A Physics Informed Neural Network (PINN) framework for fractional order modeling of Alzheimer's disease

A mixed formulation for physics-informed neural networks as a potential solver for engineering problems in heterogeneous domains: Comparison with finite element method

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

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 computationally efficient tools for addressing inverse problems in solid mechanics, but often face accuracy limitations when compared to traditional methods. We introduce a refined PINN approach that rigorously enforces certain physics constraints, improving accuracy while retaining the computational benefits of PINNs. Unlike conventional PINNs, which are trained to approximate (differential) equations, this method incorporates classical techniques, such as stress potentials, to satisfy certain physical laws. The result is a physics-enforced PINN that combines the precision of the Constitutive Equation Gap Method (CEGM) with the automatic differentiation and optimization frameworks characteristic of PINNs. Numerical comparisons reveal that the enforced PINN approach indeed achieves near-CEGM accuracy while preserving the efficiency advantages of PINNs. Validation through real experimental data demonstrates the ability of the method to accurately identify material properties and inclusion geometries in inhomogeneous samples. • A fast and accurate inverse strategy for inhomogeneous material identification. • Classical methods, such as stress potentials, can be combined with machine learning. • Enforcing physics phenomena is more accurate than teaching it to the neural-network. • Do not teach neural-networks things that are easily enforced.

Physics-informed neural networks (PINNs) have recently been developed as a novel solution approach for physical problems governed by partial differential equations (PDEs). Compared to purely data-driven methods, PINNs have the advantage of embedding physical constraints in the training process, thus increasing their reliability. Compared to traditional numerical methods for PDEs, PINNs have the advantage of being “meshless”; they are in general less accurate and more computationally expensive, but also more suitable to sparse-data assimilation and to inverse modelling, which is increasing their popularity in many scientific fields. However, hydraulic applications of PINNs in the context of free-surface flows are still in their infancy. In this work, the effectiveness of PINNs to model one-dimensional free-surface flows over non-horizontal bottom is tested. The governing PDEs are the shallow water equations (SWEs), which represent the mass and momentum conservation in free-surface flows. The inclusion of a spatially variable topography in a meshless method such as PINNs is not trivial. Here, the idea of solving the augmented system of SWEs with topography is exploited. Augmentation consists in treating the bed elevation as a conserved variable (together with water depth and unit discharge) and adding a fictitious equation to the system, which states that this variable is constant in time (i.e., its time derivative is null), while it can be variable in space (its space derivative is included in the bed slope source term). In this way, bed elevation can be easily provided with other initial conditions, and the fixed-bed constraint preserves its value in time. Different cases of unsteady flows with flat and non-flat bottom are considered, and the accuracy obtained using PINNs with augmented SWEs is checked by comparing PINNs predictions with analytical solutions. Results show that a fair accuracy for depth and velocity can be obtained, even for some challenging test cases such as the dam-break over a bottom step and the planar flow over a parabolic basin (Thacker’s test case). Moreover, it is shown that, if PINNs are applied to a case with horizontal bottom, for which topography is not strictly necessary, similar accuracy and computational time are obtained when PINNs solve standard SWEs or augmented SWEs. It can therefore be concluded that the augmentation of SWEs is a simple but promising strategy to deal with flows over complex bathymetries using PINNs, which paves the way for applications to flows over more realistic topographies.

Physics informed neural network (PINN) demonstrates powerful capabilities in solving forward and inverse problems of nonlinear partial differential equations (NLPDEs) through combining data-driven and physical constraints. In this paper, two PINN methods that adopt tanh and sine as activation functions, respectively, are used to study data-driven solutions and parameter estimations of a family of high order KdV equations. Compared to the standard PINN with the tanh activation function, the PINN framework using the sine activation function can effectively learn the single soliton solution, double soliton solution, periodic traveling wave solution, and kink solution of the proposed equations with higher precision. The PINN framework using the sine activation function shows better performance in parameter estimation. In addition, the experiments show that the complexity of the equation influences the accuracy and efficiency of the PINN method. The outcomes of this study are poised to enhance the application of deep learning techniques in solving solutions and modeling of higher-order NLPDEs.

Data assimilation (DA) refers to methodologies which combine data and underlying governing equations to provide an estimation of a complex system. Physics informed neural network (PINN) provides an innovative machine learning technique for solving and discovering the physics in nature. By encoding general nonlinear partial differential equations, which govern different physical systems such as fluid flows, to the deep neural network, PINN can be used as a tool for DA. Due to its nature that neither numerical differential operation nor temporal and spatial discretization is needed, PINN is straightforward for implementation and getting more and more attention in the academia. In this paper, we apply the PINN to several flow problems and explore its potential in fluid physics. Both the mesoscopic Boltzmann equation and the macroscopic Navier-Stokes are considered as physics constraints. We first introduce a discrete Boltzmann equation informed neural network and evaluate it with a one-dimensional propagating wave and two-dimensional lid-driven cavity flow. Such laminar cavity flow is also considered as an example in an incompressible Navier-Stokes equation informed neural network. With parameterized Navier-Stokes, two turbulent flows, one within a C-shape duct and one passing a bump, are studied and accompanying pressure field is obtained. Those examples end with a flow passing through a porous media. Applications in this paper show that PINN provides a new way for intelligent flow inference and identification, ranging from mesoscopic scale to macroscopic scale, and from laminar flow to turbulent flow.

Physics-informed neural networks modelling for systems with moving immersed boundaries: Application to an unsteady flow past a plunging foil

AsPINN: Adaptive symmetry-recomposition physics-informed neural networks

Fluid dynamics computations for tube-like geometries are crucial in biomedical evaluations of vascular and airways fluid dynamics. Physics-Informed Neural Networks (PINNs) have emerged as a promising alternative to traditional computational fluid dynamics (CFD) methods. However, vanilla PINNs often demand longer training times than conventional CFD methods for each specific flow scenario, limiting their widespread use. To address this, multi-case PINN approach has been proposed, where varied geometry cases are parameterized and pre-trained on the PINN. This allows for quick generation of flow results in unseen geometries. In this study, we compare three network architectures to optimize the multi-case PINN through experiments on a series of idealized 2D stenotic tube flows. The evaluated architectures include the ‘Mixed Network’, treating case parameters as additional dimensions in the vanilla PINN architecture; the “Hypernetwork”, incorporating case parameters into a side network that computes weights in the main PINN network; and the “Modes” network, where case parameters input into a side network contribute to the final output via an inner product, similar to DeepONet. Results confirm the viability of the multi-case parametric PINN approach, with the Modes network exhibiting superior performance in terms of accuracy, convergence efficiency, and computational speed. To further enhance the multi-case PINN, we explored two strategies. First, incorporating coordinate parameters relevant to tube geometry, such as distance to wall and centerline distance, as inputs to PINN, significantly enhanced accuracy and reduced computational burden. Second, the addition of extra loss terms, enforcing zero derivatives of existing physics constraints in the PINN (similar to gPINN), improved the performance of the Mixed Network and Hypernetwork, but not that of the Modes network. In conclusion, our work identified strategies crucial for future scaling up to 3D, wider geometry ranges, and additional flow conditions, ultimately aiming towards clinical utility.

SUMMARY In recent years, offshore tsunami observation networks equipped with ocean bottom pressure gauges (OBPGs), such as S-net, DONET and N-net, have been deployed around Japan, enabling real-time collection of high-quality tsunami data near the source. These networks make it possible to estimate the spatiotemporal variation of the tsunami wavefield using a data assimilation approach, and to predict coastal tsunamis from the initial or current tsunami wavefield. This study proposes a novel tsunami data assimilation method that uses physics-informed neural networks (PINNs) to estimate tsunami wavefields from the observed OBPG data. The neural network was optimized by minimizing the sum of the data loss, which quantifies discrepancies from the tsunami data, and the physical loss, which quantifies the satisfaction of the linear long wave equation. This was performed to ensure that the estimated results are consistent with both the observed data and the physics of tsunami propagation, even when there are limited observational data and significant noise. We first validated the effectiveness of the proposed method using synthetic S-net OBPG data from the 2011 Tohoku-oki earthquake ($M_\mathrm{w}$ 9.0) tsunami. The results confirmed that by using both data and physical constraints in the PINN optimization, the PINN could adequately assimilate the spatiotemporal distribution of the tsunami wavefield from OBPG data, even for predictions outside the network coverage area. The predicted tsunami waveforms at the coastal stations, computed from the estimated initial wavefield, showed good agreement with the actual waveforms. Next, we conducted an experiment using actual S-net OBPG data from the 2016 Fukushima-oki earthquake ($M_\mathrm{w}$ 6.9) tsunami. The initial tsunami source estimated by PINN was in good agreement with other studies based on waveform inversion, although the maximum source amplitude and maximum coastal tsunami heights were underestimated. We also conducted an experiment using N-net OBPG data from the 2024 Hyuganada earthquake ($M_\mathrm{w}$ 7.0) tsunami. The PINN could accurately estimate the initial tsunami source, even though the tsunami source of this event was located outside the N-net coverage area. Finally, we have shown that incorporating tsunami observations over time into the iterative optimization of the PINN model allows for accurate and efficient tsunami data assimilation.

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

Physics-informed neural networks (PINNs) have been employed as a new type of solver of partial differential equations (PDEs). However, PINNs suffer from two limitations that impede their further development. First, PINNs exhibit weak physical constraints that may result in unsatisfactory results for complex physical problems. Second, the differential operation using automatic differentiation (AD) in the loss function may contaminate backpropagated gradients hindering the convergence of neural networks. To address these issues and improve the ability of PINNs, this paper introduces a novel PINN, referred to as CV-PINN, based on control volumes with the collocation points as their geometric centers. In CV-PINN, the physical laws are incorporated in a reformulated loss function in the form of discretized algebraic equations derived by integrating the PDEs over the control volumes by means of the finite volume method (FVM). In this way, the physical constraints are transformed from a single local collocation point to a control volume. Furthermore, the use of algebraic discretized equations in the loss function eliminates the derivative terms and, thereby, avoids the differential operation using AD. To validate the performance of CV-PINN, several benchmark problems are solved. CV-PINN is first used to solve Poisson's equation and the Helmholtz equation in square and irregular domains, respectively. CV-PINN is then used to simulate the lid-driven cavity flow problem. The results demonstrate that CV-PINN can precisely predict the velocity distributions and the primary vortex. The numerical experiments demonstrate that enhanced physical constraints of CV-PINN improve its prediction performance in solving different PDEs.

The physics-informed neural network (PINN) is a promising approach in scientific computing. However, PINNs still face significant challenges, including high training costs, weak physical constraints, and difficulties in handling multi-scale problems. To address these challenges, this paper presents a PINN based on the finite volume method (FVM), referred to as FVM-PINN. FVM-PINN replaces the partial differential equations in the loss function with discretized equations derived using the FVM. Since the discretized equations exclude derivative terms and enforce conservation principles at more collocation points, FVM-PINN avoids the computationally intensive automatic differentiation operations in the complex computational graph and also imposes more rigorous physical constraints in the loss function. To demonstrate the performance of FVM-PINN, the forward and inverse problems of lid-driven square cavity flow are studied. The results show that FVM-PINN achieves a higher accuracy than PINN while requiring only one-tenth of the training time. Notably, it can predict the subtle, multi-scale behavior of lid-driven square cavity flow at higher Reynolds numbers, which outperforms PINN. This study also investigates the influence of different discretization schemes on the performance of FVM-PINN. It is shown that the schemes based on more physically rational profile assumptions and involving more surrounding points enhance the model's compliance with physical constraints, thereby improving the model's convergence performance and prediction accuracy. Additionally, the accuracy of the discretization schemes is also important for the accuracy of the model. These findings provide the guidelines for selecting appropriate discretization schemes when constructing the loss function of FVM-PINN for different flow problems.

• Developing physics-informed neural networks with scarce data. • Applying physics-informed neural networks for non-routine operations. • Incorporating heuristic physical knowledge into neural networks. • Improving predictive accuracy and model generalizability with meta-learning. • Removing complex correlations to predict twin roll press washer blockage. Extrapolation of process models beyond routine operations is challenging because of the complexity of chemical engineering unit operations, especially for which first-principles models may be unavailable or difficult to formulate. This study investigates to what extent a physics-informed neural network model of a twin roll press washer, incorporating only the generalized heuristic proportionality-based physical relationships that are available, can improve predictive accuracy under non-routine conditions compared to “conventional” data-driven neural network models. The methodology is applied to a case study on predicting roll speed in a twin roll press washer used in pulp and paper production, a key fault-indicating variable for which no established mechanistic or empirical correlations currently exist. To enhance model adaptability, meta-learning is used to treat physical parameters as trainable, allowing the model to adjust them during training and better align physics constraints with observed data. This approach eliminates the need for manual calibration of coefficients in parameterized differential equations, a step that is often impractical in industrial settings due to data scarcity and evolving process conditions. The proposed method achieved a mean squared error of 0.092 RPM 2 , a reduction of nearly 90% compared to purely data-driven models and 30% compared to a fixed-parameter physics-informed neural network model, without significantly increasing training time. The results reinforce the value of the physics-informed neural network modeling approach to process engineering applications and confirm the validity of the proposed novel meta-learning, simple relational physics-based approach.