Anosov flows

[westurner]

Is your feature request related to a problem? Please describe.

• STORY: Users can generalize about dynamical systems with insufficient samples (and Anosov flows)

Describe the solution you'd like

• https://www.quantamagazine.org/flow-proof-helps-mathematicians-find-stability-in-chaos-20230615/
  • (TIL about) Anosov flows, and there's a pendulum infographic, like "Deep Learning of Dynamics and Coordinates with SINDy Autoencoders" (2020) https://www.youtube.com/watch?v=WHhDgxkiR9c

Describe alternatives you've considered

• https://www.google.com/search?&q=awesome+autoencoder+site%3Agithub.com
• Quantum Algorithms, Quantum Computers
  • https://qiskit.org/ecosystem/machine-learning/tutorials/12_quantum_autoencoder.html
  • https://en.wikipedia.org/wiki/Church%E2%80%93Turing%E2%80%93Deutsch_principle
    • https://en.wikipedia.org/wiki/Quantum_complexity_theory#Simulation_of_quantum_circuits
    • If the data is known to be of a quantum statistical process, can a classical autoencoder be sufficient (like how numerical solutions to n-body gravity problems are possible but not a complete set of solutions)?
  • https://github.com/tequilahub/tequila ; expectation values w/ various simulators and real QC
  • https://github.com/topics/quantum-fluids
  • Gross-Pitaevskii equations for:
    • Bose-Einstein Condensate superfluids
    • Superfluid Quantum Gravity w/ vortices (and Bernoulli's and Navier-Stokes)
    • https://en.wikipedia.org/wiki/Quantum_fluid

      Quantum mechanical effects become significant for physics in the range of the de Broglie wavelength. For condensed matter, this is when the de Broglie wavelength of a particle is greater than the spacing between the particles in the lattice that comprises the matter.

    • https://www.researchgate.net/figure/Comparison-between-the-velocities-of-a-quantum-superfluid-vortex-with-l-1-quantum_fig2_23757600 :
    • "Quantum-like gravity waves and vortices in a classical fluid" (2009) https://arxiv.org/abs/0901.1270 https://scholar.google.com/scholar?cluster=14044130315847002901&hl=en&as_sdt=5,43&sciodt=0,43

      However, one can write in this case a system of hydrodynamic equations in terms of a continuity equation and of a two- dimensional Euler equation where the density is replaced by the height profile of the fluid surface (see, e.g., [8]). Our new proposal is therefore to build a quantum-like potential, and then a wave function, from the square root of the surface height. One can then show that the gravity waves at the surface of a fluid subjected to such a generalized ‘quantum potential’ would be solutions of a nonlinear Schr¨odinger equation whose form is typical of superfluids. Then we suggest to achieve this quantum potential in a real experiment,

    • freak waves

      These results may also be relevant to the study of freak waves, which have already been described by nonlinear Schr¨odinger equations.

  • "Gravity as a fluid dynamic phenomenon in a superfluid quantum space" (2016) https://hal.science/hal-01248015v5/preview/Fluid-Quantum-Gravity-And-Relativity_Fedi-v5_Sept2016.pdf
  • No gravitons
  • No dark energy
  • GR + Fluid Vortices
  • https://github.com/topics/lattice-boltzmann

Additional context

• https://www.google.com/search?q=anosov+flows+superfluid+quantum+gravity

[Jacob-Stevens-Haas]

I'm sorry, this question is too general. The problem section needs to be specific about something that SINDy doesn't do, as well as specifically what you'd like to see. E.g. provide an example of equations for Anosov flow, some code of how you would hope to use pysindy with those equations, and what you'd like to see as output.

My own research isn't in this direction, so I'm unable to mentally create that kind of an example from the description you provided.

I'm likely to close this issue, as there doesn't seem to be anything other than quotes and links. You need to put the work into summarizing the relevant information; it's unfair to expect us to do so.

[westurner]

I'm a bit miffed that you've suggested that you deserved a ScholarlyArticle to teach you a better approach for the problems you're modeling with PySindy and reluctant to spend more time on yours.

Are you aware of tools that model the same fluid problems as pysindy but yield solutions more algorithmically efficiently on classical computers?

FWIU, this new application of Anosov flows ~hashes stable patterns in fluids?

On Mon, Aug 7, 2023, 4:16 PM Jacob Stevens-Haas @.***> wrote:

Closed #383 https://github.com/dynamicslab/pysindy/issues/383 as completed.

— Reply to this email directly, view it on GitHub https://github.com/dynamicslab/pysindy/issues/383#event-10028359859, or unsubscribe https://github.com/notifications/unsubscribe-auth/AAAMNS3K7FVU44JQN7E4NLTXUFEJRANCNFSM6AAAAAA3CI62HQ . You are receiving this because you authored the thread.Message ID: @.***>

[Jacob-Stevens-Haas]

There's a variety of competing methods, from SINDy-adjacent-but-not-in-pysindy to ones that take a different approach, such as Gaussian Process Regression. We're more focused on the problems of noisy data, choosing the function library, and choosing the measurement coordinates than on algorithmic efficiency.

[westurner]

From "Symbolic Regression" https://en.wikipedia.org/wiki/Symbolic_regression :

By not requiring a priori specification of a model, symbolic regression isn't affected by human bias, or unknown gaps in domain knowledge. It attempts to uncover the intrinsic relationships of the dataset, by letting the patterns in the data itself reveal the appropriate models, rather than imposing a model structure that is deemed mathematically tractable from a human perspective. The fitness function that drives the evolution of the models takes into account not only error metrics (to ensure the models accurately predict the data), but also special complexity measures, [6]

Read: https://en.wikipedia.org/wiki/Discovery_system_(AI_research)

• LLM prompt:
  • "you are a computational AI physicist explaining to a"
  • "let's think step by step"
  • "in SymPy, with pytest tests"
  • "in SymPy, with Hypothesis @given decorator tests"
  • "and then assert that"
  • "in a table with citations,"
  • "in JSON-LD"
  • "with lean mathlib"
  • https://github.com/leanprover-community/mathlib/issues/17919
  • Table of SOTA math LLMs: https://github.com/nlpxucan/WizardLM/tree/main/WizardMath#comparing-wizardmath-with-the-llm-models

Anyways, Fitness functions for quantum chaotic fluid model prediction is out of scope and tangential, too: https://en.wikipedia.org/wiki/Fitness_function

-Anosov flows, degrees of curl, and fluid pattern emergence;

• are commonly emergent fluid flow patterns "hashable" after e.g. SiFT and RiFT or at a lower level.
• The subset/superset relation between Anosov flows and Lorenz curves (in fluids)

On Mon, Aug 14, 2023, 6:30 PM Jacob Stevens-Haas @.***> wrote:

There's a variety of competing methods, from SINDy-adjacent-but-not-in-pysindy https://arxiv.org/abs/2009.01006 to ones that take a different approach, such as Gaussian Process Regression https://arxiv.org/abs/2004.08376. We're more focused on the problems of noisy data, choosing the function library, and choosing the measurement coordinates than on algorithmic efficiency.

— Reply to this email directly, view it on GitHub https://github.com/dynamicslab/pysindy/issues/383#issuecomment-1678169624, or unsubscribe https://github.com/notifications/unsubscribe-auth/AAAMNS4GH2OLNECRK7YKAY3XVKRJVANCNFSM6AAAAAA3CI62HQ . You are receiving this because you authored the thread.Message ID: @.***>