Invariant Solutions Salon : The Battle of Taylor-Couette II

our totally informal, free-for-all tricontinental “morning” chat:

The Battle of Taylor-Couette II

Back in 2007 Jonathan Halcrow, John Gibson and collaborators were proud to be able to show that in a  minimal cell state space truncated to 61,506 dimensions,  turbulent  plane Couette flow at Re=400 fits neatly into the Starship Enterprise (click here).

Recently, Jonathan pointed Fable at his PhD thesis and redid it in about 15 min. Here he tells us about how that went.

Video of the Zoom meetup

Zoom IE + Predrag recap, including hyperlinks to source materials

Jonathan's used Claude AI to reproduce and extend his PhD thesis on plane Couette flow. Now he discovered 131 relative periodic orbits using a JAX-based solver that achieved approximately 30x speed improvement over existing C++ implementations. Jonathan verified his results against previous literature and developed a symbolic dynamics approach using a 4-letter alphabet based on different flow phases (streaky, roll-like, wave-like, and high-dissipation states). Predrag discussed the limitations of forward-in-time versus space-time approaches for symbolic dynamics in spatiotemporal systems. The conversation evolved into a broader discussion about computational methods, with Predrag explaining the advantages of spatiotemporal optimization approaches using space-time solutions rather than traditional time-stepping methods.

Next steps

• Try the Gauss-Newton method with adjoint of the Jacobian (as suggested by Matt Golden) to improve convergence of the solver.
• Share the JAX solver code with John for further analysis and understanding the speed of the new time-stepping algorithm.
• Consider running the solver internally at Google to leverage more computational resources for finding longer periodic orbits.
• Investigate further the relationship between equilibria in the smaller box and periodic orbits in the wider box.
• Refine the symbolic dynamics and the set of symbols used for partitioning the state space.
• Explore the possibility of pushing the solver to larger spatial domains and consider a space-time approach for finding solutions.

Summary

Plane Couette Flow Research Update

Jonathan presented his work on revisiting his 18-year-old thesis on plane Couette flow, which studied equilibrium solutions and periodic orbits in minimal turbulent cells. He used Claude to implement a new solver in Python and JAX, which was much faster than his, Gibson and Schreiber previous Fortran and c++ implementations. While his current work is preliminary and 100% verified, he is still catching up on recent literature and welcomes feedback on potential duplicates. 

JAX Implementation Results Presentation

Jonathan presented results from his implementation using JAX, which showed significant speed improvements over ChannelFlow, particularly when using GPU and batching. He successfully reproduced solutions from previous papers and his thesis, achieving agreement down to three decimal points on dissipation and leading eigenvalues. Jonathan also explored periodic orbits in a wider box geometries, finding that equilibria from the smaller box fold before reaching the wider box, with relative periodic orbits becoming dominant instead.

Periodic Orbits Discovery Presentation

Jonathan presented his work on finding periodic orbits, discussing the discovery of 131 orbits total, with most having time periods below T=120. He explained his methodology of finding orbits through long trajectories and recurrence searching, with some recent attempts at symbolic dynamics. Predrag clarified that the orbits are primarily relative periodic rather than periodic, and Jonathan confirmed that RPO21 matches an orbit previously found by Kawahara and Kita.

Turbulence Analysis and Orbits

Jonathan presented his analysis of turbulence in the plane Coutte, focusing on periodic orbits and their role in determining escape rates from turbulent states. He identified that most orbits are dominated by a single RPO21 orbit and found that the neighborhood of PO2 serves as a key gate controlling whether turbulence persists or the flow relaminarizes. Jonathan developed a symbolic dynamics approach using letters to represent different flow states, though noted it requires remembering multiple symbols for accurate predictions.

JAX 2DMHD Solver Approaches

Matt Golden shared insights about using JAX to implement Gauss-Newton instead of Newton GMRes for 2DMHD solvers, noting it offers a larger basin of convergence and is computationally efficient. Dmitriy mentioned Stokes preconditioning as another approach for speedup, though clarified it only applies to traveling waves, not periodic orbits. The discussion then shifted to Jonathan's work on partitioning state space, where he explained using maximum visited points across orbits to define letter classifications, with Chris confirming this was done using L2 distance and asking about the use of a library of solutions, to which Jonathan responded that a nearest visitations approach worked better than using periodic orbits directly.

Turbulence Prediction Methods Discussion

Chris and Jonathan discussed the effectiveness of using proximity to solutions for predicting dynamics in turbulence. Jonathan acknowledged that while the chosen points in state space showed some correlation with known turbulence patterns, a more careful analysis using periodic orbits might yield better results. 

Predrag explained that forward-in-time, one-dimensional sequence symbolic dynamics approaches are not suitable for problems with spatial translational invariance, and recommended using instead spatiotemporal symbol arrays, as demonstrated in previous work on the Kuramoto-Sivashinsky equation by Matt Gudorf and others.

Spatiotemporal Solutions and Periodic Domains

Predrag and Chris discussed spatiotemporal solutions and periodic domains in three dimensions, with Predrag explaining how solutions can be treated as building blocks or "Lego blocks" that can be combined and optimized. Predrag outlined a method involving spatiotemporal periodic domains, Bravais cells, and primitive cells to systematically find solutions. 

The conversation ended with Predrag confessing that ever since his student Freddy Christiansen outperformed him in their computations, his life has been a struggle to catch up in computational work.

Code Performance Improvement Discussion

The group discussed the performance improvements in Jonathan's code implementation, with Jonathan explaining that a factor of 4 speed-up came from changing the time-stepping algorithm while maintaining the same accuracy. Jonathan noted that the time step size difference was a key factor, though he couldn't fully explain the methodological differences between the implementations. The discussion also touched on the importance of examining solutions in state space and using appropriate coordinates to understand the dynamics of the system, though Jonathan mentioned that equilibrium solutions in the current geometry were less prominent and not as dynamically important as in smaller geometries.

Unstable Directions in Symmetric Systems

Jonathan and Predrag discussed the challenges of representing unstable directions in systems with translational symmetry. Predrag explained techniques like Poincare sectioning and slicing to simplify the analysis by removing translational invariance, suggesting these methods could help find a better representation than Jonathan's initial attempt using the unstable manifold of RPO21. They also discussed how to handle multiple trajectories in spatiotemporal cases, including the concept of tori shadowing each other in space-time.

Space-Time Optimization in Dynamical Systems

Predrag and Jonathan discussed the challenges and benefits of using space-time optimization methods in dynamical systems, particularly for the 1D Kuramoto-Sivashinsky system. Predrag explained that while forward-in-time solutions face exponential instabilities, space-time approaches using optimization algorithms can successfully find stable solutions in larger domains. Jonathan expressed interest in exploring larger spatial cells and potentially using tiling of solutions found in smaller boxes to extend their work to larger spaces.

Spatio-Temporal Problem Solving Approach

Predrag discussed the approach to solving spatio-temporal problems by considering solutions in state space rather than plumber's image boxes, and proposed using a Lagrangian approach instead of Newton's method. He explained how to identify periodic solutions and build a library of space-time tiles that can be used to extend solutions beyond a given window. Chris asked about extending solutions in temporal direction, and Predrag clarified that the approach considers solutions in multiple directions, not just forward in time.

Space-Time vs Dynamical Systems Discussion

Predrag and Chris discussed the concept of thinking in space-time versus traditional dynamical systems approaches, with Predrag explaining how periodic orbits can be viewed as solutions in space-time and how observables are evaluated on these solutions.

The Salon concluded with a discussion of Messi's retirement and Argentine World Cup show, with Predrag expressing what in Argentina is a minority opinion about the team's performance.

Links to earlier Salons and other matters of importance

We did NOT opine on criminal dimensions of claims concerning the Millennium Problem. We have some serious plumbing to focus on, so don't speak now 😉, forever hold your peace.

Alex Blumenthal Math colloquium, Sept 15, 2026

The best explanation of the mathematics behind the formulation of the problem that Predrag has heard so far is

Alex Blumenthal: Blowup for Navier-Stokes: Context, Relevance and History

              Two cleaned-up very informal Salon discussion, both to be expanded upon in later salons:

John F. Gibson, Algebraic models of plane Couette equilibria. 
Jonathan Halcrow & Fable, Invariant solutions salon, Sept 11, 2026

Invariant Solutions Salon Sept 4, 2026

Unedited raw 1 hour recording of Jonathan, Francesco and Predrag's discussion. Share sparingly - some of us take years, nay, decades to actually publish our conceptual breakthroughs.

Jonathan Halcrow:

I pointed Fable at my thesis and some of our papers and asked it to try to produce a more complete catalog of periodic orbits. It speedran my thesis in about 15 min and built a new DNS, equilibrium and RPO search code from scratch in Jax. It hooked this up to my GPU and it now runs about 1000x faster than some of our primitive human stuff did back in the day. It's reproduced most of the equlibiria we had before and seems to have found a number of RPOs (18 in the HKW box at Re=400, in the last few hours). I wanted to compare against the ones in the Channelflow database to see how many are new but it doesn't seem to exist anymore. Do you have a copy saved anywhere?

      Predrag:

I cannot find it in the 2018 version

www.channelflow.ch

Discourse.channelflow.ch

And let us praise Ashley - he started the Salon way back then 🙂

Day's Salon is a continuation of March 27, 2025 Invariant Solutions Salon:

The Battle of Taylor-Couette I

Clearly the "physical", "inertial manifold", "latent space", ...,  dimension should be much smaller than 100,000 dimensions. Predrag could think of no argument that would bring it to much below 100 dimensions. Typical Mittel-Europa defeatism.  Today we present:

In the left corner we have John Gibson. He does it by (re)thinking. Today he can do it in 19 dimensions.

In the right corner we have Alec Linot. He does it by (ML)thinking. He can do it in 18 dimensions.

Unedited raw 1 1/2 hour recording 

Francesco Fedele contributed April 2, 2025 email "4D Waleffe Model, Yann Lecun, AI-learned physical Features in Channel Turbulence...." which you can ask him to resend it to you.

Goldenfeld Rutgers seminar "Emergence and Generalization in Machine Learning"

These days the zone is flooded with Super Idiocy ("some call it SI") slop. Fuhgeddaboudit. But. The good papers are now also coming our way at inhuman pace, this is the the 3rd "must study" work of the day 🙁 .

For me, Nigel Goldenfeld's "Emergence and Generalization in Machine Learning",  arXiv.org:2607.04135, is an inspiring talk: a strong recommend. For me, also deeply related to the deterministic spatiotemporal field theory, so what follows is my discussion of these connections, not Nigel's work.

 

Li and Goldenfeld show that the machine learning takes place in the ergodic, 

or what we call anti-integrable or hyperbolic, corner of the spatiotemporal field theory. Makes sense - this anti-integrable, weakly coupled corner accommodates a huge number of configurations, exponential in the state-space volume (# of labels). Then they descend to the relatively low dimensional latent space (what physicists clumsily call "inertial manifold", perhaps) in the fictitious (learning) time by a stochastic Langevin.

For the free field theories there is a phase transition, at the massless, diffusion value of \mu=0, where real \mu is the Klein-Gordon mass, with \mu > 0 the ergodic AKA chaotic phase,  and the \mu --> \infty limit the anti-integrable limit. For nonlinear field theories, the value of critical \mu is a calculation.

As, I think, always, the ergodic phase is separated from the oscillatory phase by a singularity.

In contrast, the imaginary \mu = I k, k < 1 oscillatory phase was the setting for the original 1990 Esther Levin, Naftali Tishby, and Sara A. Solla "A Statistical Approach to Learning and Generalization in Layered Neural Networks" Gibbs theory of learning, in the noisy data polynomial fitting, or elliptic, "oscillatory" corner.

Sara explains the theory in, for example, this 2022 Les Houches lecture. As any stat mech paper looks like any other stat mech paper, here is what I believe is an important step that you might miss: the normalization constant z of eqs. (5) and (21c) leads to averaging different from the ensemble average over the Gibbs measure.

The right half of the Li & Goldenfeld plot is our home, where friends and I have since  6 AM PST, January 24, 2017 (Nigel was there) worked on recasting high-dimensional ergodicity as the spatiotemporal chaotic field theory. The left and the right phases in the plot are related to each other as is harmonic oscillator to the `temporal cat', Hooke’s wild, ‘anti-harmonic’ sister, see Appendix A. Historical context here (and search for `harmonic' throughout the text).

In stochastic setting, it's backbone is the deterministic field theory, the set of all solutions of the constraints of the theory. I believe the Li & Goldenfeld is essentially the same theory as Parisi & Wu stochastic quantization, the difference being that in ML a simple action of physics is replaced by the many parameter trained neural network of machine learning. Lippolis is working on showing how they are determined and weighted in the stochastic Langevin approach, arXiv:2510.12532. Each solution breaks translation invariance, as ergodicity should. The totality yields expectation values of observables in terms of solution-weighted averages over values of observables.

For me, another physics of ML inspiring Rutger's seminar was Mézard's "How diffusion theory is used to produce fake data",  Marc Mézard, Giulio Biroli et al arXiv.org:260?.????. I believe that Lippolis & Cvitanovic "optimal partition hypothesis" is a dynamical calculation of Mézard & Biroli "memorization–generalization transition" as a function of the noise strength in the Fokker-Planck equation.

In contrast to numerical averaging of ergodic states, in spatiotemporal field theory these expectation values are computed by exploring the state space hierarchically and exhaustively, in terms of exact prime deterministic solutions. The simplest motivational example is the Riemann (actually Euler) zeta function. 

Genug. By now Nigel has already written another paper 🙂

The wife of Steven Weinberg

Audrey: 

I will be taking baby

 

to her second conference this summer, this time in Nova Scotia.

Predrag:

What's she talking about?

Once had a lunch with Steven Weinberg and his wife. At the Niels Bohr Institute. Guys like that always went through Niels Bohr Institute, because that's on the way to the Nobel Prize.

He talked without stop in his booming voice. He's very smart (reason I became a theorist, but that's another story), and articulate. It was all about particle physics and (perhaps?) the standard model. He ignored us ignoramuses, and only addressed his wife.

Now, she's a lawyer. They had moved to Texas, as UT Austin hired her as a law professor, a part of a package deal.

She talked back as a good theorist would. She knew all the words, you would not know that she knew no theoretical physics. It's just that she's been listening to his stuff for a long time.

So, I'm expecting The Baby to give a good talk. The ignoramuses will be awed.

I'm accompanying My Baby to Eresfjord, Norway. As the Mediterranean is becoming increasingly unlivable, Norway is inundated with tourists. Norwegians are desperate. They are rich, they do not need all these tourists. 

 As Swift puts it :

"for as to those filthy Yahoos, although there were few greater lovers of mankind at that time than myself, yet I confess I never saw any sensitive being so detestable on all accounts; and the more I came near them the more hateful they grew"

 

A brief history of the Center for Nonlinear Science parties

A bit of local history: Center of Nonlinear Science (CNS) is 25 years old. From the very inception, grad students used to organize spring or fall CNS picnics in the backyard of Sara & Predrag’s home, some of the old photos and videos are here:

https://cns.gatech.edu/CNS/album.html, https://YouTu.be/9a9Fpgui3WI, https://YouTu.be/uIEod4N7fcQ. Sara & Predrag would provide food, wine & live music, grad students would do the rest - invite friends, show up early to set everything up, play with the kids, not dance https://youtu.be/xxfT4q55AhU, clean up the aftermath

(from the 2002 picnic)

and take the remaining food to Howey for later delectation. 

Then came pandemic, things fell apart, followed by the The Board of Regents of the University System of Georgia hostile takeover of Georgia Tech. Check the regents out: https://www.usg.edu/regents . Business people with no interest in or experience with higher education, hand picked by the Governor. They are not your friends. Their goal is to complicate your lives, make a buck. Case at hand - I am told that now, after 141 years of GaTech running itself, the Schools are -inter alia- not allowed to run student events off campus, and no events that are not catered by the GaTech catering monopoly.

So that’s how CNS 25th year picnic became “Predrag’s 80th”, with not a sniff of anything Georgistan Technical School to it. We are celebrating 50 years that Predrag has not touched string theory, 100 years of Schrödinger equation, 250 years of Lagrange’s proof that the Solar System is stable (spoiler: it’s not), 500 years of Battle of Mohács. Predrag loves you, and is throwing a party for you.

Now, the personpower situation is this: at 979 Rosedale Rd there is an 80 year old professor, and a famed but mobility impaired computational neuroscientist. That’s it. We had not run this picnic 25 years ago, and sure are not in shape to do it today. Your professors will not save you. Of the ½ that opened the Evite email, ½ cannot figure out what to do with the [View Invitation] link. They might show up. Or not.

It’s our picnic. Marc has agreed to coordinate the grad students brigade. Sara and I will do our part.

Inhuman Resources