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Solving partial differential equations (PDEs) is a required step in the simulation of natural and engineering systems. The associated computational costs significantly increase when exploring various scenarios, such as changes in initial or boundary conditions or different input configurations. This study proposes the Variational Physics-Informed Neural Operator (VINO), a deep learning method designed for solving PDEs by minimizing the energy formulation of PDEs. This framework can be trained without any labeled data, resulting in improved performance and accuracy compared to existing deep learning methods and conventional PDE solvers. By discretizing the domain into elements, the variational format allows VINO to overcome the key challenge in physics-informed neural operators, namely the efficient evaluation of the governing equations for computing the loss. Comparative results demonstrate VINO's superior performance, especially as the mesh resolution increases. As a result, this study suggests a better way to incorporate physical laws into neural operators, opening a new approach for modeling and simulating nonlinear and complex processes in science and engineering.

In recent years, Artificial intelligence (AI) has become ubiquitous, empowering various fields, especially integrating artificial intelligence and traditional science (AI for Science: Artificial intelligence for science), which has attracted widespread attention. In AI for Science, using artificial intelligence algorithms to solve partial differential equations (AI for PDEs: Artificial intelligence for partial differential equations) has become a focal point in computational mechanics. The core of AI for PDEs is the fusion of data and partial differential equations (PDEs), which can solve almost any PDEs. Therefore, this article provides a comprehensive review of the research on AI for PDEs, summarizing the existing algorithms and theories. The article discusses the applications of AI for PDEs in computational mechanics, including solid mechanics, fluid mechanics, and biomechanics. The existing AI for PDEs algorithms include those based on Physics-Informed Neural Networks (PINNs), Deep Energy Methods (DEM), Operator Learning, and Physics-Informed Neural Operator (PINO). AI for PDEs represents a new method of scientific simulation that provides approximate solutions to specific problems using large amounts of data, then fine-tuning according to specific physical laws, avoiding the need to compute from scratch like traditional algorithms. Thus, AI for PDEs is the prototype for future foundation models in computational mechanics, capable of significantly accelerating traditional numerical algorithms.

This study investigates different Scientific Machine Learning (SciML) approaches for the analysis of functionally graded (FG) porous beams and compares them under a new framework. The beam material properties are assumed to vary as an arbitrary continuous function. The methods consider the output of a neural network/operator as an approximation to the displacement fields and derive the equations governing beam behavior based on the continuum formulation. The methods are implemented in the framework and formulated by three approaches: (a) the vector approach leads to a Physics-Informed Neural Network (PINN), (b) the energy approach brings about the Deep Energy Method (DEM), and (c) the data-driven approach, which results in a class of Neural Operator methods. Finally, a neural operator has been trained to predict the response of the porous beam with functionally graded material under any porosity distribution pattern and any arbitrary traction condition. The results are validated with analytical and numerical reference solutions. The data and code accompanying this manuscript will be publicly available at https://github.com/eshaghi-ms/DeepNetBeam.

AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov-Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov-Arnold-Informed Neural Network (KINN) for solving forward and inverse problems. We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP regarding accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.

This paper presents the development of a complete CAD-compatible framework for structural shape optimization in 3D. The boundaries of the domain are described using NURBS while the interior is discretized with B\'ezier tetrahedra. The tetrahedral mesh is obtained from the mesh generator software Gmsh. A methodology to reconstruct the NURBS surfaces from the triangular faces of the boundary mesh is presented. The description of the boundary is used for the computation of the analytical sensitivities with respect to the control points employed in surface design. Further, the mesh is updated at each iteration of the structural optimization process by a pseudo-elastic moving mesh method. In this procedure, the existing mesh is deformed to match the updated surface and therefore reduces the need for remeshing. Numerical examples are presented to test the performance of the proposed method. The use of the movable mesh technique results in a considerable decrease in the computational effort for the numerical examples.

Flexoelectricity - the generation of electric field in response to a strain gradient - is a universal electromechanical coupling, dominant only at small scales due to its requirement of high strain gradients. This phenomenon is governed by a set of coupled fourth-order partial differential equations (PDEs), which require $C^1$ continuity of the basis in finite element methods for the numerical solution. While Isogeometric analysis (IGA) has been proven to meet this continuity requirement due to its higher-order B-spline basis functions, it is limited to simple geometries that can be discretized with a single IGA patch. For the domains, e.g., architected materials, requiring more than one patch for discretization IGA faces the challenge of $C^0$ continuity across the patch boundaries. Here we present a discontinuous Galerkin method-based isogeometric analysis framework, capable of solving fourth-order PDEs of flexoelectricity in the domain of truss-based architected materials. An interior penalty-based stabilization is implemented to ensure the stability of the solution. The present formulation is advantageous over the analogous finite element methods since it only requires the computation of interior boundary contributions on the boundaries of patches. As each strut can be modeled with only two trapezoid patches, the number of $C^0$ continuous boundaries is largely reduced. Further, we consider four unique unit cells to construct the truss lattices and analyze their flexoelectric response. The truss lattices show a higher magnitude of flexoelectricity compared to the solid beam, as well as retain this superior electromechanical response with the increasing size of the structure. These results indicate the potential of architected materials to scale up the flexoelectricity to larger scales, towards achieving universal electromechanical response in meso/macro scale dielectric materials.

Modeling fracture is computationally expensive even in computational simulations of two-dimensional problems. Hence, scaling up the available approaches to be directly applied to large components or systems crucial for real applications become challenging. In this work. we propose domain decomposition framework for the variational physics-informed neural networks to accurately approximate the crack path defined using the phase field approach. We show that coupling domain decomposition and adaptive refinement schemes permits to focus the numerical effort where it is most needed: around the zones where crack propagates. No a priori knowledge of the damage pattern is required. The ability to use numerous deep or shallow neural networks in the smaller subdomains gives the proposed method the ability to be parallelized. Additionally, the framework is integrated with adaptive non-linear activation functions which enhance the learning ability of the networks, and results in faster convergence. The efficiency of the proposed approach is demonstrated numerically with three examples relevant to engineering fracture mechanics. Upon the acceptance of the manuscript, all the codes associated with the manuscript will be made available on Github., Comment: 9 Pages, 8 Figures

The phase-field fracture free-energy functional is non-convex with respect to the displacement and the phase field. This results in a poor performance of the conventional monolithic solvers like the Newton-Raphson method. In order to circumvent this issue, researchers opt for the alternate minimization (staggered) solvers. Staggered solvers are robust for the phase-field based fracture simulations as the displacement and the phase-field sub-problems are convex in nature. Nevertheless, the staggered solver requires very large number of iterations (of the order of thousands) to converge. In this work, a robust monolithic solver is presented for the phase-field fracture problem. The solver adopts a fracture energy-based arc-length method and an adaptive under-relaxation scheme. The arc-length method enables the simulation to overcome critical points (snap-back, snap-through instabilities) during the loading of a specimen. The use of an under-relaxation scheme stabilizes the solver by preventing the divergence due to an ill-behaving stiffness matrix. The efficiency of the proposed solver is further amplified with an adaptive mesh refinement scheme based on PHT-splines within the framework of isogeometric analysis. The numerical examples presented in the manuscript demonstrates the efficacy of the solver. All the codes and data-sets accompanying this work will be made available on GitHub (https://github.com/rbharali/IGAFrac).

Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In general, in order to solve PDEs that represent real systems to an acceptable degree, analytical methods are usually not enough. One has to resort to discretization methods. For engineering problems, probably the best known option is the finite element method (FEM). However, powerful alternatives such as mesh-free methods and Isogeometric Analysis (IGA) are also available. The fundamental idea is to approximate the solution of the PDE by means of functions specifically built to have some desirable properties. In this contribution, we explore Deep Neural Networks (DNNs) as an option for approximation. They have shown impressive results in areas such as visual recognition. DNNs are regarded here as function approximation machines. There is great flexibility to define their structure and important advances in the architecture and the efficiency of the algorithms to implement them make DNNs a very interesting alternative to approximate the solution of a PDE. We concentrate in applications that have an interest for Computational Mechanics. Most contributions that have decided to explore this possibility have adopted a collocation strategy. In this contribution, we concentrate in mechanical problems and analyze the energetic format of the PDE. The energy of a mechanical system seems to be the natural loss function for a machine learning method to approach a mechanical problem. As proofs of concept, we deal with several problems and explore the capabilities of the method for applications in engineering.

We present a new physics informed neural network (PINN) algorithm for solving brittle fracture problems. While most of the PINN algorithms available in the literature minimize the residual of the governing partial differential equation, the proposed approach takes a different path by minimizing the variational energy of the system. Additionally, we modify the neural network output such that the boundary conditions associated with the problem are exactly satisfied. Compared to conventional residual based PINN, the proposed approach has two major advantages. First, the imposition of boundary conditions is relatively simpler and more robust. Second, the order of derivatives present in the functional form of the variational energy is of lower order than in the residual form. Hence, training the network is faster. To compute the total variational energy of the system, an efficient scheme that takes as input a geometry described by spline based CAD model and employs Gauss quadrature rules for numerical integration has been proposed. Moreover, we utilize the concept of transfer learning to obtain the crack path in an efficient manner. The proposed approach is used to solve four fracture mechanics problems. For all the examples, results obtained using the proposed approach match closely with the results available in the literature. For the first two examples, we compare the results obtained using the proposed approach with the conventional residual based neural network results. For both the problems, the proposed approach is found to yield better accuracy compared to conventional residual based PINN algorithms., Comment: 21 pages

This paper presents a novel methodology of local adaptivity for the frequency-domain analysis of the vibrations of Reissner-Mindlin plates. The adaptive discretization is based on the recently developed Geometry Independent Field approximaTion (GIFT) framework, which may be seen as a generalisation of the Iso-Geometric Analysis (IGA). Within the GIFT framework, we describe the geometry of the structure exactly with NURBS (Non-Uniform Rational B-Splines), whilst independently employing Polynomial splines over Hierarchical T-meshes (PHT)-splines to represent the solution field. The proposed strategy of local adaptivity, wherein a posteriori error estimators are computed based on inexpensive hierarchical $h-$refinement, aims to control the discretisation error within a frequency band. The approach sweeps from lower to higher frequencies, refining the mesh appropriately so that each of the free vibration mode within the targeted frequency band is sufficiently resolved. Through several numerical examples, we show that the GIFT framework is a powerful and versatile tool to perform local adaptivity in structural dynamics. We also show that the proposed adaptive local $h-$refinement scheme allows us to achieve significantly faster convergence rates than when using a uniform $h-$refinement.

A challenge in isogeometric analysis is constructing analysis-suitable volumetric meshes which can accurately represent the geometry of a given physical domain. In this paper, we propose a method to derive a spline-based representation of a domain of interest from voxel-based data. We show an efficient way to obtain a boundary representation of the domain by a level-set function. Then, we use the geometric information from the boundary (the normal vectors and curvature) to construct a matching C1 representation with hierarchical cubic splines. The approximation is done by a single template and linear transformations (scaling, translations and rotations) without the need for solving an optimization problem. We illustrate our method with several examples in two and three dimensions, and show good performance on some standard benchmark test problems.

We present an efficient approach of Finite Element Method (FEM)-based nonrigid image registration, in which the spatial transformation is constructed using truncated hierarchical B-splines (THB-splines). The image registration framework minimizes an energy functional using an FEM-based method and thus involves solving a large system of linear equations. This framework is carried out on a set of successively refined grids. However, due to the increased number of control points during subdivision, large linear systems are generated which are generally demanding to solve. Instead of using uniform subdivision, an adaptive local refinement scheme is carried out, only refining the areas of large change in deformation of the image. By incorporating the key advantages of THB-spline basis functions such as linear independence, partition of unity and reduced overlap into the FEM-based framework, we improve the matrix sparsity and computational efficiency. The performance of the proposed method is demonstrated on 2D synthetic and medical images.