Someone on Twitter suggested that tidal models are not understood “The tides connection to the moon should be revised.”. Unrolled thread after the “Read more” break
machine learning
QBO: Pattern recognition and signal processing
TANSTAAFL: there ain’t no such thing as a free lunch … but there’s always crumbs for the taking.
Machine learning won’t necessarily make a complete discovery by uncovering some ground-breaking pattern in isolation, but more likely a fragment or clue or signature that could lead somewhere. I always remind myself that there are infinitely many more non-linear formulations than linear ones potentially lurking in nature, yet humans are poorly-equipped to solve most non-linear relationships. ML has started to look at the tip of the non-linear iceberg and humans have to be alert when it uncovers a crumb. Recall that pattern recognition and signal processing are well-established disciplines in their own right, yet consider the situation of searching for patterns in signals hiding in the data but unknown in structure. That’s often all we are looking for — some foot-hold to start from.
Continue readingSubjectivity and Perception
What really is the color GREEN? What does it mean to different people? We actually have no idea. Now, consider The Dress on its 10 year anniversary. One can either argue with others whether it is BLUE and BLACK or WHITE and GOLD, or objectively look at it’s composition from a technical standpoint. Below is a cross-section of the dress image using the online image-picker application showing a palette from selected pixels in the image
Google Gemini Deep Research on ENSO
First evaluation of Gemini Advanced 1.5 Pro with Deep Research. Logged in with trial subscription and gave this as an initial prompt. Saved the results to a Google Docs file, and then created the following PDF. Note the top-level focus on this blog and published citations in the scope of Chapter 12 of Mathematical Geoenergy, even though the chapter was not directly cited — only via an embedded cite in a submitted ESD Ideas article, reference 1. Interesting that the Lin & Qian paper not cited.
Prompt: Explain tidal forcing behind ENSO using derivations based on reduced effective gravity on equatorial thermocline.
Tidal Gauge Differential
A climate science breakthrough likely won’t be on some massive computation but on a novel formulation that exposes some fundamental pattern (perhaps discovered by deep mining during a machine learning exercise). Over 10 years ago, I wrote on a blog post on how one can extract the ENSO signal by doing simple signal processing on a sea-level height (SLH) tidal time-series — in this case, at Fort Denison located in Sydney harbor.
The formulation/trick is to take the difference between the SLH reading and that from 2 years (24 months) prior, described here
Check the recent blog post Lunar Torque Controls All for context of how it fits in to the unified model.
The rationale for this 24 month difference is likely related to the sloshing of the ocean triggered on an annual basis. I think this is a pattern that any ML exercise would find with very little effort. After all, it didn’t take me that long to find it. But the point is that the ML configuration has to be open and flexible enough to be able to search, generate, and test for the same formulation. IOW, it may not find it if the configuration, perhaps focused on computationally massive PDEs, is too narrow. That was my comment to a RC post on applying machine learning to climate science, see the following link and subsequent quote:
Nick McGreivy commented on:
“ML-based parameterizations have to work well for thousands of years of simulations, and thus need to be very stable (no random glitches or periodic blow-ups) (harder than you might think). Bias corrections based on historical observations might not generalize correctly in the future.”
This same issue arises when using ML to simulate PDEs. The solution is to analytically calculate what the stability condition(s) is (are), then at each timestep to add some numerical diffusion that nudges the solution towards satisfying the stability condition(s). I imagine this same technique could be used for ML-based parametrizations.
Low Dimensions
“A key enabling assumption, sometimes called the manifold hypothesis [14], is that the data lie on or near a low-dimensional manifold; for physical systems with dissipation, such manifolds can often be rigorously shown to exist [15–18]. These manifolds enable a low-dimensional latent state representation, and hence, low-dimensional dynamical models. Linear manifold learning techniques, such as principal component analysis, cannot learn the nonlinear manifolds that represent most systems in nature. To do so, we require nonlinear methods, some of which are developed in [19–25] and reviewed in [26].“
Floryan, Daniel, and Michael D. Graham. “Data-driven discovery of intrinsic dynamics.” Nature Machine Intelligence 4.12 (2022): 1113-1120.
From <https://www.nature.com/articles/s42256-022-00575-4>
GC22A-04 Can ML beats chaos?
Abstract
“Chaos is typically blamed for the lack of predictability beyond a forecasting time window, which is on the order of 10 days for weather forecasting. However, on the one hand, most turbulent and chaotic systems exhibit strong coherence in the flow, such as synoptic events in weather or coherent structures in turbulence. On the other hand, most physical model might have additional structural errors that limit their capacity to correctly forecast beyond a certain time horizon, independent of chaos.
We will show in this presentation that most chaotic and turbulent flows can be predicted on relatively long range, at least longer than expected with both physical models and standard deep learning, using a combination of a reduced order model (that captures the low-dimensional coherent structures in the flow) and generative AI to obtain the crisp results and details of the flow. We will conclude stating that current AI-based weather models might not have achieved a plateau in performance yet, especially at longer time scales, and that physics-based weather model still have room for improvements. Reduced order models might not be able to beat chaos but can lead to much longer-range prediction than currently expected.“
https://agu.confex.com/agu/agu24/meetingapp.cgi/Paper/1522150
Heuristics
In the book Mathematical GeoEnergy, I mention the word heuristic or heuristics 82 times. In scientific research, it’s an important signpost because it identifies where a physical understanding is lacking, which is the point I was trying to make in a recent online discussion.
In the preface, I specifically cite the sunspot cycle as a heuristic, which I commented on:
“For example, the 11 -year sunspot cycle is considered a heuristic but not the annual or daily cycles, which are trivially explained. Tried to get ChatGPT to agree with me:
https://chat.openai.com/share/2706730c-2767-4060-b65e-08549b538d0e“chatGPT is correct.
whats your problem
I responded with two examples of heuristics that can conceivably be replaced by plausible and parsimonious physics.
Heuristic: The wobble is approximately 433 days, thought to be excited by fluctuations of mass on Earth achieving a natural resonance condition.
Physics: The wobble of precisely 433 days is a frequency side-band of the lunar draconic cycle interacting with the annual cycle. Not resonant just as the annual wobble is not resonant, and due to a forced angular momentum response.
QBO.
Heuristic: The oscillation is approximately 2+ years, thus the name “quasi-biennial”, thought to be excited by atmospheric waves.
Physics: The oscillation of 2.37 years period follows from the semi-annual oscillation at higher altitudes locked to the longer period by the lunar tidal forcing at lower altitudes of the stratosphere. The cycle is commensurate with simultaneous nodal crossings of both the moon and sun across the ecliptic plane.
In both these cases, the heuristic could be discarded and the description of the behaviors updated. Another example may be Milankovitch cycles, which replaces the heuristic of glacial cycles with a plausible physical mechanism that matches the observations. Whether Chandler, QBO, or Milankovitch will stand the test of time is another question, but these new models aren’t considered heuristics because they predict precise values for the observations. Discovering replacements for heuristics are rare nowadays as most of the heuristics (such as ocean tidal cycles) were resolved long ago. Circadian rhythms were a heuristic replaced by an encoded mechanism 20-30 years ago. Explaining predator-prey population cycles are at the heuristic stage still, IMO and may not get resolved as humans modify the environment.
The suggestion is that an unresolved heuristic is always a good candidate for a thesis topic.
NEW
Discussing relatively recent discoveries that replaced a heuristic, one that also came up is the Quantum Hall Effect. This is a subtle one because although the effect was experimentally discovered by von Klitzing, a Japanese team did roughly predict it a few years earlier, but added a caveat that they did not believe their own calculations! They also did not predict the quantization was in exact integer multiples. So that remained a heuristic for only a short time until it was physically explained (I remember giving a talk on this derivation for my solid-state physics class, which the instructor was very happy about). But then the experimental discovery of a fractional Quantum Hall Effect occurred, which apparently is still a heuristic because a strong consensus has yet to emerge on the physics behind it. Over the years, there have been at least 3 Nobel Prizes shared among 7 physicists to topological QHE research.
I bring this up because there’s a recent Quanta Magazine article dated July 18 titled “How Quantum Physicists Explained Earth’s Oscillating Weather Patterns” which describes how the QHE math can be conceivably applied to making predictions for equatorial patterns, and thus removing at least some of the heuristic nature. Geoffrey Vallis, who is an expert on geophysical fluid dynamics, is quoted in the article saying that the new result is a significant advance that will provide a “foundational understanding” of Earth’s fluid systems. The intriguing aspect is that this did not require the periodic order of a lattice — quoting from the article:
“I was surprised to see that topology could be defined in fluid systems without periodic order,” said Anton Souslov, a theoretical physicist at the University of Bath
Curry has a tweet on this, a reply tweet here because I am blocked
Canonical Cross-Validation
The only hope for a non-controlled-experiment-verified model to gain acceptance is either by (1) showing repeated success in predictions, or, precluding that due to long cycle time (2) producing rock-solid cross-validation results. Why? Let ChatGPT-4 answer:
Dante GPT-4
Trying out the Dante plug-in for GPT-4 LLM, using Mathematical GeoEnergy as a knowledge-base.
Try the prompt “What is the distribution of global variation of wind?”
The distribution of global variation of wind can be characterized by the entropic dispersion of wind speeds, which often follows a Rayleigh or Weibull distribution. This distribution shows a peak away from zero wind speeds and a rapid decline of frequency at higher velocities. Wind power distribution data from different regions, such as Ontario and northwest Germany, display the same damped exponential probability distribution function for the frequency of wind power derived from wind speed. This universal behavior in wind speed variability is influenced by the Earth’s climate and environment, which generate more states for the system to exist within, increasing the likelihood of variability.
Add “How does the BesselK distribution fit in to this?”
The BesselK distribution is another way to model the distribution of wind speeds and other natural phenomena. It is derived from the maximum entropy principle, which aims to find the most likely distribution given certain constraints. In the context of wind speed variability, the BesselK distribution can provide a good fit to the observed data, similar to the Rayleigh or Weibull distributions. The BesselK distribution has been used to model various natural processes, such as rainfall and wave heights, in addition to wind speeds. It is important to note that different distributions may be more suitable for specific regions or conditions, and the choice of the distribution should be based on the best fit to the observed data.
How is ENSO driven by lunar forcing ?
ENSO (El Niño-Southern Oscillation) is driven by lunar forcing through the interaction of lunar cycles with the Earth’s rotation rate and the seasonal cycle. The lunar cycles, such as the monthly and fortnightly cycles, are aliased due to their interaction with the annual cycle, resulting in ENSO cycles observed at the multiyear scale. The gravitational forcing from the lunar orbit and its interaction with the Sun’s yearly (lunar-perturbed) cycle can be applied to model ENSO effectively. The better the precise orbital path is obtained, the better resolved the fit.
The premise for the ENSO model is that a predictable lunar forcing leads to the erratic cycles of ENSO. The erratic nature of the cycles and any difficulty in matching these are directly the result of the physical aliasing that occurs from applying a low-frequency sampling (the yearly impulse) to a higher-frequency phenomenon (the monthly and fortnightly lunar cycles) which are incommensurate with the annual cycle.
How to solve LTE on the equatorial waveguide?
- Start with Laplace’s tidal equations for a fluid sheet of average thickness D, the vertical tidal elevation ζ, and the horizontal velocity components u and v (in the latitude φ and longitude λ directions).
- Simplify the equations by considering the small-angle approximation along the equator, where φ is close to zero.
- Reduce the set of Laplace’s tidal equations to a single equation by applying the chain rule and making an adjoint connection between a change in latitudinal forcing with a temporal change.
- Obtain a potentially highly nonlinear result by fixing φ(t) to a periodic function with a long-term mean of zero.
- Derive the horizontal longitudinal acceleration of the fluid (v) by applying the solution to Laplace’s third tidal equation in simplified form.
- Consider multiple solutions for the various standing waves, which will be used in fitting the model to the data.
- Apply the known tidal forcing along with an annual harmonic (and subharmonic) forcing and observe if the result correlates or matches the ENSO time series. By following these steps, you can generate a concise solution that has all the characteristics of the measured ENSO. Keep in mind that this approach is a simplification of the full Navier-Stokes equations and may not capture all the complexities of the equatorial waveguide.
Gist Evaluation
The Gist site on GitHub allows you to comment on posts very easily. For example, images of charts can be pasted in the discussion area. Also snippets of code can be added and updated, which is useful for neural net evaluation. The following is a link to an initial Gist area for evaluating LTE models.
