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r"""Spherical Harmonics as functions of Euler angles"""
import math
from typing import List, Tuple
import torch
from torch import fx
from sympy import Integer, Poly, diff, factorial, pi, sqrt, symbols
from e3nn.util.jit import compile_mode
from e3nn import o3, get_optimization_defaults
def _conditional_script(fn):
"""apply torch.jit.script only if jit_mode is 'script'"""
if get_optimization_defaults()["jit_mode"] == "script":
return torch.jit.script(fn)
return fn
@compile_mode("script")
class SphericalHarmonicsAlphaBeta(torch.nn.Module):
"""JITable module version of :meth:`e3nn.o3.spherical_harmonics_alpha_beta`.
Parameters are identical to :meth:`e3nn.o3.spherical_harmonics_alpha_beta`.
"""
normalization: str
_ls_list: List[int]
_lmax: int
def __init__(self, l, normalization: str = "integral") -> None:
super().__init__()
if isinstance(l, o3.Irreps):
ls = [l for mul, (l, p) in l for _ in range(mul)]
elif isinstance(l, int):
ls = [l]
else:
ls = list(l)
self._ls_list = ls
self._lmax = max(ls)
self.legendre = Legendre(ls)
self.normalization = normalization
def forward(self, alpha: torch.Tensor, beta: torch.Tensor) -> torch.Tensor:
y, z = beta.cos(), beta.sin()
sha = spherical_harmonics_alpha(self._lmax, alpha.flatten()) # [z, m]
shy = self.legendre(y.flatten(), z.flatten()) # [z, l * m]
out = _mul_m_lm([(1, l) for l in self._ls_list], sha, shy)
if self.normalization == "norm":
out.div_(
torch.cat(
[
(math.sqrt(2 * l + 1) / math.sqrt(4 * math.pi))
* torch.ones(2 * l + 1, dtype=out.dtype, device=out.device)
for l in self._ls_list
]
)
)
elif self.normalization == "component":
out.mul_(math.sqrt(4 * math.pi))
return out.reshape(alpha.shape + (shy.shape[1],))
def spherical_harmonics_alpha_beta(l, alpha, beta, *, normalization: str = "integral"):
r"""Spherical harmonics of :math:`\vec r = R_y(\alpha) R_x(\beta) e_y`
.. math:: Y^l(\alpha, \beta) = S^l(\alpha) P^l(\cos(\beta))
where :math:`P^l` are the `Legendre` polynomials
Parameters
----------
l : int or list of int
degree of the spherical harmonics.
alpha : `torch.Tensor`
tensor of shape ``(...)``.
beta : `torch.Tensor`
tensor of shape ``(...)``.
Returns
-------
`torch.Tensor`
a tensor of shape ``(..., 2l+1)``
"""
sh = SphericalHarmonicsAlphaBeta(l, normalization=normalization)
return sh(alpha, beta)
@_conditional_script
def spherical_harmonics_alpha(l: int, alpha: torch.Tensor) -> torch.Tensor:
r""":math:`S^l(\alpha)` of `spherical_harmonics_alpha_beta`
Parameters
----------
l : int
degree of the spherical harmonics.
alpha : `torch.Tensor`
tensor of shape ``(...)``.
Returns
-------
`torch.Tensor`
a tensor of shape ``(..., 2l+1)``
"""
alpha = alpha.unsqueeze(-1) # [..., 1]
m = torch.arange(1, l + 1, dtype=alpha.dtype, device=alpha.device) # [1, 2, 3, ..., l]
cos = torch.cos(m * alpha) # [..., m]
m = torch.arange(l, 0, -1, dtype=alpha.dtype, device=alpha.device) # [l, l-1, l-2, ..., 1]
sin = torch.sin(m * alpha) # [..., m]
out = torch.cat(
[
math.sqrt(2) * sin,
torch.ones_like(alpha),
math.sqrt(2) * cos,
],
dim=alpha.ndim - 1,
)
return out # [..., m]
@compile_mode("script")
class Legendre(fx.GraphModule):
# pylint: disable=abstract-method
def __init__(self, ls) -> None:
super().__init__(self, fx.Graph())
# == generate code ==
graph = self.graph
z = fx.Proxy(graph.placeholder("z", torch.Tensor))
y = fx.Proxy(graph.placeholder("y", torch.Tensor))
out = z.new_zeros(z.shape + (sum(2 * l + 1 for l in ls),))
i = 0
for l in ls:
leg = []
for m in range(l + 1):
p = _poly_legendre(l, m)
p = list(p.items())
(zn, yn), c = p[0]
x = float(c) * z**zn * y**yn
for (zn, yn), c in p[1:]:
x += float(c) * z**zn * y**yn
leg.append(x.unsqueeze(-1))
for m in range(-l, l + 1):
out.narrow(-1, i, 1).copy_(leg[abs(m)])
i += 1
graph.output(out.node, torch.Tensor)
self.recompile()
def _poly_legendre(l, m):
r"""
polynomial coefficients of legendre
y = sqrt(1 - z^2)
"""
z, y = symbols("z y", real=True)
return Poly(_sympy_legendre(l, m), domain="R", gens=(z, y)).as_dict()
def _sympy_legendre(l, m) -> float:
r"""
en.wikipedia.org/wiki/Associated_Legendre_polynomials
- remove two times (-1)^m
- use another normalization such that P(l, -m) = P(l, m)
- remove (-1)^l
y = sqrt(1 - z^2)
"""
l = Integer(l)
m = Integer(abs(m))
z, y = symbols("z y", real=True)
ex = 1 / (2**l * factorial(l)) * y**m * diff((z**2 - 1) ** l, z, l + m)
ex *= sqrt((2 * l + 1) / (4 * pi) * factorial(l - m) / factorial(l + m))
return ex
@_conditional_script
def _mul_m_lm(mul_l: List[Tuple[int, int]], x_m: torch.Tensor, x_lm: torch.Tensor) -> torch.Tensor:
"""
multiply tensor [..., l * m] by [..., m]
"""
l_max = x_m.shape[-1] // 2
out = []
i = 0
for mul, l in mul_l:
d = mul * (2 * l + 1)
x1 = x_lm[..., i : i + d] # [..., mul * m]
x1 = x1.reshape(x1.shape[:-1] + (mul, 2 * l + 1)) # [..., mul, m]
x2 = x_m[..., l_max - l : l_max + l + 1] # [..., m]
x2 = x2.reshape(x2.shape[:-1] + (1, 2 * l + 1)) # [..., mul=1, m]
x = x1 * x2
x = x.reshape(x.shape[:-2] + (d,))
out.append(x)
i += d
return torch.cat(out, dim=-1)
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