Outcomes meeting 12/10 - Poster

[amartinhuertas]

• @sghelichkhani will write a draft (bullet list) for the research aims (big picture, long term goal would be good), and the strong form of the PDE system we are actually solving. It would be good if we can talk also about how we are randomly generating the initial conditions and why this way of generating them is statistically relevant, etc. I think we should also may be mention why we focus on the temperature fields only, and neglect the information coming from the velocity and the pressure (by now).
• @rhyshawkins and @sghelichkhani provide some background about the Geoid problem as well.
• @rhyshawkins reviews the scripts to plot the geoid surface and errors. There seems to be something wrong there.
• Remove POD table (until we fully understand it). Transform LOSS table into a plot. S (can we find a better name?) is the x-axis and y-axis is the error. We can may be show a box or violin plot? https://www.geeksforgeeks.org/box-plot-visualization-with-pandas-and-seaborn/
• Remove the temperature plots of the lower animation from the poster. Increase the size of the images. Use new extra space to compare visually how do they look at different time steps, for different values of S (rows time steps, columns the value of S, just a suggestion).
  • What else?

[amartinhuertas]

@sghelichkhani FYI ... find attached the poster specification instructions poster-spec.pdf

[GiteonCaulfied]

Hi @amartinhuertas,

I've updated the poster according to the outcomes of today's meeting.

• @rhyshawkins reviews the scripts to plot the geoid surface and errors. There seems to be something wrong there.

I've checked my code again and it turned out I made a silly mistake when plotting the geoid errors that would make the error image all black, which means there's nothing wrong with @rhyshawkins 's code. I've replaced those geoid visualization in the poster and the report with correct ones, sorry about leading to some unnecessary confusion.

• Remove POD table (until we fully understand it). Transform LOSS table into a plot. S (can we find a better name?) is the x-axis and y-axis is the error. We can may be show a box or violin plot? https://www.geeksforgeeks.org/box-plot-visualization-with-pandas-and-seaborn/

The POD table has been removed and I've replaced the current LOSS table with a box plot.

• Remove the temperature plots of the lower animation from the poster. Increase the size of the images.

The lower animation (the LSTM one) has been removed and the size of the animation sheet has been increased.

... Use new extra space to compare visually how do they look at different time steps, for different values of S (rows time steps, columns the value of S, just a suggestion).

For this I think it may be better if I use row to represent different values of S (7 rows, where 6 of them are different values for S and one of them is for ground truth) and only shows 4-5 time steps in total using columns (start, 25th, 50th, 75th, end). Also, I plan to use a random simulation (e.g. the 657th simulation in the dataset) to compare the result for different values of S.

[rhyshawkins]

@GiteonCaulfied I've updated the example plotting script to use pyshtools which has the correct spherical harmonics CS phase and normalization. You will need to pip install pyshtools to use it.

[amartinhuertas]

I've checked my code again and it turned out I made a silly mistake when plotting the geoid errors that would make the error image all black, which means there's nothing wrong with @rhyshawkins 's code. I've replaced those geoid visualization in the poster and the report with correct ones, sorry about leading to some unnecessary confusion.

Ok, that makes sense.

For this I think it may be better if I use row to represent different values of S (7 rows, where 6 of them are different values for S and one of them is for ground truth) and only shows 4-5 time steps in total using columns (start, 25th, 50th, 75th, end). Also, I plan to use a random simulation (e.g. the 657th simulation in the dataset) to compare the result for different values of S.

Yes, whatever you think it is best to illustrate visually the outcome of your research.

[amartinhuertas]

Some additional comments:

• In the geoid plots, you are not showing the color palette. Thus, we cannot know, e.g., the magnitude of the errors.
• Not a big deal: can you make all figures with transparent background? png format supports transparent background.
• why $x A^T$ in the definition of the affine map, and not just $Ax$. The second is easier to grasp, I think.

[amartinhuertas]

Figure 7 caption. Replace "Best Geoid visualization" by "Visualization of most accurate Geoid prediction"

[GiteonCaulfied]

Hi @amartinhuertas @rhyshawkins ,

The poster and report has been updated.

When I was visualizing the plots of animations with different values of S, I realized that I made a mistake in my implementation that leads to the POD table we were struggle to understand during this week meeting. The mistake has been fixed and I've uploaded some brand new table for the Data loss, POD difference and relative POD difference in the last section of the Mantle Convection Chapter in my report (including the box plots).

Here's the correct result for your convenience:

Fortunately, the result now makes more sense. The animation sheet is also added to the poster. You can also find it in the following path of the repo: 2D_GIFs/larger_dataset(interpolated)/FNN_S_consecutive.gif

• In the geoid plots, you are not showing the color palette. Thus, we cannot know, e.g., the magnitude of the errors.

The color palettes are now shown for both the visualization image and the error image using @rhyshawkins 's latest code.

• Not a big deal: can you make all figures with transparent background? png format supports transparent background.

I've made some of the figures transparent and I will head for others later.

UPDATE: All figures (except for the first one) have been made transparent.

• why xAT in the definition of the affine map, and not just Ax. The second is easier to grasp, I think.

xAT is used here to be consistent with Pytorch's official document on Linear layer.

Here's the document: https://pytorch.org/docs/stable/generated/torch.nn.Linear.html

Figure 7 caption. Replace "Best Geoid visualization" by "Visualization of most accurate Geoid prediction"

The caption has been changed, thanks for the naming suggestion!

[rhyshawkins]

I've been trying to find a good general paper/reference on the geoid problem. A good short and high level introduction is the following Nature letter: Hager, B. The geoid and geodynamics. Nature 299, 104–105 (1982). https://doi.org/10.1038/299104a0

The key points are that:

• The geoid represents the shape of a gravity equipotential across the whole of Earth's surface. The units of the geoid are in metres and represent values +- some reference value. On an homogeneous planet, this surface would be 0 everywhere, but it is related to both the laterally varying density of the Earth but also heterogeneity in the Earths viscosity (see article above).
• Another insight from modelling that treats the geoid as being purely based on density variation near the surface has been shown to poorly explain the observed geoid measurements (which are now very accurate due to satellite measurements). The implication is that a large proportion of the geoid signal is due to processes within the Earth's mantle. See the following ref: https://doi.org/10.1098/rsta.1989.0038
• The forward modelling we use in the geoid problem assumes that viscosity varying in the radial direction (ie with depth) is the most important so the Earths viscosity is modelled as a 1D spherically symmetric viscosity model (the 257 input viscosity values) and computes a geoid surface in terms of spherical harmonics coefficients (the 60 output values) that can be used to construct the Geoid surface.

There is a book reference, but early work on this problem can be found in the following reference. This is essentially what the geoid code is doing but with some extra bits that I may have to talk to Sia about.

https://doi.org/10.1029/JB089iB07p05987

[GiteonCaulfied]

Hi @rhyshawkins @amartinhuertas,

A poster with more Geoid related content has been pushed to the repo. I added a few paragraphs and one figure to the geoid part of Introduction, Research Aims, Conclusion and Future Works based on the key points @rhyshawkins mentioned. I haven't updated the geoid part of the report with these background yet and I will do this soon.

[amartinhuertas]

Some comments:

• Uniform usage of periods to end sentences. You use it in some sentences, but not in all sentences. I am not sure if in posters one uses periods to finish sentences in bullet points (to check)
• Fig8: All 4 geoid plots seem the same
• Fig8: What geoid plots represent? (magnitude and units)
• Fig9: Why error on top in black? Are you using the same scale for both plots? I would use different scales for each plot, and thus different color bars.
• Fig10: It is not clear (to me) what you are comparing there and how the figure should be read. For example, given a figure, how can I now the corresponding time-step?
• Fig12: the same as Fig 10. I would explicitly put in the figure the values of S and T in the bottom and left margins. It should be easy to identify the value of S and the associated time stamp for each plot in the matrix.
• Fig 11: I would remove S=99, put it in another figure, or leave all the values together in the same plot, but zoom the initial values of S. Right now, there are two scales in the figure, and the reader cannot distinguish the differences among the first five values of S tested.

[amartinhuertas]

Add an "Acknowledgements" section with the following content:

This work was supported by computational resources provided by the Australian Government through the National Computational Infrastructure (NCI) under the ANU Merit Allocation Scheme.

[amartinhuertas]

Latent space size in the figure does not correspond to the one specified in the text. Please fix.

[GiteonCaulfied]

Hi @amartinhuertas,

I've updated the poster and pushed it to the repo.

• Uniform usage of periods to end sentences. You use it in some sentences, but not in all sentences. I am not sure if in posters one uses periods to finish sentences in bullet points (to check)

All sentences have periods now for consistence

• Fig8: What geoid plots represent? (magnitude and units)

Some description about magnitude and units has been added under the sample geoid visualization figure for clearance according to the key points listed by @rhyshawkins

• Fig8: All 4 geoid plots seem the same
• Fig9: Why error on top in black? Are you using the same scale for both plots? I would use different scales for each plot, and thus different color bars.

I've now used separate scales for all six plots, even though 4 plots seem the same, there are indeed some difference between their scales. The error on top is all black because the error is too small and I am using the same scale for both error plots, this is fixed now.

• Fig10: It is not clear (to me) what you are comparing there and how the figure should be read. For example, given a figure, how can I now the corresponding time-step?
• Fig12: the same as Fig 10. I would explicitly put in the figure the values of S and T in the bottom and left margins. It should be easy to identify the value of S and the associated time stamp for each plot in the matrix.

I've added some description under these figures to make things a bit more clear, will try to update the figures themselves such as putting in the figure the values of S and T in the bottom and left margins later.

• Fig 11: I would remove S=99, put it in another figure, or leave all the values together in the same plot, but zoom the initial values of S. Right now, there are two scales in the figure, and the reader cannot distinguish the differences among the first five values of S tested.

The S=99 is now removed and put in a separate figure. As expected, the first five values of S look better now.

Add an "Acknowledgements" section with the following content:

This work was supported by computational resources provided by the Australian Government through the National Computational Infrastructure (NCI) under the ANU Merit Allocation Scheme.

The "Acknowledgements" section has been added, thanks for the suggestion!

Latent space size in the figure does not correspond to the one specified in the text. Please fix.

Sorry about that, this is fixed now.

[amartinhuertas]

• Typo in Fig. 11 title
• Period in the caption of figures.

[GiteonCaulfied]

Hi @amartinhuertas,

I've pushed the updated report with some geoid background to the repo. Also, Fig12 and Fig10 in the poster now are added with some lines and text to make things more clear.

• Typo in Fig. 11 title
• Period in the caption of figures.

This is fixed now, period are also added to the caption of figures in the report.

[amartinhuertas]

I think we are still missing in the poster a non-expert motivation/big picture of the project.

Some ideas (to be refined):

• In computational geophysics (as in many other fields), one typically knows the continuum equations that govern a given physical phenomena (e.g. fluid flow inside the Earth's Mantle), but unfortunately, due to scarceness and/or uncertainty of data measurements, one cannot feed these equations with this data out-of-the-box to generate accurate predictions.
• To cope with this, one tries to solve a so-called inverse problem. The objective is to find a model using gradient-optimization techniques which best fits to the available data coming from partial/incomplete/noisy measurements.
• The standard approach to solve this inverse problem is the so-called adjoint method [REFERENCE]. This method is very computationally demanding, as it requires to solve a time-dependent nonlinear forward problem and an adjoint problem at each gradient-optimization step.
• To reduce computational demands, the (long-term) goal of this project is evaluate the suitability of deep learning techniques as surrogate models that enable one to approximate the forward and adjoint problems which much less computational demands.
• As a first step towards this goal, we have tackled the forward problem of two classic problems in geodynamics, namely the Geoid and 2D Mantle convection problems.