docs: add theory section to DPA3 documentation

doc/model/dpa3.md

@@ -4,26 +4,105 @@
 **Supported backends**: PyTorch {{ pytorch_icon }}, JAX {{ jax_icon }}, DP {{ dpmodel_icon }}
 :::
 
-DPA3 is an advanced interatomic potential leveraging the message passing architecture.
-Designed as a large atomic model (LAM), DPA3 is tailored to integrate and simultaneously train on datasets from various disciplines,
-encompassing diverse chemical and materials systems across different research domains.
-Its model design ensures exceptional fitting accuracy and robust generalization both within and beyond the training domain.
-Furthermore, DPA3 maintains energy conservation and respects the physical symmetries of the potential energy surface,
-making it a dependable tool for a wide range of scientific applications.
+DPA3 is an advanced interatomic potential based on message passing.
+As a large atomic model (LAM), it is designed to integrate and jointly train on datasets from different domains,
+covering diverse chemical and materials systems.
+Its architecture provides high fitting accuracy and robust generalization both within and beyond the training domain.
+DPA3 also preserves energy conservation and the physical symmetries of the potential energy surface,
+making it a reliable model for a wide range of scientific applications.
 
 Reference: [DPA3 paper](https://arxiv.org/abs/2506.01686).
 
 Training example: `examples/water/dpa3/input_torch.json`.
 
+## Theory
+
+DPA3 is a graph neural network operating on the Line Graph Series (LiGS) constructed from atomic configurations.
+
+### Line Graph Series (LiGS)
+
+Given an initial graph $G^{(1)}$ representing the atomic system, where atoms are vertices and pairs of neighboring atoms within a cutoff radius $r_c$ are edges, the line graph transform $\mathcal{L}$ constructs a new graph $G^{(2)} = \mathcal{L}(G^{(1)})$ by:
+
+1. Converting each edge in $G^{(1)}$ to a vertex in $G^{(2)}$
+1. Creating edges in $G^{(2)}$ between vertices whose corresponding edges in $G^{(1)}$ share a common vertex
+
+Recursively applying this transform generates a series of graphs $\{G^{(1)}, G^{(2)}, \ldots, G^{(K)}\}$, where $G^{(k)} = \mathcal{L}(G^{(k-1)})$. This sequence is called the Line Graph Series (LiGS) of order $K$.
+
+Geometrically, vertices in $G^{(1)}$, $G^{(2)}$, $G^{(3)}$, and $G^{(4)}$ correspond to atoms, bonds (pairs of atoms), angles (three atoms with two bonds sharing a common atom), and dihedral angles (four atoms with two angles sharing a common bond), respectively.
+
+### Message Passing on LiGS
+
+DPA3 performs message passing across all graphs in the LiGS. At layer $l$, the vertex and edge features on graph $G^{(k)}$ are denoted as $\mathbf{v}_\alpha^{(k,l)} \in \mathbb{R}^{d_v^{(k)}}$ and $\mathbf{e}_{\alpha\beta}^{(k,l)} \in \mathbb{R}^{d_e^{(k)}}$, where $\alpha$ and $\alpha\beta$ denote vertex and edge indices, and $d_v^{(k)}$, $d_e^{(k)}$ are per-graph feature dimensions (for example, in `RepFlowArgs`: $d_v^{(1)}=n_\text{dim}$, $d_e^{(1)}=e_\text{dim}$, $d_v^{(2)}=e_\text{dim}$, and $d_e^{(2)}=a_\text{dim}$).
+
+The feature update follows a recursive formulation with residual connections. We use $\text{Update}_V$ and $\text{Update}_E$ to distinguish vertex and edge update modules, respectively:
+
+**Edge updates (all graphs $G^{(k)}$):**
+Edge features are updated based on messages from connected vertices:
+
+```math
+\mathbf{e}_{\alpha\beta}^{(k,l+1)} = \mathbf{e}_{\alpha\beta}^{(k,l)} + \text{Update}_E^{(k)}\left(\mathbf{e}_{\alpha\beta}^{(k,l)}, \mathbf{v}_\alpha^{(k,l)}, \mathbf{v}_\beta^{(k,l)}\right)
+```
+
+**For $G^{(1)}$ (initial graph, vertex update):**
+Vertex features are updated through self-message and symmetrization:
+
+```math
+\mathbf{v}_\alpha^{(1,l+1)} = \mathbf{v}_\alpha^{(1,l)} + \text{Update}_V^{(1)}\left(\mathbf{v}_\alpha^{(1,l)}, \{\mathbf{e}_{\alpha\beta}^{(1,l)}\}_{\beta \in \mathcal{N}(\alpha)}\right)
+```
+
+**For $G^{(k)}$ with $k > 1$ (vertex identity):**
+The vertex feature of $G^{(k)}$ is identical to the edge feature of $G^{(k-1)}$:
+
+```math
+\mathbf{v}_\alpha^{(k,l)} = \mathbf{e}_{\alpha\beta}^{(k-1,l)}
+```
+
+where $(\alpha,\beta)$ denotes the edge in $G^{(k-1)}$ corresponding to vertex $\alpha$ in $G^{(k)}$. This identity eliminates redundant storage.
+
+The same edge update rule also applies to $G^{(1)}$ edge features $\mathbf{e}_{\alpha\beta}^{(1,l)}$ (i.e., with $k=1$ in $\text{Update}_E^{(k)}$). Therefore, these features evolve across layers and, via the $\mathbf{v}^{(2,l)}$-$\mathbf{e}^{(1,l)}$ identity, drive the updates on $G^{(2)}$.
+
+### Descriptor Construction
+
+The final vertex features of $G^{(1)}$ serve as the descriptor representing the local environment of each atom:
+
+```math
+\mathcal{D}^\alpha = \mathbf{v}_\alpha^{(1,L)}
+```
+
+where $L$ is the total number of layers.
+
+The descriptor output is then consumed by downstream fitting/model components for property prediction (e.g., energy). See the model/fitting documentation for those equations and training objectives.
+
+### Physical Symmetries and Conservative Forces
+
+DPA3 respects the physical symmetries of the potential energy surface:
+
+1. **Translational invariance**: The model depends only on relative coordinates $\mathbf{r}_{\alpha\beta} = \mathbf{r}_\beta - \mathbf{r}_\alpha$, not absolute positions.
+
+1. **Rotational invariance**: The final descriptor is rotationally invariant; intermediate equivariant representations are used internally and contracted to produce invariant atomic features.
+
+1. **Permutational invariance**: Atoms of the same chemical species are treated identically under permutation symmetry operations (re-labeling) of identical atoms.
+
+In addition, DPA3 is inherently conservative: forces are derived from energy gradients:
+
+```math
+\mathbf{F}_\alpha = -\frac{\partial E}{\partial \mathbf{r}_\alpha}
+```
+
+Virials are similarly derived from cell tensor gradients, ensuring the model is conservative and suitable for molecular dynamics simulations.
+
+### Default Configuration
+
+DPA3 uses LiGS order $K=2$ as the default configuration, which was found effective in prior work ([DPA3 paper](https://arxiv.org/abs/2506.01686)). The model supports scaling through increasing the number of layers $L$ (e.g., DPA3-L3, DPA3-L6, DPA3-L12, DPA3-L24).
+
 ## Hyperparameter tests
 
-We systematically conducted DPA3 training on six representative DFT datasets (available at [AIS-Square](https://www.aissquare.com/datasets/detail?pageType=datasets&name=DPA3_hyperparameter_search&id=316)):
-metallic systems (`Alloy`, `AlMgCu`, `W`), covalent material (`Boron`), molecular system (`Drug`), and liquid water (`Water`).
-Under consistent training conditions (0.5M training steps, batch_size "auto:128"),
-we rigorously evaluated the impacts of some critical hyperparameters on validation accuracy.
+We systematically trained DPA3 on six representative DFT datasets (available at [AIS-Square](https://www.aissquare.com/datasets/detail?pageType=datasets&name=DPA3_hyperparameter_search&id=316)): metallic systems (`Alloy`, `AlMgCu`, `W`), a covalent material (`Boron`), a molecular system (`Drug`), and liquid water (`Water`).
+Under consistent training conditions (0.5M training steps, `batch_size` = `auto:128`),
+we evaluated the impact of key hyperparameters on validation accuracy.
 
-The comparative analysis focused on average RMSEs (Root Mean Square Error) for both energy, force and virial predictions across all six systems,
-with results tabulated below to guide scenario-specific hyperparameter selection:
+The comparative analysis focused on average RMSEs (Root Mean Square Error) for energy, force, and virial predictions across the six systems.
+The results are summarized below to guide scenario-specific hyperparameter selection:
 
 | Model            | comment         | nlayers | n_dim   | e_dim  | a_dim | e_sel   | a_sel  | start_lr | stop_lr  | loss prefactors           | rmse_e (meV/atom) | rmse_f (meV/Å) | rmse_v (meV/atom) | Training wall time (h) |
 | ---------------- | --------------- | ------- | ------- | ------ | ----- | ------- | ------ | -------- | -------- | ------------------------- | ----------------- | -------------- | ----------------- | ---------------------- |
@@ -35,10 +114,10 @@ with results tabulated below to guide scenario-specific hyperparameter selection
 |                  | Large sel       | 6       | 256     | 128    | 32    | **154** | **48** | 1e-3     | 3e-5     | 0.2\|20, 100\|60, 0.02\|1 | 4.76              | 78.4           | 40.2              | 31.8                   |
 | DPA2-L6 (medium) | Default         | 6       | -       | -      | -     | -       | -      | 1e-3     | 3.51e-08 | 0.02\|1, 1000\|1, 0.02\|1 | 12.12             | 109.3          | 83.1              | 12.2                   |
 
-The loss prefactors (0.2|20, 100|60, 0.02|1) correspond to (`start_pref_e`|`limit_pref_e`, `start_pref_f`|`limit_pref_f`, `start_pref_v`|`limit_pref_v`) respectively.
+The loss prefactors (0.2|20, 100|60, 0.02|1) correspond to (`start_pref_e`|`limit_pref_e`, `start_pref_f`|`limit_pref_f`, `start_pref_v`|`limit_pref_v`), respectively.
 Virial RMSEs were averaged exclusively for systems containing virial labels (`Alloy`, `AlMgCu`, `W`, and `Boron`).
 
-Note that we set `float32` in all DPA3 models, while `float64` in other models by default.
+Note that all DPA3 models use `float32`, while other models use `float64` by default.
 
 ## Requirements of installation from source code {{ pytorch_icon }} {{ paddle_icon }}