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High-fidelity computer simulations of childbirth remain prohibitively expensive and time consuming, making them impractical for guiding decision-making during obstetric emergencies. Cheap computer simulations that preserve the accuracy of high-fidelity models can be developed using surrogate modeling. Two common approaches to surrogate modeling are physics-based reduced order modeling (ROM) and machine learning (ML), with the latter gaining popularity as the scientific computing community seeks to leverage advances from other, mostly non-physics-based, computational strategies. Although ROM and ML have been compared for various problems, to our knowledge, such a comparison for simulations of vaginal deformations is currently missing. This study provides a baseline numerical comparison between methods from these two fundamentally different approaches. Since there are many methods falling into each modeling approach, to provide a fair and natural comparison, we select a basic model from each category, with each allowing (i) a straightforward implementation in commercial software packages, and (ii) use by practitioners with limited experience in the field. As a benchmark for the numerical comparison of the ROM and ML approaches, we use the finite element (FE) modeling of the ex vivo deformations of rat vaginal tissue subjected to inflation testing to study the effect of a pre-imposed tear. From the ROM strategies, we consider a simplified Galerkin ROM (G-ROM) that is based on the linearization of the underlying nonlinear equations. From the ML strategies, we select a feed-forward neural network to create mappings from constitutive model parameters and luminal pressure values to either the FE displacement history (in which case we denote the resulting model ML) or the proper orthogonal decomposition (POD) coefficients of the displacement history (in which case we denote the resulting model POD-ML). The numerical investigation of G-ROM, ML, and POD-ML takes place in the reconstructive regime. The numerical results show that the G-ROM outperforms the ML model in terms of offline central processing unit (CPU) time for model training, online CPU time required to generate approximations, and relative error with respect to the FE models. The G-ROM achieves superior error performance to the best ML model with 11 POD basis functions. With higher-dimensional POD bases, the G-ROM achieves a relative error 3 orders of magnitude lower than that of the best ML model with an online CPU time still on the same order of magnitude as the best ML model. The POD-ML model improves on the speed performance of the ML, having online CPU times comparable to those of the G-ROM given the same size of POD bases. However, the POD-ML model does not improve on the error performance of the ML and is still outperformed by the G-ROM for POD bases of size greater than 11. This baseline numerical investigation serves as a starting point for future computer simulations that consider state-of-the-art G-ROM and ML strategies, and the in vivo geometry, boundary conditions, and material properties of the human vagina, as well as their changes during labor. [ABSTRACT FROM AUTHOR]

We propose a strongly-constrained physics-informed neural network method with the derivative information to more suitably predict soliton dynamics of the nonlinear partial differential equation. We integrate more constrained physical information into the neural network, namely we add the derivative information of solutions for the nonlinear partial differential equation to the neural network structure, and introduce adaptive weight and flexible learning rate, to accelerate the convergence of the loss function of the network. As an example, we use this new method to predict a variety of soliton solutions of higher-order nonlinear Schrödinger equation. Moreover, we introduce the derivative of soliton solution to further reflect the stability of the data set. Results show that this new method can achieve the prediction in a wider solution area than the traditional method. Compared with results in previous literatures, the accuracy of the strongly-constrained physics-informed neural network method in predicting soliton dynamics is improved by 2-12 times. Therefore, this neural network is a forward-looking technology for scientific computing and automatic modeling. [ABSTRACT FROM AUTHOR]

We present a parallel data structure for the discretization of partial differential equations which is based on distributed point objects and which enables the flexible, transparent, and efficient realization of conforming, nonconforming, and mixed finite elements. This concept is realized for elliptic, parabolic and hyperbolic model problems, and sample applications are provided by a tutorial complementing a lecture on scientific computing. The corresponding open-source software is based on this parallel data structure, and it supports multilevel methods on nested meshes and 2D and 3D as well as in space–time. Here, we present generic results on porous media applications including multilevel preconditioning and multilevel Monte Carlo methods for uncertainty quantification. [ABSTRACT FROM AUTHOR]

deal.II is a state-of-the-art finite element library focused on generality, dimension-independent programming, parallelism, and extensibility. Herein, we outline its primary design considerations and its sophisticated features such as distributed meshes, h p -adaptivity, support for complex geometries, and matrix-free algorithms. But deal.II is more than just a software library: It is also a diverse and worldwide community of developers and users, as well as an educational platform. We therefore also discuss some of the technical and social challenges and lessons learned in running a large community software project over the course of two decades. [ABSTRACT FROM AUTHOR]

Kaskade 7 is a finite element toolbox for the solution of stationary or transient systems of partial differential equations, aimed at supporting application-oriented research in numerical analysis and scientific computing. The library is written in C++ and is based on the Dune interface. The code is independent of spatial dimension and works with different grid managers. An important feature is the mix-and-match approach to discretizing systems of PDEs with different ansatz and test spaces for all variables. We describe the mathematical concepts behind the library as well as its structure, illustrating its use at several examples on the way. [ABSTRACT FROM AUTHOR]

Abstract A modified multilevel algorithm for solving the excessive storage requirements and ill-conditioning encountered in the boundary-type discretization method is proposed. The modified multilevel algorithm is an extension of the modified dual-level algorithm from dual levels to multiple levels. The method is a kernel-independent method. The core idea is the layer-by-layer calculation and then layer-by-layer correction. Making use of a multilevel structure, the original sparse matrix of the modified dual-level algorithm breaks down into a series of smaller sparse matrices corresponding to different fine meshes. The final matrix to be solved is hereby transformed to a series of smaller sparse matrices instead of a fully-populated matrix. The preconditioning effect originating from the recursive computations among the coarse mesh and fine meshes constitutes its core competitive attribute. The method evaluates far-field contributions only by the coarse mesh and uses a gradual approach to evaluate the near-field contributions. The storage requirements and computing complexity are hereby further reduced significantly. [ABSTRACT FROM AUTHOR]