We investigate the data-driven holographic transport models without
translation symmetry. Our data are chosen as the real part of the
frequency-dependent shear viscosity. We develop a radial flow equation for a
large class of holographic models, which determine the shear viscosity by the
black hole metric and the graviton mass. The latter serves as the bulk dual of
the translation symmetry breaking on the boundary. We convert the flow equation
to a Neural Ordinary Differential Equation (Neural ODE), which is a neural
network with continuous depth and produces output through a black-box ODE
solver. Given either the metric or the mass, we illustrate that the Neural ODE
can learn the other with high accuracy. Our work demonstrates the capabilities
of Neural ODEs in bulk reconstruction and applied holography.
The conclusions of the article are as follows:
- Holographic transport models without translation symmetry can be investigated using data-driven approaches.
- The real part of the frequency-dependent shear viscosity is chosen as the data for analysis.
- A radial flow equation is developed for a large class of holographic models, which determines the shear viscosity based on the black hole metric and the graviton mass.
- The flow equation is converted to a Neural Ordinary Differential Equation (Neural ODE), which is a neural network with continuous depth and solves the equation as a black-box ODE solver.
- The Neural ODE can accurately learn either the black hole metric or the graviton mass when given the other as input.
- This work demonstrates the potential of Neural ODEs in bulk reconstruction and applied holography.
Future Roadmap:
Looking ahead, there are several challenges and opportunities on the horizon:
1. Refining and expanding the dataset:
To further improve the accuracy of the Neural ODE’s learning capabilities, a more comprehensive dataset can be assembled. This could involve collecting additional data on shear viscosity under different conditions or incorporating other relevant variables.
2. Exploring more complex holographic models:
The current study focused on a large class of holographic models with translation symmetry breaking. Future research could investigate more complex models that involve additional factors or variables, potentially uncovering new insights into the behavior of holographic transport systems.
3. Fine-tuning the Neural ODE architecture:
The Neural ODE used in this study has shown promise in accurately learning the black hole metric and graviton mass. However, further optimization and fine-tuning of the neural network architecture could enhance its performance and efficiency even more.
4. Applying Neural ODEs to other areas of physics:
The success of Neural ODEs in bulk reconstruction and applied holography suggests that this approach could be applied to other areas of physics as well. Researchers could explore the potential of Neural ODEs in different scientific contexts, expanding their utility beyond holography.
5. Collaborative research and cross-disciplinary studies:
To fully leverage the capabilities of Neural ODEs and address the challenges posed by more complex holographic models, collaboration between experts in different fields, such as physics, mathematics, and computer science, would be beneficial. Cross-disciplinary studies could lead to innovative solutions and novel applications of Neural ODEs.
Conclusion:
The application of Neural ODEs to study holographic transport models without translation symmetry shows promising results. As researchers refine the methodology, explore more complex models, and apply Neural ODEs to other areas of physics, the field of applied holography is likely to advance. Collaboration and cross-disciplinary approaches can further enhance the potential of Neural ODEs in solving complex problems and uncovering new insights.