Physics-informed neural networks for inverse problems in nano-optics and metamaterials

The central aim of this work is to investigate the effective integration of mechanistic knowledge and modern deep learning techniques for modelling and forecasting infectious disease dynamics. The research is driven by the need to bridge the interpretability and theoretical rigour of classical epidemic models with the expressive power and adaptability of data-driven approaches. By embedding physical principles into neural network architectures, this work develops Physics-Informed Deep Learning (PIDL) frameworks that address the limitations of both purely mechanistic and purely data-driven models, and demonstrates how this paradigm enables interpretable and effective modelling of infectious disease dynamics. In Chapter 3, this work presents a neural networks architecture informed by the Susceptible-Infectious-Recovered (SIR) model, framed within the paradigm of Physics-Informed Neural Networks (PINNs). This approach incorporates Ordinary Differential Equations (ODEs) into the learning objective, enabling the model to learn from observational data while being constrained by known epidemiological dynamics. A series of numerical experiments on synthetic data—generated using an extended Susceptible- Asymptomatic-Infectious-Recovered-Dead (SAIRD) model—demonstrates the robustness of the method. The model achieves superior forecasting performance compared to traditional numerical solvers using inferred parameters and effectively identifies key epidemiological quantities, even under limited data conditions. Application to COVID-19 reported data from Germany further validates the framework, showing that simplified physical priors can enhance both predictive accuracy and interpretability. This study highlights the feasibility of partial physics-guided learning as a practical approach for short-term epidemic forecasting and parameter identification. Chapter 4 builds on this foundation by extending the modelling framework to capture the spatial and temporal heterogeneity inherent in epidemic spread. A novel Causal Spatiotemporal Graph Neural Network (CSTGNN) hybrid framework is proposed, which integrates a Spatio-Contact SIR (SCSIR) model within a deep learning architecture. This framework captures human mobility-driven regional transmission dynamics through a spatiotemporal graph neural network that incorporates graph structure learning, temporal trend decomposition, and causal inference modules. The model is validated on longitudinal COVID-19 data from both China and Germany, spanning varied seasons and public health policy regimes. Experimental results demonstrate that the proposed approach consistently outperforms baseline models in predictive accuracy and gener- alisability, particularly in settings with strong spatial heterogeneity. Moreover, the framework yields interpretable epidemiological indicators—such as effective reproduction numbers and dynamic contact matrices—that align closely with real-world observations. This extension marks a significant advancement in unifying mechanistic epidemic modelling with scalable, data-driven spatiotemporal graph learning architectures. Taken together, these components establish a coherent trajectory in the development of PIDL methods for epidemiological modelling. Ranging from single-region deterministic frameworks based on ordinary differential equations to graph-based spatiotempo- ral architectures, this work proposes a unified methodology for incorporating domain-specific knowledge across multiple scales. The contributions lie not only in the formulation of hybrid models, but also in demonstrating how physical knowledge can be embedded into deep learning at varying depths: from soft constraint formulations, as exemplified by the PINNs, to deeply integrated mechanistic structures, as realised in the CSTGNN. These multi-level strategies collectively enhance model accuracy, robustness, and interpretability. From a theoretical perspective, this work provides a principled formulation of PIDL for infectious disease systems. The results offer strong evidence that mechanistic assumptions, even when approximate or incomplete, can serve as powerful inductive biases in neural network training. In doing so, they help to mitigate the common pitfalls of overfitting, instability, and poor generalisability typically observed in unconstrained data-driven models. Furthermore, this work explores various forms of physical integration—from hard-coded differential equation constraints to modularised structural priors—thereby contributing to the methodological diversity of the field. Algorithmically, this work incorporates a range of deep learning techniques tailored to spatiotemporal epidemic modelling, including temporal convolutional networks, graph-based representation learning, causal spatiotemporal graph neural networks, and specialised modules for temporal and spatial feature extraction. Through extensive experimentation, this work demonstrates that PIDL approaches consistently achieve accurate forecasting, robust parameter estimation, and valuable epidemiological insight, even in the face of real-world data limitations.

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

Comparison of Physics Informed Neural Networks and Finite Element Method Solvers for advection-dominated diffusion problems

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.

Physics-informed neural networks (PINNs) provide a new approach to solving partial differential equations (PDEs), while the properties of coupled physical laws present potential in surrogate modeling. However, the accuracy of PINNs in solving forward problems needs to be enhanced, and solving inverse problems relies on data samples. The smoothed finite element method (S-FEM) can obtain high-fidelity numerical solutions, which are easy to solve for the forward problems of PDEs, but difficult to solve for the inverse problems. To the best of the authors’ knowledge, there has been no prior research on coupling S-FEM and PINN. In this paper, a novel approach that couples S-FEM and PINN is proposed. The proposed approach utilizes S-FEM to synthesize high-fidelity datasets required for PINN inversion, while also improving the accuracy of data-independent PINN in solving forward problems. The proposed approach is applied to solve linear elastic and elastoplastic forward and inverse problems. The computational results demonstrate that the coupling of the S-FEM and PINN exhibits high precision and convergence when solving inverse problems, achieving a maximum relative error of 0.2% in linear elasticity and 5.69% in elastoplastic inversion by using approximately 10,000 data points. The coupling approach also enhances the accuracy of solving forward problems, reducing relative errors by approximately 2–10 times. The proposed coupling of the S-FEM and PINN offers a novel surrogate modeling approach that incorporates knowledge and data-driven techniques, enabling it to solve both forward and inverse problems associated with PDEs with high levels of accuracy and convergence.

Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

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.

The accurate modeling of water and heat transport in soils is crucial for both geo-environmental and geothermal engineering. Traditional modeling methods are problematic because they require well-defined boundaries and initial conditions. Recently, physics-informed neural networks (PINNs), which incorporate partial differential equations (PDEs) to solve forward and inverse problems, have attracted increasing attention in machine learning research. In this study, we applied PINNs to tackle hydraulic and thermal transport coupling forward problems in silty sands. A fully connected deep neural network was utilized for training. This neural network model leverages automatic differentiation to apply the governing equations as constraints, based on the mathematical approximations established by the neural network itself. We conducted forward problems and compared the solutions derived from PINNs with those from Finite Element Method (FEM) simulations. The forward problem results demonstrate the PINNs model’s capability in predicting hydraulic transport, heat transport, and thermal–hydraulic coupling in silty sands under various boundary conditions. The PINNs exhibited great performance in simulating the thermal–hydraulic coupling problem. The accuracy of the PINNs solutions shows its potential for simulation in geotechnical engineering.

Neural network-augmented differentiable finite element method for boundary value problems

Physics Informed Neural Networks (PINNs) have been rarely applied to solve multiphysics systems due to the inherent challenges in optimizing their complex loss functions, which typically incorporate multiple physics-based terms. This study presents a multistage PINN approach designed to efficiently solve coupled multiphysics systems with strong interdependencies. The multistage PINN progressively increases the complexity of the physical system being modeled, enabling more effective capture of coupling between different physics. The computational merits of this approach are demonstrated through two illustrative applications: prediction of asphalt aging and modeling of lid-driven cavity flow. Quantitative and qualitative comparisons with standard PINN and adaptive weight PINN approaches demonstrate the enhanced precision and computational efficiency of the proposed algorithm. The multistage PINN achieves a reduction in training time of more than 90% compared to standard PINNs while maintaining better alignment with the finite element method (FEM) solutions. The improvement in computational efficiency, coupled with enhanced accuracy, positions the multistage PINN as a powerful tool for addressing complex multiphysics problems across various engineering disciplines. The method’s ability to handle interactions between multiple physical processes, such as diffusion, chemical reactions, and fluid dynamics, makes it suitable for simulating long-term material behavior and complex fluid systems.

Physics-informed neural entangled-ladder network for inhalation impedance of the respiratory system

This dissertation investigates the use of energy functionals derived from the variational formulation of partial differential equations (PDEs) as a basis for training Physics-Informed Neural Networks (PINNs). The work begins by revisiting the role of PDEs in modeling physical and biological phenomena, emphasizing the importance of variational principles as a mathematical foundation for obtaining weak solutions and for the development of numerical methods such as the Finite Element Method (FEM). It also reviews essential concepts from Machine Learning, including statistical learning theory, supervised learning, deep neural networks, and the training process based on empirical risk minimization. In this context, the use of automatic differentiation emerges as a key tool for computing gradients efficiently in high-dimensional models. Building on these theoretical elements, we explore the Functional PINN (Fun-PINN), a model that replaces the traditional residual-based training with the minimization of an energy functional directly associated with the PDE, which reduces the order of derivatives required during training, leading to gains in computational efficiency. We also introduce the Self-Adaptive Functional PINN (SA-Fun-PINN), which dynamically adjusts the relative importance of the energy and boundary terms during training. Both models were evaluated against classical PINNs and Self-Adaptive PINNs (SA-PINNs) in numerical experiments designed to assess convergence, accuracy, and computational efficiency. Two test cases were considered: one with a smooth solution, based on Laplace\'s equation, and another with a more oscillatory profile, based on the Poisson equation, allowing performance to be analyzed under increasing levels of complexity. The Fun-PINNs achieved stable training dynamics and offered meaningful reductions in runtime, while the self-adaptive version further improved accuracy with lower computational cost compared to SA-PINNs. In particular, the functional approach demonstrated superior performance in the oscillatory case. However, considering Fun-PINNs as a whole, these experiments do not allow us to claim results superior to those obtained with PINNs; rather, they show that Fun-PINNs are a viable alternative that also yields good results. Overall, this work highlights how classical mathematical concepts such as variational formulations and energy minimization can be effectively integrated with modern machine learning techniques. The proposed functional models combine theoretical consistency with practical efficiency, establishing a promising direction for future research on more complex PDEs, including inverse problems and hybrid approaches that link PINNs with traditional numerical solvers.

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 (PINNs) represent a groundbreaking approach wherein neural networks (NNs) integrate model equations, such as Partial Differential Equations (PDEs), within their architecture. This innovation has become instrumental in solving diverse problem sets including PDEs, fractional equations, integral-differential equations, and stochastic PDEs. It's a versatile multi-task learning framework that tasks NNs with fitting observed data while simultaneously minimizing PDE residuals. This paper delves into the landscape of PINNs, aiming to delineate their inherent strengths and weaknesses. Beyond exploring the fundamental characteristics of these networks, this review endeavors to encompass a wider spectrum of collocation-based physics-informed neural networks, extending beyond the core PINN model. Variants like physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN) constitute pivotal aspects of this exploration. The study accentuates a predominant focus in research on tailoring PINNs through diverse strategies: adapting activation functions, refining gradient optimization techniques, innovating neural network structures, and enhancing loss function architectures. Despite the extensive applicability demonstrated by PINNs, surpassing classical numerical methods like Finite Element Method (FEM) in certain contexts, the review highlights ongoing opportunities for advancement. Notably, there are persisting theoretical challenges that demand resolution, ensuring the continued evolution and refinement of this revolutionary approach.

There is increasing interest in using ultrasound shear wave elasticity imaging to study tissues described as incompressible, transversely isotropic (ITI) materials, such as skeletal muscle. In silico modeling helps us predict and understand shear wave behavior in complex materials like the ITI model, which supports two shear polarizations with different, direction-dependent propagation speeds. Existing techniques, the finite element method (FEM) and Greens functions, are computationally expensive and generate large file sizes. Physics-informed neural networks (PINNs) is a relatively novel technique to solve partial differential equations and produces solutions that are compressed, analytic, and free of space-time discretization. Here, we solve the 3D wave equation for an ITI material using PINNs and show that solutions match FEM simulations to first order for material parameters based on skeletal muscle. Estimated shear wave speeds for the PINN and FEM solutions differed by an average of 4.7%. Unlike the FEM simulation, the PINN solution had no reflection artifacts at the boundaries. Second-order differences in frequency content and amplitude distribution suggest the need for further validation. PINNs can enable rapid exploration of the complex shear wave behavior in ITI materials and can be extended to different material models by adjusting the wave equation and initial conditions.