Bridging the Gap for Sparse Observations: Data‐Driven Versus Physics‐Informed Neural Networks for a Transient Flow Problem in a Heterogeneous Porous Medium

Simulating solute transport in heterogeneous porous media poses computational challenges due to the high-resolution meshing required for traditional solvers. To overcome these challenges, this study explores a mesh-free method based on deep learning to accelerate solute transport simulation. We employ Physics-informed Neural Networks (PiNN) with a periodic activation function to solve solute transport problems in both homogeneous and heterogeneous porous media governed by the advection-dispersion equation. Unlike traditional neural networks that rely on large training datasets, PiNNs use strong-form mathematical models to constrain the network in the training phase and simultaneously solve for multiple dependent or independent field variables, such as pressure and solute concentration fields. To demonstrate the effectiveness of using PiNNs with a periodic activation function to resolve solute transport in porous media, we construct PiNNs using two activation functions, sin and tanh, for seven case studies, including 1D and 2D scenarios. The accuracy of the PiNNs’ predictions is then evaluated using absolute point error and mean square error metrics and compared to the ground truth solutions obtained analytically or numerically. Our results demonstrate that the PiNN with sin activation function, compared to tanh activation function, is up to two orders of magnitude more accurate and up to two times faster to train, especially in heterogeneous porous media. Moreover, PiNN’s simultaneous predictions of pressure and concentration fields can reduce computational expenses in terms of inference time by three orders of magnitude compared to FEM simulations for two-dimensional cases.

Indoor airflow field reconstruction using physics-informed neural network

Identifying optimal architectures of physics-informed neural networks by evolutionary strategy

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.

The characterization of in-place material properties is important for quality control and condition assessment of the built infrastructure. Although various methods have been developed to characterize structural materials in situ, many suffer limitations and cannot provide complete or desired characterization, especially for inhomogeneous and complex materials such as concrete and rock. Recent advances in machine learning and artificial neural networks (ANN) can help address these limitations. In particular, physics-informed neural networks (PINN) portend notable advantages over traditional physics-based or purely data-driven approaches. PINN is a particular form of ANN, where physics-based equations are embedded within an ANN structure in order to regularize the outputs during the training process. This paper reviews the fundamentals of PINN, notes its differences from traditional ANN, and reviews applications of PINN for selected material characterization tasks. A specific application example is presented where mechanical wave propagation data are used to characterize in-place material properties. Ultrasonic data are obtained from experiments on long rod-shaped mortar and glass samples; PINN is applied to these data to extract inhomogeneous wave velocity data, which can indicate mechanical material property variations with respect to length.

Modeling density-driven flow in porous media by physics-informed neural networks for CO2 sequestration

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

Satellite-based research on global sea surface wind is essential for understanding and monitoring the air-sea interface dynamical processes, driving the need for accurate and efficient assessment. Spaceborne scatterometers, which are among the most relevant sensors for sea surface wind observation, play a vital role in obtaining global ocean surface wind information. However, a significant challenge associated with polar-orbiting satellites lies in data gaps caused by their orbital paths, resulting in missing observations between swaths. While several satellite-derived sea surface wind products have been developed using data assimilation (DA) techniques, these existing methods are time-consuming and require large amounts of diverse data, rendering them computationally expensive. Additionally, the iterative steps of variational algorithms can only perform linear or weakly nonlinear adjustments to the governing equations, which may pose challenges given the highly non-linear nature of these equations.In this study, we leveraged physics-informed neural networks (PINNs) techniques to reconstruct large-scale sea surface wind fields with sparse scatterometer observations from different satellites, integrating observations and filling gaps. By incorporating physical constraints into the loss function, specifically the Navier-Stokes equations, we efficiently fill the data gaps and reconstruct wind fields that not only match observational data but also adhere to physical principles. Another objective of this work is to introduce the wind speed gradient and direction parallel consistent constraints into the loss function in order to enhance the detail of the reconstructed wind field and increase the accuracy of both wind speed and direction. Structurally, the PINN resembles a fully connected neural network (FCNN), offering the advantage of automatic feature extraction. Our model not only extracts valuable information from existing data but also uncovers complex patterns and correlations in data that are difficult for traditional algorithms to capture. This approach provides a novel perspective and an alternative methodology for wind field reconstruction.The results show that PINNs can reconstruct wind fields that closely resemble realistic wind patterns, capturing large-scale structures while preserving fine-scale details, thanks to the introduction of wind speed gradient and direction parallel consistent constraints. The training time for this model is about 3 hours, using only a single GPU core. This efficiency is partly due to the fact that PINNs do not rely on ensemble methods or large datasets to produce results. Unlike traditional DA methods, PINN does not depend on an initial best-guess field for assimilating observations. While we use only a small amount of scatterometer observation data, no initial field is required to complete the reconstruction. Additionally, since the PINN represents a continuous and differentiable function, it can produce outputs at any spatial or temporal point within the training domain.Recognizing their potential for forecast models and data integration, PINNs offer a promising approach for accurate sea surface wind field reconstruction and could serve as an effective alternative to current methods.

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.

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

The Burgers equation is a well-established test case in the computational modeling of several phenomena, such as fluid dynamics, gas dynamics, shock theory, cosmology and others. In this work, we present the application of physics-informed neural networks (PINNs) with a transfer learning approach using the θ-scheme to solve the Burgers' equation. The proposed approach consists of searching for a discrete solution in time through a sequence of artificial neural networks (ANNs). At each time step, the previous ANN transfers its learning to the next network model, which learns the solution in the current time by minimizing a loss function based on the θ-scheme approximation of the Burgers' equation. To test this approach, we present its application to two benchmark problems with known analytical solutions. Compared to usual PINN models, the proposed approach has the advantage of requiring smaller neural network architectures with similar accurate results and potentially decreasing computational costs.

Today, an engineer's success is largely based on their skill at employing Finite Element Analysis (FEA), a standard engineering modeling tool to numerically assess the behavior of structures. So far, improvements in computational power and FEA element formulations have supported significant advancements in this field. FEA can be used to provide accurate and fast answers to many engineering problems. However, FEA models are still associated with prohibitive time costs when solving complex scenarios. Unlike traditional computer algorithms, Machine Learning, by minimizing pre-established loss functions, can ascertain patterns and make predictions without being explicitly programmed for a given task. This paper presents initial results on improving structural analysis tools by harnessing the power of Machine Learning algorithms, namely Deep Artificial Neural Networks (ANNs) and Physics-Informed Neural Networks (PINNs), with the final objective of substituting or accelerating FEA in accurately predicting the structural behavior within an engineering domain. It is found that ANNs can accurately predict the stress, strain, and displacement maps for a cantilevered rectangular plate under a concentrated load. ANNs demonstrate that, with a large training dataset, efficient and accurate predictions can be achieved although only for specific problems. These algorithms could be enhanced with the governing differential equations to improve convergence and speed. PINNs show that, despite being less efficient than FEA, the method shows more generalization potential than ANNs. Both results show the potential for machine learning algorithms to enhance traditional computational mechanics methods.

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.