Potential Solver Research Articles

Self-propulsion performance prediction in calm water based on RANS/TEBEM coupling method

This paper proposes a hybrid RANS/TEBEM method to conduct self-propulsion prediction in calm water with accuracy and improve its efficiency by means of larger time step. In the coupling procedure, the rotating propeller is represented by time-averaged, spatial inhomogeneous body force field obtained from the potential flow solver based on Taylor Expansion Boundary Element Method (TEBEM) in RANS simulation. Effective wake is evaluated by subtracting induced velocities computed by potential solver from total velocities in RANS solver at a coupling plane upstream of propeller plane. The influence of coupling plane position and grid density on the numerical results is also investigated. Through KCS and KVLCC2 cases, it is shown that the predicted value of propeller revolution speed is within 4% of the experimental value, verifying the robustness and reliability of this method.

Acceleration of a wave-structure interaction solver by the Parareal method

Potential flow theory-based solvers are commonly used in ocean engineering to investigate the interactions between ocean waves and floating bodies. Depending on assumptions, several methods have been proposed. Among them, the Weak-Scatterer method is an interesting trade-off in the sense that this approach is not limited in theory by the small wave amplitudes and small body motions assumptions of linear methods. Moreover, this approach is in practice more stable than the fully non-linear methods. An implementation of the Weak-Scatterer method is the WS-CN code (Letournel, 2015; Chauvigné, 2016; Wuillaume, 2019).The computational time of the WS-CN code which is considered in the present study is relatively long for engineering purposes. In order to reduce it, the present paper presents an implementation of the Parareal method in the WS-CN code. The Parareal method is an algorithm for parallelizing a simulation in time that can accelerate the complete simulation (Lions, 2001) . This is a key difference in comparison to other acceleration techniques which have been studied in the literature (e.g. the Fast Multipole Method (FMM), the precorrected Fast Fourier Transform (pFFT) method, … ). To the authors’ knowledge, the present study is the first to couple the Parareal method to a potential flow theory-based wave-structure interaction solver. It is shown that the method can significantly reduce the computational time for small wave steepness, but that the performance decreases rapidly with increasing steepness.

Space Weather with Quantified Uncertainties: Improving Space Weather Predictions with Data-Driven Models of the Solar Atmosphere and Inner Heliosphere

To address Objective II of the National Space Weather Strategy and Action Plan “Develop and Disseminate Accurate and Timely Space Weather Characterization and Forecasts” and US Congress PROSWIFT Act 116–181, our team is developing a new set of open-source software that would ensure substantial improvements of Space Weather (SWx) predictions. On the one hand, our focus is on the development of data-driven solar wind models. On the other hand, each individual component of our software is designed to have accuracy higher than any existing SWx prediction tools with a dramatically improved performance. This is done by the application of new computational technologies and enhanced data sources. The development of such software paves way for improved SWx predictions accompanied with an appropriate uncertainty quantification. This makes it possible to forecast hazardous SWx effects on the space-borne and ground-based technological systems, and on human health. Our models include (1) a new, open-source solar magnetic flux model (OFT), which evolves information to the back side of the Sun and its poles, and updates the model flux with new observations using data assimilation methods; (2) a new potential field solver (POT3D) associated with the Wang–Sheeley–Arge coronal model, and (3) a new adaptive, 4-th order of accuracy solver (HelioCubed) for the Reynolds-averaged MHD equations implemented on mapped multiblock grids (cubed spheres). We describe the software and results obtained with it, including the application of machine learning to modeling coronal mass ejections, which makes it possible to improve SWx predictions by decreasing the time-of-arrival mismatch. The tests show that our software is formally more accurate and performs much faster than its predecessors used for SWx predictions.

Development and application of a mesh generator intended for unsteady vortex-lattice method simulations of wind turbines and wind farms

Abstract. In the last decades, the unsteady vortex-lattice method (UVLM) has gained a lot of acceptance to study large onshore–offshore wind turbines (WTs). Furthermore, and due to the development of more powerful computers, parallelization strategies, and algorithms like the fast multipole method, it is possible to use vortex-based methods to analyze and simulate wind farms (WFs). However, UVLM-based solvers require structured meshes, which are generally very tedious to build using classical mesh generators, such as those utilized in the context of finite element methods (FEMs). Wind farm meshing is further complicated by the large number of design parameters associated with the wind turbine (coning angle, tilt angle, blade shape, etc.), farm layout, modeling of the terrain topography (for onshore WFs), and modeling of the sea level surface (for offshore WFs), which makes the use of FEM-oriented meshing tools almost inapplicable. In the literature there is a total absence of meshing tools when it comes to building aerodynamic grids of WTs and WFs to be used along with UVLM-based solvers. Therefore, in this work, we present a detailed description of the geometric modeling and computational implementation of an interactive UVLM-oriented mesh generator, named UVLMeshGen, developed entirely in MATLAB® and easily adaptable to GNU OCTAVE, for wind turbines and onshore–offshore wind farms. The meshing tool developed here consists of (i) a geometric processor in charge of designing and discretizing an entire wind farm and (ii) an independent module in charge of computing the kinematics for the entire WF. The output data provided by the UVLMeshGen consist of nodal coordinates and connectivity arrays, making it especially attractive and useful to be used by other flow potential solvers using vortices, sources and sinks, or dipoles/doublets, among others. The work is completed by providing a series of aerodynamic results related to WTs and WFs to show the capabilities of the mesh generator, without going into detailed discussions of wind turbine aerodynamics, which are not the focus of this paper. The meshing tool developed here is freely available under a Creative Commons Attribution 4.0 International License (Roccia, 2023).

Towards High-Performance Linear Potential Flow BEM Solver with Low-Rank Compressions

The interaction of water waves with floating bodies can be modeled with linear potential flow theory and numerically solved with the boundary element method (BEM). This method requires the construction of dense matrices and the resolution of the corresponding linear systems. The cost of this method in terms of time and memory grows at least quadratically with the size of the mesh, and the resolution of large problems (such as large farms of wave energy converters) can, thus, be very costly. Approximating some blocks of the matrix with data-sparse matrices can limit this cost. While matrix compression with low-rank blocks has become a standard tool in the larger BEM community, the present paper provides its first application (to our knowledge) to linear potential flows. In this paper, we assess how efficiently low-rank blocks can approximate interaction matrices between distant meshes when using the Green function of linear potential flow. Due to the complexity of this Green function, a theoretical study is difficult, and numerical experiments are used to test the approximation method. Typical results on large arrays of floating bodies show that 99% of the accuracy can be reached with 10% of the coefficients of the matrix.

Proposição de Problemas: entendimentos

Abstract The problem posing has gained prominence in curriculum documents and current research, in which many questions are still open, and are not manifested clearly and coherently in school practices. This article aims to identify and establish understandings for the terms formulation, elaboration, creation, and problem posing to support the context of Brazilian research and practice in the field of Mathematics Education. To this end, a literature review is presented, resuming historical aspects and constituting a theoretical reference about the implications on the context of mathematics learning and students' comprehensive education. Then, based on the results obtained from a systematic study review and moving forward in the constitution of a theoretical body knowledge, we present the meanings attributed to these expressions in several Brazilian and international studies and then present the definitions we assume: we understand that problem creation involves the processes of problem formulation and elaboration and it is inserted in the problem posing which advances to present the created problem to a potential solver. The relevance of these definitions covers both contexts of educational practices that contemplate the association with problem solving and research in Mathematics Education, with conditions and reflections on the collection and analysis of data, in order to deepen the understanding of specific aspects related to problem posing to be considered in future studies.

Proficiency Level and Challenges of Grade 11 Public School Students on Contemporary Issues

Contemporary issues is essential for K-12 learners to develop their awareness of the social conditions as potential problem solvers. Hence, proficiency assessment is necessary to check their competency acquisition. This study aims to analyze the Grade 10 students’ proficiency and challenges on contemporary issues among public schools in Central Philippines in the importance of studying contemporary issues, political rights, human rights, and educational rights relative to sex and strand. Utilizing a descriptive-comparative approach, a validated and reliability tested questionnaire and checklist were administered to 261 students using beginning, developing, approaching proficiency, proficiency, and advanced responses. In data analysis, mean, standard deviation, frequency count, percentage, rank, Mann Whitney, and Kruskal Wallis were employed. Generally, they have approaching proficiency in all areas indicating that the competencies were not fully acquired. With this, improvement on instruction is critical. Comparatively, there was a difference when grouped to sex and strand where females and TVL students rated higher proficiency than their counterparts. Meanwhile, they are challenged on their motivation to learn, traditional teaching strategies, time management in accomplishing tasks, inadequate learning facilities, and insufficient interactive learning opportunities. Given the results, these imply the importance of improving the teachers’ instructional strategies in ensuring the students’ quality proficiency in Contemporary issues.

OSSP-PTA: An Online Stochastic Stepping Policy for PTA on Reinforcement Learning

DC analysis is essential and still quite challenging in large-scale nonlinear circuit simulation. Pseudo transient analysis (PTA) is a widely-used and has great potential solver in the industry. However, the PTA convergence and simulation efficiency is still seriously affected by its stepping policy. This paper proposes an online stochastic stepping policy (OSSP) for PTA based on deep reinforcement learning (DRL). To achieve better policy evaluation and stronger stepping exploration ability, the dual soft Actor-Critic agents work with the proposed valuation splitting and online momental scaling, enabling our OSSP to intelligently encode PTA iteration status and online further adjust forward and backward time-step size for unseen test circuits without human intervention and domain knowledge, trained solely by RL from self-search. Our public sample buffer and priority sampling are also introduced to overcome the sparsity and imbalance of sample data. Numerical examples demonstrate that the proposed OSSP achieves a significant efficiency speedup (up to 47.0X less NR iterations) and convergence enhancement on unseen test circuits compared with the previous iter-based and SER-based stepping methods, in just one stepping iteration.

210Pb-dating of recent sediments with the χ-mapping CF and CSAR models. On the attractors

Conventional 210Pb-dating models use assumptions on sedimentary conditions that allow for analytical formulations. The novel χ-mapping models use numerical methods to generate and test a large number (∼106) of potential solvers. Empirical data (excess 210Pb vs. mass depth profile) serve to attract the solver that minimizes the χ function (the attractor), and it has been assumed that it also defines the most likely chronology. This work aims to test this assumption in a deep way. In synthetic and varved sediments, the performance of each solver can be quantified through a parameter ξa accounting for the deviation of the model and the true ages. This work studies the complex relationships between χ and ξa using the constant flux (χ-CF) and the constant sediment accumulation rate (CSAR) models, which operate in a parametric 3D space. The full mapping of the 3D χ function serves to find the absolute minimum, for the graphical representation of the complex topology of the attractors, which is model-specific, and for plotting clouds of chronological lines from solvers with varying χ values. The minimum value of ξa (the best chronology) is achieved for a wide range of χ values, including the region of the absolute minimum. In complex cases, tiny changes in χ can result in quite different chronologies. Alternative attractors that include a reference date and an objective function are studied. The results provide guidelines for strengthening the 210Pb-based chronologies.

210Pb-based dating of recent sediments with χ-mapping versions of the CFCS, CIC, CF and TERESA models

The 210Pb-based method allows for absolute age determination in recent sediments on the centennial scale. The most common sedimentary scenarios involve ideal deposition without any further redistribution. The conventional models apply to such conditions by adding specific assumptions that allow analytical formulations: constant flux (CF), constant initial concentration (CIC), and constant flux with constant sedimentation (CFCS). The χ-mapping models are presented as a novel methodological approach. They generate a large number (105 to 106) of potential solvers, consisting of sets of n (the number of slices in the core) values of initial activity concentrations, A0, and sedimentation rates, w. Empirical data serve to attract the solver that minimizes a χ function after encountering the best sorting along the core of the A0 and w values. The specific assumptions of the above models are reformulated in terms of the statistical distributions of A0 and w. The performance of the χ-mapping versions is demonstrated with a set of cores from the literature with independent chronologies. The main advantages compared to known analytical formulations are: i) χ-mapping provides a more robust fitting method. ii) It overcomes the problem of age reversals often found in the CIC model. iii) The CF model successfully applies to cores with unsteady and/or incomplete inventories. iv) The TERESA model (time estimates from random entries of sediments and activities), without analytical counterpart, is the only model that can manage continuous trends of change in the fluxes of unsupported 210Pb. The work discusses the conditions for compensating model errors in the chronologies, and the use of the palaeorecords of fluxes, A0 and w. The methodology can be extended to other assumptions (models) and provides novel and insightful views on the 210Pb-based dating of recent sediments.

Simulation of Dawn‐To‐Dusk Electric Field in the Jovian Inner Magnetosphere via Region 2‐Like Field‐Aligned Current

AbstractThe presence of the dawn‐to‐dusk electric field of about 4 mV/m in the Jovian inner magnetosphere and its response to the enhancement of the solar wind dynamic pressure are still a mystery of the rotation‐dominated Jovian magnetosphere. Previous studies have suggested that magnetosphere‐ionosphere (M‐I) coupling via Region 2‐like (R2‐like) field‐aligned current (FAC) could be the origin of the Jovian dawn‐to‐dusk electric field. This study investigates whether the dawn‐to‐dusk electric field is formed from this scenario by using a Jovian ionosphere model and a two‐dimensional ionospheric potential solver. Our results show that the dawn‐dusk asymmetry in the ionospheric potential form even at middle latitudes and that the dawn‐to‐dusk electric field is induced in the inner magnetosphere if the electric potential is mapped to the magnetospheric equatorial plane. Around the Io orbit, the calculated electric field strength for the ionosphere without meteoroid influx is too large, 200 mV/m at dawn and 88 mV/m at dusk. One of the solutions is to consider long‐lived meteoric ions in the Jovian ionosphere, which reduce the electric field strength to 15 mV/m at dawn and 12 mV/m at dusk. The model also shows that the electric field strength increases with the intensity of R2‐like FAC, consistent with its response to the solar wind dynamic pressure observed by the Hisaki satellite.

Modeling E‐Region Artificial Periodic Inhomogeneity

AbstractArtificial periodic inhomogeneity or API experiments were conducted at the HAARP facility in Gakona, Alaska, in October 2022. The experiments concentrated on measuring ionospheric irregularities induced in the E‐region. The irregularities exhibited characteristics regarding their occurrence altitudes, rise and fall times, and Doppler shifts comparable to results from experiments conducted previously at HAARP and elsewhere. The irregularities also occurred in discrete altitude bands. Seeking to quantify these results, we constructed a simple, one‐dimensional fluid model which includes the effects of HF wave heating (direct and indirect) together with electron and ion cooling and thermal conduction, ion production, loss, and diffusion. Critically, the model includes a potential solver and can represent the ambipolar electric field. The model produced API irregularities in three distinct altitude bands which decayed according to the ambipolar diffusion rate.

Does Open Innovation Open Doors for Underrepresented Groups to Contribute to Technology Innovation?: Evidence from a Space Robotics Challenge

Diversity, equity, and inclusion (DEI) are increasingly being recognized as important policy goals for organizations across government and the industry. Improved DEI has been linked to both substantive improvement in innovation performance and societal good. However, despite a stated emphasis on DEI, progress has not kept up with aspirations. One indirect policy approach that holds promise is wider adoption of Open Innovation (OI) as part of an innovation toolkit. Proponents contend that OI reduces barriers to entry and garners productive contributions from diverse contributors. While there is anecdotal support for the diversifying potential of OI, so far, there is a dearth of empirical evidence connecting OI to DEI with consideration of performance outcomes, beyond `winners´. To study this link directly, this article leverages data from a previously conducted unique field experiment that explicitly tracked the population of potential solvers and their performance on a National Aeronautics and Space Administration (NASA) space robotics problem. We found that while OI attracted different solvers than the reference internal workforce, there was important variation in both the extent and direction of the observed differences, with respect to attributes of DEI. For instance, OI attracted proportionally fewer female solvers than the already male-dominated space workforce; and that proportion decreased further among solvers providing quality solutions. On the other hand, OI proved effective at granting access to an international pool of young professionals with potentially novel perspectives. Overall, our findings suggest OI can be an effective tool for achieving some diversity policy goals, but it is not well-suited for achieving all stated aspects of diversity. Therefore, we suggest a more targeted approach to matching the opportunities for OI to achieve specific policy objectives.

Optimization of the Hydrodynamic Performance of a Wave Energy Converter in an Integrated Cylindrical Wave Energy Converter-Type Breakwater System

AbstractWave energy converters (WECs) are built to extract wave energy. However, this kind of device is still expensive for commercial utilization. To cut down the cost of WECs by sharing the construction cost with breakwaters, an integrated cylindrical WEC-type breakwater system that includes a cylindrical WEC array in front of a very long breakwater is proposed to extract wave energy and attenuate incident waves. This paper aims to optimize the performance of the integrated cylindrical WEC-type breakwater system. A computational fluid dynamics tool, openfoam®, and a potential flow theory-based solver, HAMS®, are utilized. openfoam® provides viscosity corrections to a modified version of HAMS® in order to accurately and efficiently predict the integrated system’s performance. Parametric studies are conducted to optimize the integrated system, and a novel setup with an extra arc structure is found to significantly improve the performance of the integrated system.

Physics and Accuracy of Dual-Solver Simulations of Rotors in Ground Effect

To characterize the interactions between rotorcraft and ground obstacles when tackling modern and emerging problems such as shipboard landings and urban air mobility, first-principles computational models offer the ability to perform design and analytical studies while resolving detailed flow features. However, the cost of Navier–Stokes methods with very high resolution is, for the most part, only tractable for research use cases. Conversely, midfidelity potential-based methods are cost-effective but may not capture important physics that occur near the rotor blades. Hybrid approaches that couple Navier–Stokes and potential solvers have recently been developed, which provide the ability to resolve complex physics with a Navier–Stokes solution while employing a potential solver to resolve far-wake effects for which it is well-suited. This research discusses the impact of new improvements to and best practices for the hybrid Navier–Stokes/free-wake solver OVERFLOW-CHARM applied to rotors in ground effect. Predictions of rotor performance and flow features are compared with experimental data for microscale and subscale rotors. Rotor performance is predicted within 6% of a conventional computational fluid dynamics simulation at approximately 20% of the computational cost and predicted flow features correlate well to experimental flow visualization.

A hybrid numerical model for simulating aero-elastic-hydro-mooring-wake dynamic responses of floating offshore wind turbine

It demands many computational resources to model the coupled responses of a floating offshore wind turbine (FOWT), especially for its aero-elastic-hydro-mooring-wake dynamics and their interaction. In this paper, a new hybrid numerical model for FOWT systems is developed, which is based on the hybrid potential-viscous flow model called qaleFOAM. In this model, the aerodynamics of wind turbine are solved by the unsteady actuator line method (UALM); the elastic responses of the turbine blade are calculated by the Legendre spectral finite element model (BeamDyn); the hydrodynamics of the floating platform are dealt with by the combination of the fully nonlinear potential solver and a two-phase Navier–Stokes solver; the mooring dynamics are considered with the Lumped Mass Mooring Model (MoorDyn), and the turbine wake is solved with the large eddy simulation (LES) model. This newly formulated model can deal with wind, wave, mooring dynamics, platform motions, and turbine structural dynamics involved in the FOWT system. To demonstrate the capability of the present model, various cases with different complexities are investigated and compared with the experimental data and other numerical results. Then, the model is applied to simulation of a semi-submersible FOWT system, subjected to a regular wave and a uniform wind. The prediction of the aerodynamic performance, blade tip deflection, platform motion responses, and mooring line tension loads show good agreements with the results from other methods. In addition, the phenomenon of the coupled effects between the dynamic responses of platform, blade deformation and wake flow are captured reasonably well.

STEPS 4.0: Fast and memory-efficient molecular simulations of neurons at the nanoscale

Recent advances in computational neuroscience have demonstrated the usefulness and importance of stochastic, spatial reaction-diffusion simulations. However, ever increasing model complexity renders traditional serial solvers, as well as naive parallel implementations, inadequate. This paper introduces a new generation of the STochastic Engine for Pathway Simulation (STEPS) project (http://steps.sourceforge.net/), denominated STEPS 4.0, and its core components which have been designed for improved scalability, performance, and memory efficiency. STEPS 4.0 aims to enable novel scientific studies of macroscopic systems such as whole cells while capturing their nanoscale details. This class of models is out of reach for serial solvers due to the vast quantity of computation in such detailed models, and also out of reach for naive parallel solvers due to the large memory footprint. Based on a distributed mesh solution, we introduce a new parallel stochastic reaction-diffusion solver and a deterministic membrane potential solver in STEPS 4.0. The distributed mesh, together with improved data layout and algorithm designs, significantly reduces the memory footprint of parallel simulations in STEPS 4.0. This enables massively parallel simulations on modern HPC clusters and overcomes the limitations of the previous parallel STEPS implementation. Current and future improvements to the solver are not sustainable without following proper software engineering principles. For this reason, we also give an overview of how the STEPS codebase and the development environment have been updated to follow modern software development practices. We benchmark performance improvement and memory footprint on three published models with different complexities, from a simple spatial stochastic reaction-diffusion model, to a more complex one that is coupled to a deterministic membrane potential solver to simulate the calcium burst activity of a Purkinje neuron. Simulation results of these models suggest that the new solution dramatically reduces the per-core memory consumption by more than a factor of 30, while maintaining similar or better performance and scalability.

Efficiency and accuracy of the domain and functional decomposition strategies for the wave-structure interaction problem

Extreme waves can be modelled in a numerical wave tank (NWT) as a focusing wave in order to generate events that are crucial for designing any offshore structures. The focused wave generation implemented in the completely nonlinear potential solver Higher Order Spectral method (HOS-NWT) with the time-reversal technique was used in the present work. The OpenFOAM based solvers foamStar and foamStarSWENSE which use the domain decomposition and the functional decomposition approach, respectively, were used to couple with a HOS-NWT to perform the focusing wave interaction with the structure (in this case, a cylinder). The incident waves from HOS-NWT are blended into a foamStar and foamStarSWENSE subdomain, where the waves will propagate and interact with the structure within their domain. In the SWENSE methodology, the total field in the computational domain is divided into the incident field and a complementary field. foamStar solves for the total flow fields, and foamStarSWENSE solves for the complementary (total minus incident) flow fields. The Volume of Fluid (VoF) method was selected to capture the interface. 2D parametric studies for focused wave generation were carried out with both solvers, and 3D wave interactions for different mesh types (coarse, medium, and fine) were investigated to understand the convergence for such transient type problems. For verification, the case’s uncertainty was quantified using the Richardson extrapolation approach, and validation was then done by comparing the results to the experiment. The computational efficiency of the two coupling techniques was also compared, and recommendations for solver improvements are made. In general, when solving the problem of wave generation and propagation in 2D and 3D conditions, foamStarSWENSE surpassed foamStar in accuracy. On the other hand, when the simulation included a structure in the domain, the accuracy of the solution in foamStar and foamStarSWENSE was nearly the same, but the foamStar performed better in terms of computational cost. Overall, both coupling approaches generate a realistic depiction of focused wave interaction and force over the cylinder at a minimum computational cost.

Petrophysical Regression regarding Porosity, Permeability, and Water Saturation Driven by Logging-Based Ensemble and Transfer Learnings: A Case Study of Sandy-Mud Reservoirs

From a general review, most petrophysical models applied for the conventional logging interpretation imply that porosity, permeability, or water saturation mathematically have a linear or nonlinear relationship with well logs, and then arguing the prediction of these three parameters actually is accessible under a regression of logging sequences. Based on this knowledge, ensemble learning technique, partially developed for fitting problems, can be regarded as a solution. Light gradient boosting machine (LightGBM) is proved as one representative of the state-of-the-art ensemble learning, thus adopted as a potential solver to predict three target reservoir characters. To guarantee the predicting quality of LightGBM, continuous restricted Boltzmann machine (CRBM) and Bayesian optimization (Bayes) are introduced as assistants to enhance the significance of input logs and the setting of employed hyperparameters. Thereby, a new hybrid predictor, named CRBM-Bayes-LightGBM, is proposed for the prediction task. To validate the working performance of the proposed predictor, the basic data derived from the member of Chang 8, Jiyuan Oilfield, Ordos Basin, Northern China, is collected to launch the corresponding experiments. Additionally, to highlight the validating effect, three sophisticated predictors, including k-nearest neighbors (KNN), support vector regression (SVR), and random forest (RF), are introduced as competitors to implement a contrast. Since ensemble learning models universally will cause an underfitting issue when dealing with a small-volumetric dataset, transfer learning in this circumstance will be employed as an aided technique for the core predictor to achieve a satisfactory prediction. Then, three experiments are purposefully designed for four validated predictors, and given a comprehensive analysis of the gained experimented results, two critical points are concluded: (1) compared to three competitors, LightGBM-cored predictor has capability to produce more reliable predicted results, and the reliability can be further improved under a usage of more learning samples; (2) transfer learning is really functional in completing a satisfactory prediction for a small-volumetric dataset and furthermore has access to perform better when serving for the proposed predictor. Consequently, CRBM-Bayes-LightGBM combined with transfer learning is solidly demonstrated by a stronger capability and an expected robustness on the prediction of porosity, permeability, and water saturation, which then clarify that the proposed predictor can be viewed as a preferential selection when geologists, geophysicists, or petrophysicists need to finalize a characterization of sandy-mud reservoirs.

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

Physics informed neural networks (PINNs) are capable of finding the solution for a given boundary value problem. Here, the training of the network is equivalent to the minimization of a loss function that includes the governing (partial differential) equations (PDE) as well as initial and boundary conditions. We employ several ideas from the finite element method (FEM) to enhance the performance of existing PINNs in engineering problems. The main contribution of the current work is to promote using the spatial gradient of the primary variable as an output from separated neural networks. Later on, the strong form (given governing equation) which has a higher order of derivatives is applied to the spatial gradients of the primary variable as the physical constraint. In addition, the so-called energy form of the problem (which can be obtained from the weak form) is applied to the primary variable as an additional constraint for training. The proposed approach only required up to first-order derivatives to construct the physical loss functions. We discuss why this point is beneficial through various comparisons between different models. The mixed formulation-based PINNs and FE methods share some similarities. While the former minimizes the PDE and its energy form at given collocation points utilizing a complex nonlinear interpolation through a neural network, the latter does the same at element nodes with the help of shape functions. We focus on heterogeneous solids and check the performance of the proposed PINN model against the solution from FEM on two prototype problems: elasticity and the Poisson equation (steady-state diffusion problem). It is concluded that by properly designing the network architecture in PINN, the deep learning model has the potential to solve the unknowns in a heterogeneous domain without any available initial data from other sources. Finally, discussions are provided on the combination of PINN and data from physical FE simulations for a fast and accurate design of composite materials in future developments.