. The results were compared against a mathematical based multiphase flow measurement approach and validated through a controlled loop test using oil, water, and gas. The PINN embeds fundamental fluid dynamics principles into its learning process, ensuring that the neural network adheres to the governing physical laws of motion, continuity, and conservation. Unlike traditional neural networks that rely solely on data-driven learning, PINNs incorporate fluid dynamics driven differential equations, to enhance model accuracy and reliability. This study evaluates the accuracy, efficiency, and repeatability of PINNs by comparing their results against traditional mathematical-based multiphase flow measurement (MPFM) technology and controlled test cases. The developed PINN successfully solved the Navier- Stokes (N-S) equations in their closest approximation to the original form, with minimal assumptions and simplifications, to estimate multiphase flow rates, integrating fundamental fluid dynamics principles into its learning process. The results demonstrated that the PINN-based approach achieved accuracy comparable to traditional mathematical multiphase flow measurement (MPFM) technologies, while significantly improving computational efficiency. During controlled loop tests with oil, water, and gas, the PINN model consistently captured flow dynamics with high precision, validating its effectiveness under real-world conditions. Moreover, the PINN method exhibited strong generalization capabilities, adapting to various flow regimes without the need for extensive empirical calibration. Compared to conventional MPFM technologies, PINN reduced reliance on predefined correlations and offered greater adaptability to changing fluid properties. The repeatability of results confirmed the model's reliability, positioning PINNs as a promising alternative for complex multiphase flow measurement applications. These findings highlight the potential of PINNs as an innovative, physics-guided, and data-efficient solution, effectively bridging the gap between traditional physics-based methods and modern machine learning techniques. This paper presents an innovative application of Physics-Informed Neural Networks (PINNs) for multiphase flow measurement, offering a data-driven yet physics-guided alternative to traditional mathematical MPFM technologies. By embedding fundamental physical laws into the neural network training process, the proposed approach enhances accuracy, computational efficiency, and adaptability, while minimizing reliance on extensive empirical calibration. This advancement significantly improves real-time flow monitoring, equipping practicing engineers with a more reliable, data-efficient, and scalable solution for complex multiphase flow measurement in industrial applications.

A physics-informed neural network (PINN) uses physics-augmented loss functions, e.g., incorporating the residual term from governing partial differential equations (PDEs), to ensure its output is consistent with fundamental physics laws. However, it turns out to be difficult to train an accurate PINN model for many problems in practice. In this article, we present a novel perspective of the merits of learning in sinusoidal spaces with PINNs. By analyzing behavior at model initialization, we first show that a PINN of increasing expressiveness induces an initial bias around <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">flat output functions</i> . Notably, this initial solution can be very close to satisfying many physics PDEs, i.e., falling into a local minimum of the PINN loss that only minimizes PDE residuals, while still being far from the true solution that jointly minimizes PDE residuals and the initial and/or boundary conditions. It is difficult for gradient descent optimization to escape from such a local minimum trap, often causing the training to stall. We then prove that the sinusoidal mapping of inputs—in an architecture we label as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sf</i> -PINN—is effective to increase input gradient variability, thus avoiding being trapped in such deceptive local minimum. The level of variability can be effectively modulated to match high-frequency patterns in the problem at hand. A key facet of this article is the comprehensive empirical study that demonstrates the efficacy of learning in sinusoidal spaces with PINNs for a wide range of forward and inverse modeling problems spanning multiple physics domains.

Physics-informed neural networks (PINN) have recently become attractive for solving partial differential equations (PDEs) that describe physics laws. By including PDE-based loss functions, physics laws such as mass balance are enforced softly in PINN. This paper investigates how mass balance constraints are satisfied when PINN is used to solve the resulting PDEs. We investigate PINN’s ability to solve the 1D saturated groundwater flow equations (diffusion equations) for homogeneous and heterogeneous media and evaluate the local and global mass balance errors. We compare the obtained PINN’s solution and associated mass balance errors against a two-point finite volume numerical method and the corresponding analytical solution. We also evaluate the accuracy of PINN in solving the 1D saturated groundwater flow equation with and without incorporating hydraulic heads as training data. We demonstrate that PINN’s local and global mass balance errors are significant compared to the finite volume approach. Tuning the PINN’s hyperparameters, such as the number of collocation points, training data, hidden layers, nodes, epochs, and learning rate, did not improve the solution accuracy or the mass balance errors compared to the finite volume solution. Mass balance errors could considerably challenge the utility of PINN in applications where ensuring compliance with physical and mathematical properties is crucial.

3D concrete printing (3DCP) is an innovative construction method that extrudes cementitious materials layer-by-layer to fabricate building components based on a digital model. 3DCP has gained increasing adoption globally for projects like buildings, bridges, retaining walls, and stormwater management systems. However, 3DCP places stringent demands on the rheological properties of the printable cementitious materials. Rheology is critical for the flow and buildability of fresh concrete during extrusion. The mixture must have a low yield stress for pumping, moderate viscosity to hold its shape after deposition, minimal bleeding and segregation of aggregates, as well as responsive rheology that can transition from fluid to solid state as layers are printed sequentially. In this dissertation, we aim to solve the challenges mentioned above including: (1) accurate and efficient quantification of rheological and thixotropic properties of cementitious materials, (2) evaluation of flow behavior of fresh concrete during extrusion. Using traditional experimental methods to evaluate rheological properties of cement paste is labor-intensive and time-consuming. Numerical simulations like Finite Element Method (FEM) or Computational Fluid Dynamics (CFD) can help increase the speed of finding the most suitable material for 3DCP. However, these simulation methods are costly in computing power, making them less desirable for simulating the dynamic and complex process of 3DCP. Moreover, mesh-based approaches are not suitable for complex geometries as the mesh generation would be challenging with curved boundaries, small features, holes or thin regions. Physics-Informed Neural Network (PINN) is introduced in the work as an innovative solution to simulate and model the whole process of 3DCP. It can leverage the flexibility and computational efficiency of data-based neural networks, which can accelerate the learning and convergence speed. At the same time, Partial Differential Equations (PDEs) would be embedded in the network structure to increase the accuracy of its prediction. The integrated governing physical laws can help the neural network understanding the underlying physics of the 3DCP process, potentially increasing the accuracy and reliability of simulations. A significant part of the research is dedicated to understanding the rheological and thixotropic behavior of cementitious materials, which is critical for the material design in 3DCP. We developed a Rheology-informed Neural Network, RheologyNet, where we embedded the rheological constitutive laws into the loss function. These physical laws can help the network focus on simulating the rheological properties of cementitious materials. Moreover, a thixotropic evaluation system has been established based on the proposed model to evaluate the thixotropic property of cement paste. This study is presented in Chapter 2. Then, we proposed a Navier-Stokes Informed Neural Network (NSINN) to study the flow behavior of cement paste in the 3DCP barrel and nozzle since it involves complicated Multiscale or Multiphysics behaviors, such as anisotropy and non-uniform shear rate distribution. The Navier Stokes Equation and the rheological constitutive equations are coupled and embedded into the NSINN architecture to simulate the flow behavior in the barrel during 3D-printing extrusion process. This approach demonstrates the potential of PINNs to provide high-accuracy simulations of velocity and pressure fields in a computationally efficient manner compared to traditional mesh-based models. Also, we used the NSINN to learn the relationship between the nozzle size and the printing quality. The details of the study are presented in Chapter 3. At last, a multi-subnetwork PINN architecture (MultiSubPINN) is proposed to address the limitations of traditional PINNs. This is because traditional PINNs used Fully-connected Neural Network (FNN) as main bone. It has problems like different outputs must share parameters within the network structure. However, this parameter sharing nature might lead to potential errors and decrease the model performance. The main architecture of this network is consisted of multiple separated sub-networks and the loss function of each sub-network is calculated separately to update the parameters in each sub-network. By doing this, we can ensure that the optimization for one output doesn't negatively impact others, therefore providing a more flexible, scalable, and potentially more accurate alternative for the evaluation. Besides, using randomly distributed training points in the computational domain for PINNs is a common way. However, many physical problems exhibit non-uniform behavior, with certain regions of the domain having more complex dynamics or sharper gradients than others. Randomly distributed training points might not adequately capture these regions, leading to a model that misrepresents the underlying physics in critical areas. We proposed a Physical-based sampling strategy to optimize the training of the PINNs. This study is presented in Chapter 4. In summary, this dissertation has established a new platform that is enabled by interpretable NN to accurately quantify cement paste’s rheological properties and simulate the flow behavior of 3D concrete printing process. The findings suggest that PINNs can not only improve the efficiency and accuracy of simulations but also provide a more profound understanding of the material behaviors and process dynamics critical to 3DCP. The novel PINN structure developed from this dissertation is a powerful general platform that could be applicable to a variety of material domains for complex rheological behavior predictions.

A novel sequential method to train physics informed neural networks for Allen Cahn and Cahn Hilliard equations

Adaptive transfer learning for PINN

Solving the two‐dimensional Shallow Water Equations (SWE) is a fundamental problem in flood simulation technology. In recent years, physics‐informed neural networks (PINNs) have emerged as a novel methodology for addressing this problem. Given their advantages in parallel computing, potential for data assimilation and parameter calibration, and the rapid advancement of artificial intelligence, it is crucial to investigate both the capabilities and limitations of PINNs. While current research has demonstrated the significant potential of PINNs, many aspects of this new approach remain to be explored. In this paper, we employ PINNs enhanced by dimensional transformation and Neuron‐wise Local Adaptive Activation Function techniques to validate its effectiveness in solving two‐dimensional free surface flow with rainfall on terrain topography. The SWE primarily exist in two forms: the primitive form and the conservative form. Through theoretical analysis and experimental validation, we demonstrate that a hybrid primitive‐conservative form offers superior performance. Additionally, in our cases, we find that incorporating the energy conservation law, specifically the entropy condition, does not improve accuracy and sometimes destabilizes training due to an over‐weighting of the mass conservation law. Furthermore, we have developed an open‐source module on the PINNacle platform for solving SWE using PINNs, which includes over 10 case studies and various equation forms, to promote research and application in this field.

In recent years, successful applications of deep learning (DL) have inspired scientists to explore the possibilities of applying DL approaches to modeling scientific problems. Existing studies have revealed that to bake the physics into the DL makes a good supplement to the traditional numerical methods (e.g., finite element, finite volume method) which primarily rely on partial differential equations (PDEs). While DL models are ordinarily trained in a purely data-driven manner, integrating physics into them for simulating scientific problems has several benefits such as (i) physics constraints could regularize the over-parameterized model and hence mitigate the overfitting issue commonly seen in DL; (ii) physics information could also effectively reduce the amount of data needed for training the model; (iii) the resultant physics-informed DL models feature better interpretability and generalizability compared with the conventional black-box model. Furthermore, the powerful expressiveness of the deep network, guaranteed by the universal approximation theorem, makes it a suitable approximator for the solution to a physical system. In this dissertation, we develop two different DL architectures (or approaches), one being continuous scheme-based while the other discrete scheme-based, that leverage physics knowledge for modeling scientific problems. Through comprehensive numerical experiments, we demonstrate the proposed models can be used in solving general PDEs, establishing predictive data-driven models for dynamical systems, identifying the parameters in governing PDEs or even discovering the entire governing PDEs of dynamical systems from scarce and noisy measurements. The continuous model roots on the physics-informed neural network (PINN) which uses a fully connected neural network (FCNN) to approximate the physical fields of a system globally. This model is mesh-free as the residual of the physics (e.g., PDEs, initial/boundary values) is evaluated on a set of collocation points within the physical domain. Several applications including the forward simulations, data-driven simulations and solving inverse problems are presented to exemplify the advantages of PINN over traditional numerical methods. However, the original PINN suffers from inaccurate initial/boundary values due to the weak enforcement of the initial/boundary conditions (I/BCs). To overcome this issue, we propose an improved PINN model by utilizing multiple deep neural networks (DNNs) to construct the solution. Through a DNN pre-trained to represent the initial/boundary values, the approximated solution would obey the given I/BCs forcibly. With several numerical examples, we show that the improved PINN is characterized with much better accuracy on the I/BCs. Though the PINN shows great promise in data-driven modeling and solving inverse problems, some inherent limitations of PINN still exist, such as (i) the solution might lacks fine-scale details due to the global approximation of FCNN; (ii) high computational expense caused by the FCNN it roots on; (iii) incapability to incorporate existing PDE terms (e.g., $\Delta u$) into the network architecture. To overcome these drawbacks, this dissertation also proposes a discrete model - Physics-encoded Recurrent Convolutional Neural Network (PeRCNN) which recurrently updates the solution (or state variable) for time marching. Specifically, it utilizes convolutional neural network (CNN) to capture the spatial patterns of the solution while the recurrent network mimics the forward Euler scheme (or Runge-Kutta scheme) in numerical methods. PeRCNN is a mesh-based and discrete model due to the discretization in time and spatial dimension. The local connectivity of CNN makes PeRCNN more computationally efficient. In addition, the coercive encoding mechanism of physics in PeRCNN, fundamentally different from the PINN relying on soft penalty, ensures the network to rigorously obey given physics. The proposed PeRCNN is successfully applied to solving general PDEs, the data-driven modeling of dynamical systems and the data-driven discovery of governing PDEs from scarce and noisy measurements. Comparisons with the state-of-the-art DL models demonstrate that the proposed PeRCNN possesses excellent computational efficiency, accuracy and generalizability. --Author's abstract

Glucocorticoid-induced hypertension affects over 30% of treated patients, yet its underlying mechanisms remain unclear, particularly how glucocorticoids regulate renin within the renin-angiotensin-aldosterone system (RAAS). Modeling these dynamics is difficult because only four dose-response measurements are available at a single 24-h timepoint (36 observations total), while the system depends on roughly eleven biochemical parameters spanning minutes-long receptor interactions to days-long protein secretion. Classical parameter estimation becomes unreliable in this extremely underdetermined setting, and purely data-driven methods offer limited biological interpretability. In this paper, we introduce a physics-informed neural network (PINN) framework that integrates ELISA measurements from As4.1 juxtaglomerular cells, ordinary differential equations describing glucocorticoid receptor signaling and renin transcription, and automatic differentiation to enforce mechanistic constraints. By systematically tuning synthetic-data weights (SW in {0.2, 0.3, 0.5}), we identify an intermediate value of SW = 0.3 that provides the best overall balance between predictive accuracy, accepted ensemble size, and biologically plausible parameter estimates among the tested configurations. The framework uses adaptive constraint scheduling with a plateau ramp to reduce premature convergence and introduces calibrated plausibility thresholds reflecting experimental noise. The accepted PINN ensemble (n = 5, 50% success rate) achieved R2 = 0.803, compared with 0.759 for the SW = 0.5 baseline and −0.220 for the ODE-only baseline, with RMSE = 0.024. Key learned parameters (IC50 = 2.925 ± 0.012 mg/dL, Hill = 1.950 ± 0.009) are biologically plausible within the model assumptions, and the best single accepted model attained R2 = 0.891. Information criteria favored the PINN over the ODE model, with improvements of approximately 77× (AIC) and 5.9× (BIC). Despite training on a single timepoint, the PINN also infers full 48-h trajectories and reproduces non-monotonic dose-response behavior. This work presents, to our knowledge, the first PINN framework for glucocorticoid-mediated renin regulation and should be interpreted as a proof-of-concept approach for integrating sparse biomedical data with mechanistic constraints. The inferred parameters and temporal dynamics are best viewed as model-dependent, hypothesis-generating estimates rather than validated biological quantities.

Flood forecasting is usually performed by means of physically based models or conceptual models. Data-driven methods such as Artificial Neural Networks (ANN) are a promising solution as they can extract complex non-linear dependencies between inputs and outputs with robustness, generalization, and computational efficiency. However, the ANNs still rank below the classical method of numerical solvers for Saint Venant Equations in terms of accuracy. To bridge the accuracy gap, this work focuses on devising a Physics Informed Neural Network (PINN) model for spatial-temporal scale flood forecasting based on the Saint Venant Equations. The ANN architecture is extended to PINN with the implementation of the custom Partial Differential Equation (PDE) loss. This procedure allows to quickly steer the learning algorithm toward a true solution. A synthetic dataset was developed by performing numerical simulations of the Saint Venant Equations for training and evaluation of the PINN model. The results demonstrate that the PINN model performs better than the ANN Models and is suitable for water depth forecasting.

Error homogenization in physics-informed neural networks for modeling in manufacturing