补强中英文模型文档的原理层说明与公式表达，完善序列建模路线的技术叙述

1. 在 CNN、ResNet、BiLSTM、mLSTM 页面补充核心计算形式与结构动机，包括卷积扫描、残差连接、LSTM 门控与矩阵记忆更新。
2. 在 Transformer、Linear Transformer、iTransformer 页面补充标准注意力、线性注意力近似与通道维建模视角，强化不同注意力路线的差异说明。
3. 在 Mamba、RWKV 页面补充状态空间递推与递归加权聚合的高层公式，明确其长程序列建模逻辑。
4. 在 DNA foundation model、MAE、fusion 页面补充预训练表示流水线、重建目标与融合数学形式，完善表示学习与配对交互层的说明。
5. 中英文文档同步扩写，保留原有页面结构与历史内容，不引入构建产物。
6. 重新构建 Sphinx 文档，确认页面可生成且没有新的文档 warning。

Author: billzi2016

doc/source/en/BiLSTMModel.rst

@@ -23,6 +23,32 @@ Architecture summary
 - Hidden width: 256 per direction
 - Output: concatenated forward/backward final states
 
+At each step, an LSTM updates gates and memory through:
+
+.. math::
+
+   i_t = \sigma(W_i x_t + U_i h_{t-1} + b_i)
+
+.. math::
+
+   f_t = \sigma(W_f x_t + U_f h_{t-1} + b_f)
+
+.. math::
+
+   o_t = \sigma(W_o x_t + U_o h_{t-1} + b_o)
+
+.. math::
+
+   \tilde{c}_t = \tanh(W_c x_t + U_c h_{t-1} + b_c)
+
+.. math::
+
+   c_t = f_t \odot c_{t-1} + i_t \odot \tilde{c}_t, \qquad
+   h_t = o_t \odot \tanh(c_t)
+
+The bidirectional encoder runs this recurrence in both directions so that each
+output representation can reflect both upstream and downstream context.
+
 Why this helps for EPI
 ++++++++++++++++++++++
 
@@ -33,6 +59,9 @@ not confined to a single motif window. A BiLSTM can model:
 - local-to-mid-range contextual accumulation;
 - asymmetric sequence signals that plain pooling may wash out.
 
+This matters when the functional meaning of a motif depends on order, nearby
+context, or the sequence of several local events rather than on isolated hits.
+
 Strengths
 +++++++++
 
@@ -55,4 +84,3 @@ benchmark. It helps answer whether explicit sequential recurrence remains useful
 after introducing Transformer-style, state-space, and foundation-model routes.
 
 .. image:: ../img/div.png
-

doc/source/en/CNNModels.rst

@@ -19,6 +19,17 @@ layers are naturally suited for:
 - offering relatively stable optimization and efficient training;
 - providing strong baseline performance without requiring external pretraining.
 
+At the operator level, a 1D convolution behaves like a trainable motif scanner:
+
+.. math::
+
+   h_t = \sigma\!\left(\sum_{c=1}^{C}\sum_{i=0}^{k-1} w_{c,i}\,x_{c,t+i} + b\right)
+
+where :math:`x` is the multi-channel sequence input, :math:`k` is the kernel
+width, and :math:`h_t` is the activation at position :math:`t`. Each filter can
+be interpreted as a learned detector for short sequence patterns and their
+local variants.
+
 M1: CNN single-branch baseline
 ++++++++++++++++++++++++++++++
 
@@ -36,6 +47,10 @@ Interpretation:
   already produce non-trivial predictive signal.
 - It is intentionally simple and easy to compare against more expressive models.
 
+Because it avoids pair fusion and avoids recurrent or attention mechanisms, M1
+is the cleanest answer to a foundational question: how far can local feature
+detectors and hierarchical pooling go on their own?
+
 M2: CNN dual-branch baseline
 ++++++++++++++++++++++++++++
 
@@ -52,6 +67,15 @@ Why it matters:
 - Separate branches let the model preserve region-specific representations
   before interaction modeling.
 
+Its computation can be summarized as:
+
+.. math::
+
+   h_e = f_e(x_e), \qquad h_p = f_p(x_p), \qquad z = \mathrm{Fuse}(h_e, h_p)
+
+This makes the architecture explicitly separate the two encoding problems
+before asking a fusion layer to model compatibility, asymmetry, or synergy.
+
 M3: k-mer embedding plus CNN
 ++++++++++++++++++++++++++++
 
@@ -69,6 +93,16 @@ Why use k-mers:
 - They can sometimes make biologically meaningful subsequence patterns easier to
   capture than raw one-hot channels alone.
 
+Formally, the input representation becomes:
+
+.. math::
+
+   s = (t_1, \dots, t_n), \qquad e_i = E[t_i]
+
+where :math:`t_i` is a k-mer token and :math:`E` is the embedding table. The
+CNN then operates over token vectors rather than raw nucleotide channels, which
+places M3 between classical motif CNNs and language-model style representations.
+
 Shared strengths
 ++++++++++++++++
 
@@ -95,4 +129,3 @@ work. For that reason, it plays two roles at once:
 - practical baseline against which all later architectures are evaluated.
 
 .. image:: ../img/div.png
-

doc/source/en/DNAFoundationModels.rst

@@ -27,6 +27,17 @@ Two usage modes
   head;
 - finetune mode: update the backbone with a smaller learning rate.
 
+In representation terms, the pipeline is:
+
+.. math::
+
+   \text{DNA sequence} \rightarrow \text{tokenizer} \rightarrow \text{pretrained backbone}
+   \rightarrow \text{sequence embedding} \rightarrow \text{task head}
+
+The scientific question is whether broad genomic pretraining has already
+captured regulatory priors that a project-level dataset would struggle to learn
+from scratch.
+
 Why multiple backbones are useful
 +++++++++++++++++++++++++++++++++
 
@@ -46,4 +57,3 @@ learning. It asks whether external genomic prior knowledge can outperform or
 complement task-specific training from raw sequence encodings.
 
 .. image:: ../img/div.png
-

doc/source/en/FusionStrategies.rst

@@ -16,13 +16,39 @@ Supported strategies
 - ``bilinear``
 - ``concat_sub_mul``
 
+If enhancer and promoter embeddings are written as :math:`h_e` and :math:`h_p`,
+the main fusion forms can be summarized as:
+
+.. math::
+
+   \mathrm{concat}: [h_e; h_p]
+
+.. math::
+
+   \mathrm{add}: h_e + h_p
+
+.. math::
+
+   \mathrm{subtract}: h_e - h_p
+
+.. math::
+
+   \mathrm{multiply}: h_e \odot h_p
+
+.. math::
+
+   \mathrm{concat\_sub\_mul}: [h_e; h_p; h_e - h_p; h_e \odot h_p]
+
 Why fusion matters
 ++++++++++++++++++
 
 Even a strong encoder can underperform if the pairwise interaction layer is too
 weak. Fusion determines how much relational information between enhancer and
 promoter embeddings is exposed to the classifier.
 
+This is why fusion should not be treated as a minor implementation detail. It
+defines what kinds of pair structure the classifier is allowed to see directly.
+
 Interpretation of the main options
 ++++++++++++++++++++++++++++++++++
 
@@ -43,4 +69,3 @@ balanced representation-rich option without requiring the parameter overhead of
 full bilinear interaction.
 
 .. image:: ../img/div.png
-

doc/source/en/LinearTransformerModel.rst

@@ -21,13 +21,28 @@ Project implementation highlights
 - Attention: ELU+1 kernel feature map
 - Pooling: mean pooling over sequence positions
 
+The key approximation replaces softmax attention with feature maps:
+
+.. math::
+
+   \mathrm{Attn}(Q, K, V) \approx
+   \frac{\phi(Q)\big(\phi(K)^\top V\big)}
+        {\phi(Q)\big(\phi(K)^\top \mathbf{1}\big)}
+
+where :math:`\phi(\cdot)` is a positive kernel feature map such as ELU+1. This
+reorders the computation so sequence length scaling becomes linear in style.
+
 Why it matters for EPI
 ++++++++++++++++++++++
 
 This model asks an important question: can we keep the global interaction flavor
 of attention while scaling better to long DNA sequences than a standard
 Transformer?
 
+That question matters directly for genomic inputs because long-range regulatory
+dependencies are scientifically relevant, but full quadratic attention becomes
+costly exactly in the length regime where those dependencies matter most.
+
 Strengths
 +++++++++
 
@@ -44,4 +59,3 @@ Limitations
   relevant to the task.
 
 .. image:: ../img/div.png
-

doc/source/en/MAEModel.rst

@@ -15,6 +15,17 @@ The model uses a Masked Autoencoder style workflow:
 3. reconstruct the masked content;
 4. reuse the learned encoder for supervised EPI prediction.
 
+The reconstruction objective can be written as:
+
+.. math::
+
+   \mathcal{L}_{\mathrm{MAE}} =
+   \frac{1}{|\mathcal{M}|}\sum_{i \in \mathcal{M}}
+   \ell(\hat{x}_i, x_i)
+
+where :math:`\mathcal{M}` is the masked patch set and :math:`\ell` measures how
+well the decoder recovers hidden sequence content.
+
 Why this is different from M13
 +++++++++++++++++++++++++++++++
 
@@ -40,6 +51,10 @@ Implementation logic
 - Pretraining objective: masked reconstruction
 - Finetuning objective: paired enhancer-promoter classification
 
+This means the encoder is first optimized to model sequence structure without
+labels, then repurposed as a supervised feature extractor for the downstream
+interaction task.
+
 Project role
 ++++++++++++
 
@@ -51,4 +66,3 @@ This model gives the documentation a full representation-learning ladder:
 - in-project self-supervised pretraining.
 
 .. image:: ../img/div.png
-

doc/source/en/MambaModel.rst

@@ -20,6 +20,17 @@ Project implementation
 - Fallback runtime: PyTorch approximation that preserves pipeline usability
 - Output: sequence mean pooling after stacked blocks
 
+At a high level, the selective state-space update can be summarized as:
+
+.. math::
+
+   h_t = \bar{A}(x_t) h_{t-1} + \bar{B}(x_t) x_t, \qquad
+   y_t = C(x_t) h_t
+
+Unlike fixed-coefficient state-space models, the transition depends on the
+current input, which is why Mamba can adapt its memory behavior to sequence
+content.
+
 Why this matters in practice
 ++++++++++++++++++++++++++++
 
@@ -43,4 +54,3 @@ The fallback path is useful for portability and pipeline validation, but should
 not be treated as identical to the optimized implementation.
 
 .. image:: ../img/div.png
-

doc/source/en/RWKVModel.rst

@@ -24,13 +24,28 @@ Project implementation highlights
 - Channel component: per-position feed-forward style mixing
 - Stabilization: log-space handling in the recurrent weighting path
 
+Its weighted key-value accumulation can be viewed schematically as a decayed
+running summary:
+
+.. math::
+
+   s_t = \alpha_t \odot s_{t-1} + \beta_t \odot v_t, \qquad
+   y_t = \frac{s_t}{z_t}
+
+where the decay terms control how much old evidence is retained and the
+normalization path keeps the recurrent aggregation numerically stable.
+
 Why this is relevant for EPI
 ++++++++++++++++++++++++++++
 
 Enhancer-promoter prediction needs more than isolated motif hits. RWKV offers an
 alternative way to accumulate sequence evidence across long contexts while
 avoiding classic quadratic attention.
 
+This makes RWKV attractive when the task depends on gradual evidence
+accumulation across many positions rather than only on a few sharp token-token
+links.
+
 Strengths
 +++++++++
 
@@ -45,4 +60,3 @@ Limitations
 - requires careful numerical treatment for stable sequence accumulation.
 
 .. image:: ../img/div.png
-

doc/source/en/ResNetModels.rst

@@ -18,6 +18,20 @@ line in the most classical way:
 - deepen the network substantially;
 - use skip connections to stabilize optimization and preserve gradient flow.
 
+Residual formulation
+++++++++++++++++++++
+
+The defining idea of a residual block is to learn a correction rather than a
+full replacement transform:
+
+.. math::
+
+   y = F(x; W) + x
+
+Here, :math:`x` is the block input and :math:`F(x; W)` is the learned residual
+branch. The identity shortcut creates a direct information and gradient path,
+which is why much deeper convolutional stacks remain trainable.
+
 Relationship to earlier resource limits
 +++++++++++++++++++++++++++++++++++++++
 
@@ -52,6 +66,22 @@ Compared with a plain CNN stack, a residual network offers:
 - better preservation of lower-level features while building higher-level
   abstractions.
 
+In a typical block, the residual path can be written as:
+
+.. math::
+
+   F(x; W) = \mathrm{Conv}_2(\sigma(\mathrm{BN}(\mathrm{Conv}_1(x))))
+
+and when shape changes require adjustment, the shortcut becomes:
+
+.. math::
+
+   y = F(x; W) + W_s x
+
+with :math:`W_s` denoting a projection on the skip path. This gives the model a
+clean way to preserve motif-scale evidence while progressively adding higher
+level context.
+
 Why ResNet still matters for genomic sequence modeling
 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 
@@ -62,6 +92,13 @@ For regulatory DNA tasks, deeper convolution can be useful when the aim is to:
 - stay within the convolutional inductive-bias family without jumping directly
   to attention or state-space formalisms.
 
+From a genomic modeling standpoint, this naturally supports a hierarchy:
+
+- early layers detect motif-like sequence signatures;
+- middle layers compose motifs into local regulatory modules;
+- deeper layers summarize wider context without erasing the earlier evidence
+  path.
+
 Documentation role
 ++++++++++++++++++
 
@@ -70,4 +107,3 @@ active registry is centered on ``M1-M14``, ResNet should be treated as a
 documented, expected branch of the project rather than an obsolete footnote.
 
 .. image:: ../img/div.png
-