SphereUDE

SphereUDE.jl is a Julia package for non-parametric regression of data supported in three-dimensional spheres. It implements a simple universal differential equation (UDE) that naturally constrains trajectories to lie on the surface of a sphere. This has an important application in Paleomagnetism, where the objective is to fit Apparent Polar Wander Paths (APWPs) to reconstruct continents' past motion. In addition to sphere regression, SphereUDE.jl implements a series of improvements over previous modeling methods, such as

• Explicit sphere constraint that allows for univesality of regression
• Regularization on the path to incorporate physical priors
• Incorporation of temporal and spatial uncertainties
• Uncertainty quantification capabilities

To train a model with new unobserved data, we need to define the data, parameters, and regularization we want to use. Data are defined as

data = SphereData(times=times_samples, 
                  directions=X_true, 
                  kappas=nothing, 
                  L=L_true)

where times correspond to an array of the sampled times where we observed the three-dimensional vectors in directions. We can further add an array kappa to specify uncertainty in the directions according to the Fisher distribution in the sphere.

It is possible to add different types of regularizations at the same time by specifying an array of the type Regularization, which specifies the type of regularization being used and the diff_mode that specifies the underlying automatic differentiation machinery being used to compute the gradients.

regs = [Regularization(order=1, power=1.0, λ=0.001, diff_mode="Finite Differences"), 
        Regularization(order=0, power=2.0, λ=0.1, diff_mode="Finite Differences")]

Finally, the parameters include the regularization together with other customizable training parameters:

params = SphereParameters(tmin=0.0, tmax=100.0, 
                          reg=regs, 
                          u0=[0.0, 0.0, -1.0], ωmax=1.0, reltol=1e-12, abstol=1e-12,
                          niter_ADAM=1000, niter_LBFGS=600)

Training is finally being done with

results = train(data, params, rng, nothing)

with rng a random seed used for the initial setup of the neural network.

Here there is a simple example for the reconstruction of two solid rotations using SphereUDE.jl.

To install SphereUDE in a given environment, just do in the REPL:

julia> ] # enter Pkg mode
(@v1.10) pkg> activate MyEnvironment # or activate whatever path for the Julia environment
(MyEnvironment) pkg> add SphereUDE

To make plots using Matplotlib, Cartopy, and PMagPy, we install both PyCall.jl and PyPlot.jl and execute Python code directly from Julia. To do this setup manually, you can follow the next steps.

• Create a Python conda environment, based on this conda environment file, with all the required packages using conda env create -f environment.yml.
• Inside the Julia REPL, install both PyCall.jl and PyPlot.jl with ] add PyCall, Pyplot.
• Specify the Python path of the new environment with ENV["PYTHON"] = ..., where you should complete the path of the Python installation that shows when you do conda activate SphereUDE, which python.
• Inside the Julia REPL, execute Pkg.build("PyCall") to re-build PyCall with the new Python path.

You are ready to use Python from your Julia session!