A MOVING MESH FINITE ELEMENT METHOD AND ITS APPLICATIONS IN MALARIA TRANSMISSION MODEL

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

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.

Moving mesh finite element simulation for phase-field modeling of brittle fracture and convergence of Newton's iteration

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

<p>LEMTA - UMR 7563 CNRS-UL - Universite de Lorraine, Nancy, France</p><p><br>Gravity currents [Ungarish, 2009] are common finescale structures in coastal areas, where they are involved in the vertical transport of physical and biogeological tracers. These stratified flows frequently carry a suspended solid phase that controls the average properties of the flow, such as the geometry, velocity and apparent viscosity leading to a rich dynamic whose optimization/prediction remains a major scientific issue.</p><p>A better understanding of the particle-flow coupling requires an accurate characterization of the particle volume fraction, fluid density, velocity and pressure fields. The experimental apparatus available at LEMTA allows high accuracy instantaneous measurements of density fields in laboratory experiments with the light attenuation technique (LAT) as described in a companion presentation, while the joint measurement of velocity and pressure fields remains beyond the reach of known metrological techniques.</p><p>To address this issue we propose an original approach based on the Physics Informed Neural Network (PINN) technique [Raissi 2020], which extracts the complete hydrodynamic variable set from the sole observation of the density field. The accuracy of this approach is evaluated against the well-documented test case of the lock-exchange configuration in both laminar and turbulent regimes. As a validation step, the PINN technique is applied on numerical simulations outputs with the well-established Nek5000 flow solver with convincing results. Then, an application on experimental data from LAT experiments will be shown, illustrating the robustness of this promising paradigm.</p>

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.

The generalization of Physics-Informed Neural Networks (PINNs) used to solve the inhomogeneous Helmholtz equation in a simplified three-dimensional room is investigated. PINNs are appealing since they can efficiently integrate a partial differential equation and experimental data by minimizing a loss function. However, a previous study experienced limitations in acoustics regarding the source term. A challenging but realistic excitation case is a confined (e.g., single-point) excitation area, yielding a smooth spatial wave field periodically with the wavelength. Compared to studies using smooth (unrealistic) sound excitation, the network’s generalization capabilities regarding a realistic sound excitation are addressed. Different methods like hyperparameter optimization, adaptive refinement, Fourier feature engineering, and locally adaptive activation functions with slope recovery are tested to tailor the PINN’s accuracy to an experimentally validated finite element analysis reference solution computed with openCFS. The hyperparameter study and optimization are conducted regarding the network depth and width, the learning rate, the used activation functions, and the deep learning backends (PyTorch 2.5.1, TensorFlow 2.18.0 1, TensorFlow 2.18.0 2, JAX 0.4.39). A modified (feature-engineered) PINN architecture was designed using input feature engineering to include the dispersion relation of the wave in the neural network. For smoothly (unrealistic) distributed sources, it was shown that the standard PINNs and the feature-engineered PINN converge to the analytic solution, with a relative error of 0.28% and 2×10−4%, respectively. The locally adaptive activation functions with the slope lead to a relative error of 0.086% with a source sharpness of s=1 m. Similar relative errors were obtained for the case s=0.2 m using adaptive refinement. The feature-engineered PINN significantly outperformed the results of previous studies regarding accuracy. Furthermore, the trainable parameters were reduced to a fraction by Bayesian hyperparameter optimization (around 5%), and likewise, the training time (around 3%) was reduced compared to the standard PINN formulation. By narrowing this excitation towards a single point, the convergence rate and minimum errors obtained of all presented network architectures increased. The feature-engineered architecture yielded a one order of magnitude lower accuracy of 0.20% compared to 0.019% of the standard PINN formulation with a source sharpness of s=1 m. It outperformed the finite element analysis and the standard PINN in terms time needed to obtain the solution, needing 15 min and 30 s on an AMD Ryzen 7 Pro 8840HS CPU (AMD, Santa Clara, CA, USA) for the FEM, compared to about 20 min (standard PINN) and just under a minute of the feature-engineered PINN, both trained on a Tesla T4 GPU (NVIDIA, Santa Clara, CA, USA).

In the past few years, approaches such as physics informed neural networks (PINNs) have been applied to a variety of applications that can be modeled by linear and nonlinear ordinary and partial differential equations. Specifically, this work builds on the application of PINNs to a SIRD (susceptible, infectious, recovered, and dead) compartmental model and enhances it to build new mathematical models that incorporate transportation between populations and their impact on the dynamics of infectious diseases. Our work employs neural networks capable of learning how diseases spread, forecasting their progression, and finding their unique parameters. We show how these approaches are capable of predicting the behavior of a disease described by governing differential equations that include parameters and variables associated with the movement of the population between neighboring cities. We show that our model validates real data and also how such PINNs based methods predict optimal parameters for given datasets.

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.

A finite element analysis for the temperature field produced by a moving heat source

Assessing the performance of physics-informed neural networks for tumor growth prediction under noisy and sparse data conditions.

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.

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.

Physics-informed neural networks for mesh deformation with exact boundary enforcement

Deep learning techniques have found diverse applications, including solving partial differential equations (PDEs). This chapter observes the utilization of physics-informed neural networks (PINNs) for solving a range of PDEs related to fluid dynamics applications. PINNs harness the inherent physical insights embedded within PDEs by integrating them as regularization terms, thereby optimizing the performance of neural networks. Notably, PINNs prove effective in handling higher-dimensional differential equations, offering substantial reductions in computational costs. The current chapter focuses on the application of PINNs to solve specific instances of fluid dynamics problems, including second-order porous medium and third-order Korteweg-de Vriesequations. The proposed methodology incorporates the Adam optimizer and employs the “tanh” activation function. The achieved mean squared error (MSE) losses are reported as https://www.w3.org/1998/Math/MathML" display="inline">4.972656×10−6, 16.58, https://www.w3.org/1998/Math/MathML" display="inline">2.73382×10−5, and https://www.w3.org/1998/Math/MathML" display="inline">9.083319×10−5. To validate the efficacy of our approach, a comparative analysis with ground truth solutions is presented. This contribution offers insights into the cutting-edge applications of machine learning in solving differential equations that represent fluid dynamics phenomena.