Solution of conservative-form transport equations with physics-informed neural network

As Li-ion batteries become more essential in today's economy, tools need to be developed to accurately and rapidly diagnose a battery's internal state-of-health. Using a Li-ion battery's (high-rate) voltage response, it is proposed to determine a battery's internal state through Bayesian calibration.However, Bayesian calibration is notoriously slow and requires thousands of model runs. To accelerate parameter inference using Bayesian calibration, a surrogate model is developed to replace the underlying physics-based Li-ion model. Developing a surrogate model for rapid Bayesian calibration analysis is discussed for both the single particle model (SPM) and the pseudo two-dimensional (P2D) model.Surrogate models are constructed using physics-informed neural networks (PINNs) that encode the influence of internal properties on observed voltage responses. In practice, a neural network can be trained by: 1) using simulation results of the physics-based model (i.e., a data-loss approach); 2) using the residuals of the governing equations themselves (i.e., a physics-loss approach); or 3) using a combination of simulation results and governing equation residuals. In the present work, PINNs are developed using a variety of training losses and neural network architectures. In this analysis, it is shown that a PINN surrogate model can be reliably trained with only physics-informed loss. However, using a coupled data-informed and physics-loss approach produced the most accurate PINNs. Figure~\\ref{fig:spm_2d} illustrates the absolute relative errors of trained PINN networks using several different training losses and neural network architectures.After determining a consistent training strategy for both the SPM and P2D PINN surrogate models, the PINNs are extended to determine additional internal state-of-health parameters. As more and more parameters were introduced, the PINN training suffered from ``the curse of dimensionality", which was mitigated by using a hierarchical training approach (where a PINN trained with fewer variable model parameters was used to train a PINN with more variable model parameters).Next, the high-dimensionality PINN surrogates are then integrated into Bayesian calibration schemes to identify internal Li-ion battery properties from experimentally measured voltages. Interpreting the high-dimensional parameter posteriors is discussed with respect to model error, parameter prior choices, and experimental errors. Figure 1

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.

Simulating convective heat transfer using traditional numerical methods requires explicit definition of thermal boundary conditions on all boundaries of the domain, which is almost impossible to fulfill in real applications. Here, we address this ill-posed problem using machine learning techniques by assuming that we have some extra measurements of the temperature at a few locations in the domain, not necessarily located on the boundaries with the unknown thermal boundary condition. In particular, we employ physics-informed neural networks (PINNs) to represent the velocity and temperature fields while simultaneously enforce the Navier-Stokes and energy equations at random points in the domain. In PINNs, all differential operators are computed using automatic differentiation, hence avoiding discretization in either space or time. The loss function is composed of multiple terms, including the mismatch in the velocity and temperature data, the boundary and initial conditions, as well as the residuals of the Navier-Stokes and energy equations. Here, we develop a data-driven strategy based on PINNs to infer the temperature field in the prototypical problem of convective heat transfer in flow past a cylinder. We assume that we have just a couple of temperature measurements on the cylinder surface and a couple more temperature measurements in the wake region, but the thermal boundary condition on the cylinder surface is totally unknown. Upon training the PINN, we can discover the unknown boundary condition while simultaneously infer the temperature field everywhere in the domain with less than 5% error in the Nusselt number prediction. In order to assess the performance of PINN, we carried out a high fidelity simulation of the same heat transfer problem (with known thermal boundary conditions) by using the high-order spectral/hp-element method (SEM), and quantitatively evaluated the accuracy of PINN’s prediction with respect to SEM. We also propose a method to adaptively select the location of sensors in order to minimize the number of required temperature measurements while increasing the accuracy of the inference in heat transfer.

We characterize and remedy a failure mode that may arise from multi-scale dynamics with scale imbalances during training of deep neural networks, such as physics informed neural networks (PINNs). PINNs are popular machine-learning templates that allow for seamless integration of physical equation models with data. Their training amounts to solving an optimization problem over a weighted sum of data-fidelity and equation-fidelity objectives. Conflicts between objectives can arise from scale imbalances, heteroscedasticity in the data, stiffness of the physical equation, or from catastrophic interference during sequential training. We explain the training pathology arising from this and propose a simple yet effective inverse Dirichlet weighting strategy to alleviate the issue. We compare with Sobolev training of neural networks, providing the baseline of analytically ε-optimal training. We demonstrate the effectiveness of inverse Dirichlet weighting in various applications, including a multi-scale model of active turbulence, where we show orders of magnitude improvement in accuracy and convergence over conventional PINN training. For inverse modeling using sequential training, we find that inverse Dirichlet weighting protects a PINN against catastrophic forgetting.

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

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

In recent years, the artificial intelligence computing paradigm represented by physics-informed neural networks (PINNs) has received great attention in the field of plasma numerical simulation. However, the plasma chemical system considered in related research is relatively simplified, and the research on solving the more complex multi-particle low-temperature fluid model based on PINNs is still blank. In more complex chemical systems, the coupling relationship between particle densities and between particle densities and mean electron energy become more intricate. Therefore, the applicability of PINNs in dealing with sophisticated reaction systems needs further exploring and improving. In this work, we propose a general PINN framework (source term decoupled PINNs, Std-PINNs) for solving multi-particle low-temperature plasma fluid model. By introducing equivalent positive ions and replacing each particle transport equation with the current continuity equation as a physical constraint, Std-PINN splits the entire solution process into the training processes of two neural networks, realizing the decoupling of the source term of the heavy particle transport equation from the electron density and mean electron energy, which greatly reduces the complexity of neural network training. In this work, the application of Std-PINNs to solving multi-particle low-temperature plasma fluid models is demonstrated through two classic discharge cases with different complexity of reaction systems (low-pressure argon glow discharge and atmospheric-pressure helium glow discharge) and the performance of Std-PINN is compared with that of conventional PINN and finite element method (FEM). The results show that the training results output from the traditional PINN are completely incorrect due to the strong coupling correlation of each physical variable through the source terms of each particle transport equation, while the <i>L</i><sub>2</sub> relative error between Std-PINN and FEM results can reach up to ~10<sup>–2</sup> , thus verifying the feasibility of Std-PINN in simulating multi-particle plasma fluid model. Std-PINN expands the application of deep learning method to modeling complex physical systems and provides new ideas for conducting low-temperature plasma simulations. In addition, this study provides novel insights into the field of artificial intelligence scientific computing: the mathematical form that describes the state of a physical system is not unique. By introducing equivalent physical variables, equations suitable for neural network solutions can be derived and combined with observable data to simplify problems.

Physics-informed neural networks (PINNs) are emerging as a promising artificial intelligence approach for solving complex two-phase flow simulations. A critical challenge in these simulations is an accurate representation of the gas–liquid interface using interface tracking methods. While numerous studies in conventional computational fluid dynamics (CFD) have addressed this issue, there remains a notable absence of research within the context of PINNs-based two-phase flow simulations. Therefore, this study aims to develop a robust and generic PINNs for two-phase flow by incorporating the governing equations with three advanced interface tracking methods—specifically, the Volume of Fluid, Level Set, and Phase-Field method—into an improved PINN framework that has been previously proposed and validated. To further enhance the performance of the PINNs in simulating two-phase flow, the phase field constraints, residual connection and the time divide-and-conquer strategies are employed for restricting neural network training within the scope of physical laws. This self-adaptive and time divide-and-conquer (AT) PINNs is then optimized by minimizing both the residual and loss terms of partial differential equation. By incorporating the three different interface tracking methods, it efficiently handles high-order derivative terms and captures the phase interface. The case of single rising bubble in two-phase flow is simulated to validate the robustness and accuracy of the AT PINNs. The simulation's accuracy is evaluated by comparing its performance in terms of velocity, pressure, phase field, center of mass, and rising velocity with that of conventional PINNs and CFD benchmarks. The results indicate that the AT PINNs coupled with these interface tracking methods offers a satisfactory performance in simulating rising bubble phenomenon.

This thesis investigates two physics informed neural network (PINN) approaches for two mechanistic problems via the performance metrics of model parameter estimation and system state prediction. The PINNs are compared to Bayesian inference (BI) tested under similar conditions. The first PINN studied is the loss-based PINN, which embeds physics into a standard feedforward neural network within the loss function of this network. The second PINN investigated is the hybrid-recurrent physics informed neural network, which uses a recurrent neural network (RNN) that embeds physics in RNN architecture. The BI used as a point of comparison uses a Transitional Markov Chain Monte Carlo algorithm to estimate the posterior distributions. The applications studied are a compartmental model for an infectious disease, and a mass-spring-damper model. It is found that BI performs better in ideal data conditions and the PINNs perform better in weak data conditions.

In recent engineering applications using deep learning, physics-informed neural network (PINN) is a new development as it can exploit the underlying physics of engineering systems. The novelty of PINN lies in the use of partial differential equations (PDE) for the loss function. Most PINNs are implemented using automatic differentiation (AD) for training the PDE loss functions. A lesser well-known study is the use of finite difference method (FDM) as an alternative. Unlike an AD based PINN, an immediate benefit of using a FDM based PINN is low implementation cost. In this paper, we propose the use of finite difference method for estimating the PDE loss functions in PINN. Our work is inspired by computational analysis in electromagnetic systems that traditionally solve Laplace’s equation using successive over-relaxation. In the case of Laplace’s equation, our PINN approach can be seen as taking the Laplacian filter response of the neural network output as the loss function. Thus, the implementation of PINN can be very simple. In our experiments, we tested PINN on Laplace’s equation and Burger’s equation. We showed that using FDM, PINN consistently outperforms non-PINN based deep learning. When comparing to AD based PINNs, we showed that our method is faster to compute as well as on par in terms of error reduction.

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

Physics-informed neural networks (PINNs) represent a transformative approach to data models by incorporating known physical laws into neural network training, thereby improving model generalizability, reduce data dependency, and enhance interpretability. Like many other fields in engineering and science, the analysis of physiological signals has been influenced by PINNs in recent years. This manuscript provides a comprehensive overview of PINNs from various perspectives in the physiological signal analysis domain. After exploring the literature and screening the search results, more than 40 key studies in the related domain are selected and categorized based on both practically and theoretically significant perspectives, including input data types, applications, physics-informed models, and neural network architectures. While the advantages of PINNs in tackling forward and inverse problems in physiological signal contexts are highlighted, challenges such as noisy inputs, computational complexity, loss function types, and overall model configuration are discussed, providing insights into future research directions and improvements. This work can serve as a guiding resource for researchers exploring PINNs in biomedical and physiological signal processing, paving the way for more precise, data-efficient, and clinically relevant solutions.

Physics-Informed Neural Network for modeling and predicting temperature fluctuations in proton exchange membrane electrolysis

Indoor airflow field reconstruction using physics-informed neural network