Project: s41467-024-49516-2 | Function: message (python)

system: You are an expert Python programmer implementing algorithms from scientific papers.
Complete the masked function based on the algorithm description.
Follow Python best practices and maintain consistent code style.
Return **only** the complete Python function code.

user: Complete this Python function:

**DESCRIPTION:** Implements the core message passing mechanism with attention-based vector-scalar interactions for the ViSNet architecture. Computes attention coefficients between protein residues using query-key products modulated by distance cutoffs and edge features.

**KEY TERMS:** {
  "q_i": "query features from source nodes i",
  "k_j": "key features from target nodes j",
  "v_j": "value features from target nodes j",
  "dk": "edge attribute projection for attention modulation",
  "dv": "edge attribute projection for value modulation",
  "attn": "attention coefficients with distance cutoff weighting"
}

**CONTEXT CODE:**         # print("k",k.shape)
        # print("v",v.shape)

        vec1, vec2, vec3 = torch.split(self.vec_proj(vec), self.hidden_channels, dim=-1)
        # print(self.vec_proj(vec).shape) #把256通过linear变成768，然后分成3份，每份256，
        # 没有bias，意味着nn中只有乘法没有加法
        # print("vec1",vec1.shape) #vec1 shape是x.size(0), ((self.lmax + 1) ** 2) - 1, hidden_channels
        # print("vec2",vec2.shape) #vec2 shape是x.size(0), ((self.lmax + 1) ** 2) - 1, hidden_channels
        # print("vec3",vec3.shape) #vec3 shape是x.size(0), ((self.lmax + 1) ** 2) - 1, hidden_channels
        vec_dot = (vec1 * vec2).sum(dim=1)  #每个位置的元素相乘，然后在第一维上求和
        # print(vec1,vec2,vec1 * vec2,vec_dot)
        # print("vec_dot",vec_dot.shape) 38*256 if x.size(0)=38 and hidden_channels=256
        vec_dot = self.act(self.v_dot_proj(vec_dot))  # 过一个nn和激活函数

        dk = (
            self.act(self.dk_proj(f_ij)).reshape(-1, self.num_heads, self.head_dim)        #边属性投影1，转为和x一样的shape
            if self.dk_proj is not None
            else None
        )
        dv = (
            self.act(self.dv_proj(f_ij)).reshape(-1, self.num_heads, self.head_dim)          # 边属性投影2，转为和x一样的shape
            if self.dv_proj is not None
            else None
        )

        # propagate_type: (q: Tensor, k: Tensor, v: Tensor, vec: Tensor, dk: Tensor, dv: Tensor, r_ij: Tensor, d_ij: Tensor)
        x, vec_out = self.propagate(  # 在propagate中调用message和update，并不显式编辑propgate函数，
        # 运行时，qkv分别是每一条边的起点，终点的qkv，最终使用message和edge_update达到更新节点和边的效果
            edge_index,
            q=q,
            k=k,
            v=v,
            vec=vec,
            dk=dk,
            dv=dv,
            r_ij=r_ij,
            d_ij=d_ij,
            size=None,
        )
        # edge_updater_type: (vec: Tensor, d_ij: Tensor, f_ij: Tensor)
        o1, o2, o3 = torch.split(self.o_proj(x), self.hidden_channels, dim=1)  
        # output投影，分成3份，其中2份用来更新dx，1份用来更新dvec，dvec同时与vec_out也有关
        dx = vec_dot * o2 + o3
        dvec = vec3 * o1.unsqueeze(1) + vec_out
        if not self.last_layer:
            df_ij = self.edge_updater(edge_index, vec=vec, d_ij=d_ij, f_ij=f_ij)
            return dx, dvec, df_ij
        else:
            return dx, dvec, None

    def message(self, q_i, k_j, v_j, vec_j, dk, dv, r_ij, d_ij):  #i是源节点，j是目标节点
        """Implements the core message passing mechanism with attention-based vector-scalar interactions for the ViSNet architecture."""
        raise NotImplementedError("Function has been masked for testing purposes")
    
    def edge_update(self, vec_i, vec_j, d_ij, f_ij):

        w1 = self.vector_rejection(self.trg_proj(vec_i), d_ij)   # 把源节点的vec投影到边向量上，得到w1，
        # 边向量有方向，因此源节点出发是d_ij，目标节点出发是-d_ij
        w2 = self.vector_rejection(self.src_proj(vec_j), -d_ij) # 把目标节点的vec投影到边向量上，得到w2
        w_dot = (w1 * w2).sum(dim=1)  # 把w1和w2的每个元素相乘，然后求和，得到w_dot
        w_dot = self.act(self.w_dot_proj(w_dot))  # 把w_dot过一个nn

        # vector_rejection的内容：
        # def vector_rejection(self, vec, d_ij):
        # vec_proj = (vec * d_ij.unsqueeze(2)).sum(dim=1, keepdim=True)
        # # print(vec_proj.shape,'vec_proj shape')
        # # vec 78, 3, 256, d_ij,78, 3, vec_proj 78, 1, 256
        # # 先用边向量乘vec（节点的vector属性，过了神经网络），
        # # 然后求和，然后再乘边向量，然后再用节点的vector属性减去这个值
        # return vec - vec_proj * d_ij.unsqueeze(2)


        f1, f2 = torch.split(
            self.act(self.f_proj(f_ij)),
            self.hidden_channels,
            dim=1
        )
        
        return f1 * w_dot + f2  #对边属性做残差操作，加上源节点和目标节点vec的投影


    def aggregate(
        self,
        features: Tuple[torch.Tensor, torch.Tensor],
        index: torch.Tensor,
        ptr: Optional[torch.Tensor],
        dim_size: Optional[int],
    ) -> Tuple[torch.Tensor, torch.Tensor]:
        x, vec = features
        x = scatter(x, index, dim=self.node_dim, dim_size=dim_size)
        vec = scatter(vec, index, dim=self.node_dim, dim_size=dim_size)
        return x, vec

    def update(
        self, inputs: Tuple[torch.Tensor, torch.Tensor]
    ) -> Tuple[torch.Tensor, torch.Tensor]:
        return inputs


Complete the function implementation replacing "raise NotImplementedError".