pde_sphere.py

# coding=utf-8

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import torch
import torch.nn as nn
import torch_harmonics as th
from torch_harmonics.quadrature import _precompute_longitudes

import math
import numpy as np


class SphereSolver(nn.Module):
    """
    Solver class on the sphere. Can solve the following PDEs:
    - Allen-Cahn equation
    - Ginzburg-Landau equation
    
    Parameters
    -----------
    nlat : int
        Number of latitude points
    nlon : int
        Number of longitude points
    dt : float
        Time step size
    lmax : int, optional
        Maximum l mode for spherical harmonics, by default None
    mmax : int, optional
        Maximum m mode for spherical harmonics, by default None
    grid : str, optional
        Grid type ("equiangular", "legendre-gauss", "lobatto"), by default "equiangular"
    radius : float, optional
        Radius of the sphere, by default 1.0
    coeff : float, optional
        Coefficient for the PDE, by default 0.001
    """

    def __init__(self, nlat, nlon, dt, lmax=None, mmax=None, grid="equiangular", radius=1.0, coeff=0.001):
        super().__init__()

        # time stepping param
        self.dt = dt

        # grid parameters
        self.nlat = nlat
        self.nlon = nlon
        self.grid = grid

        # physical sonstants
        self.register_buffer('radius', torch.as_tensor(radius, dtype=torch.float64))
        self.register_buffer('coeff', torch.as_tensor(coeff, dtype=torch.float64))

        # SHT
        self.sht = th.RealSHT(nlat, nlon, lmax=lmax, mmax=mmax, grid=grid, csphase=False)
        self.isht = th.InverseRealSHT(nlat, nlon, lmax=lmax, mmax=mmax, grid=grid, csphase=False)

        self.lmax = lmax or self.sht.lmax
        self.mmax = lmax or self.sht.mmax

        # compute gridpoints
        if self.grid == "legendre-gauss":
            cost, _ = th.quadrature.legendre_gauss_weights(self.nlat, -1, 1)
        elif self.grid == "lobatto":
            cost, _ = th.quadrature.lobatto_weights(self.nlat, -1, 1)
        elif self.grid == "equiangular":
            cost, _ = th.quadrature.clenshaw_curtiss_weights(self.nlat, -1, 1)

        # apply cosine transform and flip them
        lats = -torch.arcsin(cost)
        lons = _precompute_longitudes(self.nlon)

        self.lmax = self.sht.lmax
        self.mmax = self.sht.mmax

        l = torch.arange(0, self.lmax).reshape(self.lmax, 1).cdouble()
        l = l.expand(self.lmax, self.mmax)
        # the laplace operator acting on the coefficients is given by l (l + 1)
        lap = - l * (l + 1) / self.radius**2
        invlap = - self.radius**2 / l / (l + 1)
        invlap[0] = 0.

        # register all
        self.register_buffer('lats', lats)
        self.register_buffer('lons', lons)
        self.register_buffer('l', l)
        self.register_buffer('lap', lap)
        self.register_buffer('invlap', invlap)

    def grid2spec(self, u):
        """
        Convert spatial data to spectral coefficients.
        
        Parameters
        -----------
        u : torch.Tensor
            Spatial data tensor
            
        Returns
        -------
        torch.Tensor
            Spectral coefficients
        """
        return self.sht(u)

    def spec2grid(self, uspec):
        """
        Convert spectral coefficients to spatial data.
        
        Parameters
        -----------
        uspec : torch.Tensor
            Spectral coefficients tensor
            
        Returns
        -------
        torch.Tensor
            Spatial data
        """
        return self.isht(uspec)

    def dudtspec(self, uspec, pde='allen-cahn'):
        """
        Compute the time derivative of spectral coefficients for different PDEs.
        
        Parameters
        -----------
        uspec : torch.Tensor
            Spectral coefficients
        pde : str, optional
            PDE type ("allen-cahn", "ginzburg-landau"), by default "allen-cahn"
            
        Returns
        -------
        torch.Tensor
            Time derivative of spectral coefficients
            
        Raises
        ------
        NotImplementedError
            If PDE type is not supported
        """
        if pde == 'allen-cahn':
            ugrid = self.spec2grid(uspec)
            u3spec  = self.grid2spec(ugrid**3)
            dudtspec = self.coeff*self.lap*uspec + uspec - u3spec
        elif pde == 'ginzburg-landau':
            ugrid = self.spec2grid(uspec)
            u3spec  = self.grid2spec(ugrid**3)
            dudtspec = uspec + (1. + 2.j)*self.coeff*self.lap*uspec - (1. + 2.j)*u3spec
        else:
            raise NotImplementedError(f"PDE type {pde} not implemented")

        return dudtspec

    def randspec(self):
        """
        Generate random spectral data on the sphere.
        
        Returns
        -------
        torch.Tensor
            Random spectral coefficients
        """
        rspec = torch.randn_like(self.lap) / 4 / torch.pi
        return rspec


    def plot_griddata(self, data, fig, cmap='twilight_shifted', vmax=None, vmin=None, projection='3d', title=None, antialiased=False):
        """
        Plot data on the sphere grid. Requires cartopy for 3d plots.
        
        Parameters
        -----------
        data : torch.Tensor
            Data to plot
        fig : matplotlib.figure.Figure
            Figure to plot on
        cmap : str, optional
            Colormap name, by default 'twilight_shifted'
        vmax : float, optional
            Maximum value for color scaling, by default None
        vmin : float, optional
            Minimum value for color scaling, by default None
        projection : str, optional
            Projection type ("mollweide", "3d"), by default "3d"
        title : str, optional
            Plot title, by default None
        antialiased : bool, optional
            Whether to use antialiasing, by default False
            
        Returns
        -------
        matplotlib.collections.QuadMesh
            The plotted image object
            
        Raises
        ------
        NotImplementedError
            If projection type is not supported
        """
        import matplotlib.pyplot as plt

        lons = self.lons.squeeze() - torch.pi
        lats = self.lats.squeeze()

        if data.is_cuda:
            data = data.cpu()
            lons = lons.cpu()
            lats = lats.cpu()

        Lons, Lats = np.meshgrid(lons, lats)

        if projection == 'mollweide':

            #ax = plt.gca(projection=projection)
            ax = fig.add_subplot(projection=projection)
            im = ax.pcolormesh(Lons, Lats, data, cmap=cmap, vmax=vmax, vmin=vmin)
            # ax.set_title("Elevation map of mars")
            ax.grid(True)
            ax.set_xticklabels([])
            ax.set_yticklabels([])
            plt.colorbar(im, orientation='horizontal')
            plt.title(title)

        elif projection == '3d':

            import cartopy.crs as ccrs

            proj = ccrs.Orthographic(central_longitude=0.0, central_latitude=25.0)

            #ax = plt.gca(projection=proj, frameon=True)
            ax = fig.add_subplot(projection=proj)
            Lons = Lons*180/math.pi
            Lats = Lats*180/math.pi

            # contour data over the map.
            im = ax.pcolormesh(Lons, Lats, data, cmap=cmap, transform=ccrs.PlateCarree(), antialiased=antialiased, vmax=vmax, vmin=vmin)
            plt.title(title, y=1.05)

        else:
            raise NotImplementedError(f"Projection {projection} not implemented")

        return im

    def plot_specdata(self, data, fig, **kwargs):
        """
        Plot spectral data by converting to spatial data first.
        
        Parameters
        -----------
        data : torch.Tensor
            Spectral data to plot
        fig : matplotlib.figure.Figure
            Figure to plot on
        **kwargs
            Additional arguments passed to plot_griddata
            
        Returns
        -------
        matplotlib.collections.QuadMesh
            The plotted image object
        """
        return self.plot_griddata(self.isht(data), fig, **kwargs)