docs: improve formula documentation based on review

- dipole_fitting: add 'an' before 'activation function'
- polarizability_fitting: add R^i shape, clarify local-frame tensor,
  explain symmetry preservation in A^T S A form

Author: njzjz-bot

deepmd/dpmodel/fitting/dipole_fitting.py

@@ -50,7 +50,7 @@ class DipoleFitting(GeneralFitting):
         \mathbf{M}^i = \mathcal{L}^{(n)} \circ \mathcal{L}^{(n-1)} \circ \cdots \circ \mathcal{L}^{(0)}(\mathcal{D}^i),
 
     where :math:`\mathcal{D}^i` is the descriptor and each layer :math:`\mathcal{L}^{(k)}`
-    is a fully connected layer with activation function.
+    is a fully connected layer with an activation function.
 
     Parameters
     ----------

deepmd/dpmodel/fitting/polarizability_fitting.py

@@ -45,24 +45,27 @@ class PolarFitting(GeneralFitting):
     r"""Fitting rotationally equivariant polarizability of the system.
 
     The polarizability tensor :math:`\boldsymbol{\alpha} \in \mathbb{R}^{3 \times 3}` is
-    computed from the fitting network output and the rotation matrix:
+    computed from the fitting network output and the rotation matrix
+    :math:`\mathbf{R}^i \in \mathbb{R}^{m_1 \times 3}` from the descriptor:
 
     **Diagonal fitting** (when `fit_diag=True`):
 
     .. math::
         \boldsymbol{\alpha}^i = \mathbf{R}^{i,T} \cdot \mathrm{diag}(\mathbf{p}^i) \cdot \mathbf{R}^i,
 
-    where :math:`\mathbf{p}^i \in \mathbb{R}^{m_1}` is the diagonal elements predicted by
-    the fitting network.
+    where :math:`\mathbf{p}^i \in \mathbb{R}^{m_1}` is the diagonal elements of the
+    local-frame polarizability predicted by the fitting network.
 
     **Full matrix fitting** (when `fit_diag=False`):
 
     .. math::
         \boldsymbol{\alpha}^i = \mathbf{R}^{i,T} \cdot \mathbf{P}^i \cdot \mathbf{R}^i,
 
-    where :math:`\mathbf{P}^i = \frac{1}{2}(\hat{\mathbf{P}}^i + \hat{\mathbf{P}}^{i,T}) \in \mathbb{R}^{m_1 \times m_1}`
-    is the symmetrized output of the fitting network (:math:`\hat{\mathbf{P}}^i` being the raw output),
-    ensuring :math:`\boldsymbol{\alpha}^i` is a symmetric tensor.
+    where :math:`\hat{\mathbf{P}}^i \in \mathbb{R}^{m_1 \times m_1}` is the raw output of
+    the fitting network, and :math:`\mathbf{P}^i = \frac{1}{2}(\hat{\mathbf{P}}^i + \hat{\mathbf{P}}^{i,T})`
+    is the symmetrized version. Since :math:`\mathbf{P}^i` is symmetric, the resulting
+    :math:`\boldsymbol{\alpha}^i` is guaranteed to be a symmetric tensor (matrix products
+    of the form :math:`A^T S A` preserve symmetry).
 
     The fitting network is: