by jsendak | Mar 14, 2024 | Computer Science

**Expert Commentary:**

## Preconditioning Techniques for Space-Time Isogeometric Discretization of the Heat Equation

This review article discusses preconditioning techniques based on fast-diagonalization methods for the space-time isogeometric discretization of the heat equation. The author analyzes three different formulations: the Galerkin approach, a discrete least-square method, and a continuous least-square method.

One of the key challenges in solving the heat equation using fast-diagonalization techniques is that the heat differential operator cannot be simultaneously diagonalized for all uni-variate operators acting on the same direction. However, the author highlights that this limitation can be overcome by introducing an additional low-rank term.

The use of arrow-head like factorization or inversion by the Sherman-Morrison formula is proposed as a suitable approach for dealing with this additional low-rank term. These techniques can significantly speed up the application of the operator in iterative solvers and aid in the construction of an effective preconditioner.

The review further highlights that the proposed preconditioners show exceptional performance on the parametric domain. Additionally, they can be easily adapted and retain good performance characteristics even when the parametrized domain or the equation coefficients are not constant.

Overall, the article provides valuable insights into the challenges of fast-diagonalization methods for the heat equation and presents effective preconditioning techniques that can enhance the efficiency and accuracy of solving the heat equation using space-time isogeometric discretization.

Further research in this area could focus on investigating the performance of these preconditioning techniques on more complex systems or extending them to other types of partial differential equations. Additionally, exploring the potential of combining these techniques with other numerical methods or algorithms could contribute to further advancements in solving heat equation problems.

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by jsendak | Feb 6, 2024 | AI

This paper proposes a meshless deep learning algorithm, enriched physics-informed neural networks (EPINNs), to solve dynamic Poisson-Nernst-Planck (PNP) equations with strong coupling and…

In this article, a groundbreaking approach to solving dynamic Poisson-Nernst-Planck (PNP) equations with strong coupling is introduced. The authors present an innovative meshless deep learning algorithm called enriched physics-informed neural networks (EPINNs). By combining the power of deep learning with the principles of physics, EPINNs demonstrate exceptional capabilities in accurately solving complex PNP equations. This novel method holds great promise in various scientific and engineering fields where understanding and predicting dynamic phenomena is crucial.

## Exploring the Potential of Enriched Physics-Informed Neural Networks (EPINNs) in Solving Dynamic PNP Equations

Traditional numerical methods for solving dynamic Poisson-Nernst-Planck (PNP) equations with strong coupling and non-linearities often rely on mesh-based approaches. However, a recent study proposes a promising alternative – enriched physics-informed neural networks (EPINNs).

EPINNs utilize deep learning algorithms to solve complex mathematical problems while incorporating physical laws and constraints. This groundbreaking approach combines the strengths of physics-based models with the computational power of neural networks, opening up new possibilities for solving challenging PNP problems.

The dynamic nature of PNP equations arises in various fields, including electrochemistry, bioengineering, and semiconductor devices. Traditionally, researchers have relied on finite element methods or finite difference methods to solve these equations. Such techniques require dividing the domain into a mesh to discretize and approximate the continuous system. However, this process can be computationally expensive and challenging for systems with irregular geometries or evolving interfaces.

EPINNs offer a potential solution to these challenges. By training a neural network using data from the system’s boundary conditions and initial conditions, EPINNs can accurately predict the system’s behavior and provide solutions without the need for a predefined mesh structure.

The key idea behind EPINNs is to incorporate physical laws as constraints into the training process. By enforcing these constraints during training, the neural network learns to generate solutions that satisfy the underlying physics of the problem. This integration of physics-based knowledge helps enhance the accuracy and reliability of the predictions.

### Potential Benefits of EPINNs in Solving Dynamic PNP Equations:

**Improved Efficiency:** EPINNs eliminate the need for mesh generation and significantly reduce computational time, making them well-suited for real-time simulations and optimizations.
**Flexibility in Geometry:** EPINNs can handle systems with irregular geometries, complex interfaces, and evolving boundaries efficiently. This flexibility expands the scope of problems that can be accurately solved.
**Enhanced Accuracy:** By incorporating physical constraints into the training process, EPINNs produce solutions that adhere to the laws of physics governing the system, leading to more accurate predictions and results.
**Generalizability:** Once trained, EPINNs can efficiently handle diverse problem instances without the need for retraining. This generalization capability saves time and computational resources.

While EPINNs show great promise, additional research and development are needed to fully explore their potential. The performance of EPINNs may vary depending on the complexity of the problem and the availability of training data. Further improvements in training algorithms, network architectures, and integration with existing numerical methods are essential to harness the full power of EPINNs.

In conclusion, enriched physics-informed neural networks (EPINNs) offer an innovative approach to solving dynamic PNP equations by combining deep learning algorithms with physical principles. Through their ability to incorporate domain knowledge as constraints and eliminate the need for mesh-based discretization, EPINNs show great potential in enhancing efficiency, accuracy, and flexibility in solving complex problems. As further advancements and refinements are made, EPINNs could become an invaluable tool for researchers in a variety of fields.

“EPINNs bridge the gap between physics-based modeling and data-driven machine learning, empowering us to solve complex problems in a more efficient and accurate manner.”

nonlinearities. The authors highlight the limitations of traditional numerical solvers, such as finite element methods, in accurately capturing complex physical phenomena in dynamic systems. In response to these challenges, they introduce EPINNs as a promising alternative that combines the power of deep learning with the principles of physics.

EPINNs leverage the concept of physics-informed neural networks (PINNs), which integrate physical laws into the neural network architecture. This allows the model to not only learn from data but also respect the underlying physics governing the system. By incorporating the PNP equations into the network’s loss function, EPINNs enable the simultaneous optimization of both data-driven and physics-driven objectives.

One key advantage of EPINNs is their meshless nature. Traditional numerical methods rely on discretizing the problem domain into a grid, which can be computationally expensive and challenging to adapt to complex geometries. In contrast, EPINNs do not require a predefined mesh, making them more flexible and efficient for solving problems with irregular boundaries or evolving geometries.

Moreover, EPINNs excel at handling strong coupling and nonlinearities present in dynamic PNP equations. These phenomena often occur in various scientific domains, such as electrochemistry and biological systems, where traditional methods struggle to accurately capture their intricate behavior. By leveraging deep learning techniques, EPINNs can capture complex nonlinear relationships in the data, leading to more accurate predictions and improved understanding of the underlying physics.

Looking ahead, there are several exciting possibilities for further enhancing EPINNs. One area of exploration could be the incorporation of additional physical constraints or domain-specific knowledge into the network architecture. This could further improve the model’s ability to generalize across different scenarios and provide more robust solutions.

Another avenue for future research could involve investigating ways to reduce the computational cost of training EPINNs. While their meshless nature offers advantages in terms of flexibility, it also introduces challenges related to scalability and efficiency. Developing techniques to accelerate training and inference processes could make EPINNs even more practical for real-time applications or large-scale simulations.

Furthermore, exploring the transferability of EPINNs to other types of partial differential equations (PDEs) would be valuable. Although this paper focuses on dynamic PNP equations, the underlying principles of EPINNs could potentially be extended to solve a wide range of PDEs encountered in diverse scientific and engineering fields.

In conclusion, this paper introduces EPINNs as a promising meshless deep learning algorithm for solving dynamic PNP equations with strong coupling and nonlinearities. By combining the power of deep learning with physics-informed neural networks, EPINNs offer a novel approach to accurately capture complex physical phenomena. The potential applications of EPINNs are vast, and further research in this area holds the promise of advancing our understanding and modeling capabilities in various scientific disciplines.

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by jsendak | Jan 31, 2024 | GR & QC Articles

In the framework of the extended Einstein-aether-axion theory we study the model of a two-level aetheric control over the evolution of a spatially isotropic homogeneous Universe filled with axionic dark matter. Two guiding functions are introduced, which depend on the expansion scalar of the aether flow, equal to the tripled Hubble function. The guiding function of the first type enters the aetheric effective metric, which modifies the kinetic term of the axionic system; the guiding function of the second type predetermines the structure of the potential of the axion field. We obtained new exact solutions of the total set of master equations of the model (with and without cosmological constant), and studied in detail four analytically solvable submodels, for which both guiding functions are reconstructed and illustrations of their behavior are presented.

## Examining the Conclusions of the Text: Future Roadmap, Challenges, and Opportunities

The extended Einstein-aether-axion theory introduces a two-level control mechanism for the evolution of a spatially isotropic homogeneous Universe filled with axionic dark matter. This theory relies on the introduction of two guiding functions, which are dependent on the expansion scalar of the aether flow, equivalent to the tripled Hubble function. The first guiding function affects the aetheric effective metric, altering the kinetic term of the axionic system, while the second guiding function determines the structure of the potential of the axion field.

This study has successfully derived new exact solutions for the master equations of the model, both with and without a cosmological constant. Additionally, four submodels have been analyzed in detail, allowing for the reconstruction of both guiding functions and providing visual representations of their behavior.

### Future Roadmap

Building upon this work, future research should consider several areas:

**Further Exploration of Guiding Functions:** Investigating the behavior of different guiding functions under various conditions and exploring their impact on the overall evolution of the Universe.
**Cosmological Implications:** Analyzing the cosmological consequences of the derived solutions and submodels, such as their implications for dark matter distribution, expansion dynamics, and large-scale structures formation.
**Numerical Simulations:** Utilizing numerical methods to simulate and validate the obtained analytical solutions, allowing for a more extensive exploration of parameter space and verification of the model’s predictions.
**Observational Tests:** Proposing observational tests and experiments to validate or reject the extended Einstein-aether-axion theory. This could involve analyzing observational data from cosmic microwave background radiation, large-scale structure surveys, and other cosmological probes.
**Theoretical Extensions:** Considering possible extensions or modifications to the current theory to incorporate additional physical phenomena, such as the inclusion of other types of dark matter or dark energy components.

### Potential Challenges

Despite the promising findings and potential opportunities, there are several challenges that may arise during future investigations:

**Complexity:** The extended Einstein-aether-axion theory is inherently intricate, potentially leading to complex equations and calculations. This complexity can pose challenges in both analytical and numerical approaches.
**Data Limitations:** Obtaining precise observational data and measurements for testing the predictions of the theory may present challenges due to limitations in current observational capabilities and experimental constraints.
**Model Verification:** Validating the model’s predictions through observational tests and experiments may require sophisticated data analysis techniques and close collaboration between theorists and observational cosmologists.
**Theoretical Consistency:** Ensuring the theoretical consistency and compatibility of the extended Einstein-aether-axion theory with other well-established theories in cosmology and particle physics poses a significant challenge, requiring rigorous theoretical calculations and checks.

### Opportunities on the Horizon

The successful derivation of new exact solutions, coupled with the analytical exploration of submodels, presents several opportunities for future advancements in cosmology:

**Deeper Understanding:** Further research can provide a deeper understanding of the complex interplay between aether flow, axionic dark matter, and the overall evolution of the Universe. This understanding may unravel additional insights into fundamental questions in cosmology.
**Novel Observational Signatures:** Exploring the consequences of the extended Einstein-aether-axion theory could lead to the prediction and discovery of unique observational signatures within cosmic microwave background radiation, large-scale structures, and other cosmological observations.
**Alternative Descriptions:** The extended Einstein-aether-axion theory offers an alternative description of the Universe’s evolution and the behavior of dark matter. These alternative descriptions may challenge and expand our current theoretical framework.
**Applications Beyond Cosmology:** The theoretical foundations and techniques developed within the framework of this theory may find applications beyond cosmology, potentially impacting fields such as particle physics and general relativity.

In summary, the extended Einstein-aether-axion theory provides a novel approach to understanding the evolution of a spatially isotropic homogeneous Universe filled with axionic dark matter. Further research should focus on exploring guiding functions, investigating cosmological implications, conducting numerical simulations, proposing observational tests, and considering theoretical extensions. Although challenges such as complexity, data limitations, model verification, and theoretical consistency may arise, opportunities for deeper understanding, novel observational signatures, alternative descriptions, and broader applications exist on the horizon.

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by jsendak | Jan 22, 2024 | GR & QC Articles

We consider the thermodynamic properties of an exact black hole solution

obtained in Weyl geometric gravity theory, by considering the simplest

conformally invariant action, constructed from the square of the Weyl scalar,

and the strength of the Weyl vector only. The action is linearized in the Weyl

scalar by introducing an auxiliary scalar field, and thus it can be

reformulated as a scalar-vector-tensor theory in a Riemann space, in the

presence of a nonminimal coupling between the Ricci scalar and the scalar

field. In static spherical symmetry, this theory admits an exact black hole

solution, which generalizes the standard Schwarzschild-de Sitter solution

through the presence of two new terms in the metric, having a linear and a

quadratic dependence on the radial coordinate, respectively. The solution is

obtained by assuming that the Weyl vector has only a radial component. After

studying the locations of the event and cosmological horizons of the Weyl

geometric black hole, we investigate in detail the thermodynamical (quantum

properties) of this type of black holes, by considering the Hawking

temperature, the volume, the entropy, specific heat and the Helmholtz and Gibbs

energy functions on both the event and the cosmological horizons. The Weyl

geometric black holes have thermodynamic properties that clearly differentiate

them from similar solutions of other modified gravity theories. The obtained

results may lead to the possibility of a better understanding of the properties

of the black holes in alternative gravity, and of the relevance of the

thermodynamic aspects in black hole physics.

According to the article, the authors have examined the thermodynamic properties of an exact black hole solution in Weyl geometric gravity theory. They have used the simplest conformally invariant action, constructed from the square of the Weyl scalar and the strength of the Weyl vector. By linearizing the action in the Weyl scalar and introducing an auxiliary scalar field, the theory can be reformulated as a scalar-vector-tensor theory in a Riemann space with a nonminimal coupling between the Ricci scalar and the scalar field.

In static spherical symmetry, this theory gives rise to an exact black hole solution that generalizes the standard Schwarzschild-de Sitter solution. The metric of the black hole solution includes two new terms that have linear and quadratic dependencies on the radial coordinate.

The authors then investigate the thermodynamic properties of this type of black hole. They analyze the locations of the event and cosmological horizons of the Weyl geometric black hole and study the quantum properties by considering the Hawking temperature, volume, entropy, specific heat, and Helmholtz and Gibbs energy functions on both horizons.

They find that Weyl geometric black holes have distinct thermodynamic properties that differentiate them from similar solutions in other modified gravity theories. These results may contribute to a better understanding of black holes in alternative gravity theories and the importance of thermodynamic aspects in black hole physics.

## Future Roadmap

To further explore the implications of Weyl geometric gravity theory and its black hole solutions, future research can focus on:

- Extension to other geometries: Investigate whether the exact black hole solutions hold for other types of symmetries, such as rotating or more general spacetimes.
- Quantum aspects: Consider the quantum properties of Weyl geometric black holes in more detail, such as evaluating the quantum fluctuations and their effects on the thermodynamics.
- Comparison with observations: Study the observational consequences of Weyl geometric black holes and compare them with astrophysical data, such as gravitational wave signals or observations of black hole shadows.
- Generalizations and modifications: Explore possible generalizations or modifications of the Weyl geometric theory that could lead to new insights or more accurate descriptions of black holes.

## Potential Challenges

During the research and exploration of the future roadmap, some challenges that may arise include:

- Complexity of calculations: The calculations involved in studying the thermodynamic properties of black holes in Weyl geometric gravity theory can be mathematically complex. Researchers will need to develop precise techniques and numerical methods to handle these calculations reliably.
- Data availability: Obtaining accurate astrophysical data for comparison with theoretical predictions can be challenging. Researchers may need to depend on simulated data or future observations to test their theoretical models.
- New mathematical tools: Investigating alternative gravity theories often requires the development and application of new mathematical tools. Researchers may need to collaborate with mathematicians or utilize advanced mathematical techniques to address specific challenges.

## Potential Opportunities

Despite the challenges, there are potential opportunities for researchers exploring the thermodynamics of Weyl geometric black holes:

- New insights into black hole physics: The distinct thermodynamic properties of Weyl geometric black holes offer a unique perspective on black hole physics. By understanding these properties, researchers can gain new insights into the nature of black holes and their behavior in alternative gravity theories.
- Applications in cosmology: The study of black holes in alternative gravity theories like Weyl geometric gravity can have implications for broader cosmological models. Researchers may discover connections between black hole thermodynamics and the evolution of the universe.
- Interdisciplinary collaborations: Exploring the thermodynamics of Weyl geometric black holes requires expertise from various fields, including theoretical physics, mathematics, and astrophysics. Collaborations between researchers from different disciplines can lead to innovative approaches and solutions to research challenges.

**In conclusion,** the research presented in the article provides valuable insights into the thermodynamic properties of black hole solutions in Weyl geometric gravity theory. The future roadmap outlined here aims to further explore these properties, address potential challenges, and take advantage of the opportunities that arise from studying Weyl geometric black holes.

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by jsendak | Jan 21, 2024 | GR & QC Articles

We investigate quasitopological black holes in $(2+1)$ dimensions in the

context of electromagnetic-generalized-quasitopological-gravities (EM-GQT). For

three different families of geometries of quasitopological nature, we study the

causal structure and their response to a probe scalar field. To linear order,

we verify that the scalar field evolves stably, decaying in different towers of

quasinormal modes. The studied black holes are either charged geometries

(regular and singular) or a regular Ba~nados-Teitelboim-Zanelli (BTZ)-like

black hole, both coming from the EM-GQT theory characterized by nonminimal

coupling parameters between gravity and a background scalar field. We calculate

the quasinormal modes applying different numerical methods with convergent

results between them. The oscillations demonstrate a very peculiar structure

for charged black holes: in the intermediate and near extremal cases, a

particular scaling arises, similar to that of the rotating BTZ geometry, with

the modes being proportional to the distance between horizons. For the single

horizon black hole solution, we identify the presence of different quasinormal

families by analyzing the features of that spectrum. In all three considered

geometries, no instabilities were found.

Based on our investigation, we have concluded that the quasitopological black holes in $(2+1)$ dimensions in the context of electromagnetic-generalized-quasitopological-gravities (EM-GQT) exhibit stable evolution of a probe scalar field. We have studied three different families of quasitopological geometries and have found that the scalar field decays in different towers of quasinormal modes.

The black holes we have examined can be classified as either charged geometries (regular and singular) or a regular Bañados-Teitelboim-Zanelli (BTZ)-like black hole. These black holes are derived from the EM-GQT theory, which includes nonminimal coupling parameters between gravity and a background scalar field.

In our calculations of the quasinormal modes, we have employed various numerical methods, all yielding convergent results. The oscillations of the modes in charged black holes exhibit a unique structure. In the intermediate and near extremal cases, a scaling proportional to the distance between horizons emerges, similar to that observed in the rotating BTZ geometry.

For the single horizon black hole solution, we have identified the presence of different quasinormal families by analyzing the characteristics of the spectrum. Importantly, we did not find any instabilities in any of the three considered geometries.

## Future Roadmap

### Challenges:

- Further investigation is needed to understand the causal structure and response of other fields, such as electromagnetic fields, to these quasitopological black holes in EM-GQT theory. The study of other probe fields may reveal additional insights and properties.
- Exploring the thermodynamic properties of these black holes can provide valuable information about their entropy, temperature, and thermodynamic stability. This analysis could involve studying thermodynamic quantities and phase transitions.
- Investigating the stability of these black holes under perturbations beyond linear order could uncover additional behavior and help to determine their long-term evolution.

### Opportunities:

- The peculiar scaling observed in the oscillations of charged black holes could lead to new understandings of their underlying physical mechanisms. Further exploration of this scaling effect and its implications may offer insights into the connection between charge and geometry.
- The identification of different quasinormal families in the single horizon black hole solution presents an opportunity for studying the distinct characteristics and dynamics of these families. This information could contribute to a deeper understanding of black hole spectra in general.
- Extending the study to higher dimensions and different theories of gravity could provide valuable comparisons and insights into the behavior of quasitopological black holes across different contexts. Such investigations could include theories with additional matter fields or modified gravity theories.

**In conclusion,** the examination of quasitopological black holes in $(2+1)$ dimensions in the context of electromagnetic-generalized-quasitopological-gravities (EM-GQT) has revealed stable evolution and unique characteristics. While there are still challenges to address and opportunities to explore, this research lays the foundation for further expanding our understanding of these intriguing black hole solutions.

Reference:

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