A Fuzzy Logic-Enhanced Optimizer for RNA Binding Site Prediction Using CNN-GCN Architectures

Ribonucleic acid (RNA)-binding proteins (RBPs) are integral to post-transcriptional regulation in eukaryotic cells, orchestrating a wide array of biological processes including RNA splicing, transport, stability, localization, and translation control [1, 2]. By forming ribonucleoprotein complexes (RNPs) through interactions with coding and non-coding RNAs, RBPs establish dynamic regulatory networks that fine-tune gene expression in a context-dependent manner [3, 4]. Disruption in RBP-RNA interactions has been implicated in a variety of pathophysiological conditions, ranging from neurodegenerative disorders like amyotrophic lateral sclerosis (ALS) and frontotemporal dementia to tumorigenesis in various cancers [5, 6].

Conventional experimental methods for RBP identification, such as cross-linking immunoprecipitation followed by sequencing (CLIP-seq) [7, 8], RNA immunoprecipitation sequencing (RIP-seq) [9], and electrophoretic mobility shift assays (EMSAs) [10], have provided foundational insights into RNA–protein interactions. However, these techniques are inherently labor-intensive, low-throughput, and require significant experimental optimization, which limits their scalability in large-scale or dynamic cellular contexts.

In response to these limitations, computational approaches—especially those based on machine learning—have emerged as viable alternatives for predicting RBP binding sites directly from nucleotide sequences. Among these, deep learning methods have shown remarkable promise due to their ability to automatically learn complex patterns from raw biological data without the need for manual feature engineering [11-13]. Specifically, convolutional neural networks (CNNs) have been widely adopted to capture local sequence motifs relevant to RBP binding, as demonstrated by DeepCLIP [14], which combines CNN and long short-term memory (LSTM) layers to model both spatial and contextual dependencies in sequence data.

Building upon the success of CNN-based models, recent studies have explored the incorporation of graph-based architectures such as graph convolutional networks (GCNs), which allow the integration of structural relationships between RNA nucleotides into the prediction framework. For instance, DeepPN [15] combines CNNs and GCNs to simultaneously leverage both local sequence context and the underlying RNA secondary structure, achieving improved predictive performance without explicit structural input.

In addition to these neural models, ensemble learning methods like HydRA [16] and DeepFusion [17] have further extended the scope of RBP prediction by integrating diverse data modalities, including protein sequence features and evolutionary information, to enhance classification accuracy and generalizability. NucleicNet [18], for example, frames RBP prediction as a residue-level classification task, offering fine-grained insights into protein-RNA recognition mechanisms at atomic resolution.

Despite remarkable progress in the development of deep learning models for RNA–protein binding site prediction, these methods often fall short in capturing the full complexity of RNA–protein interactions. Such interactions are governed by a wide array of biochemical, topological, and thermodynamic factors that are difficult to model using standard neural architectures. Moreover, many state-of-the-art models lack interpretability, functioning as “black boxes” that hinder biological validation and limit translational applicability. This poses a significant barrier, especially in biomedical contexts where model explainability is crucial for generating actionable insights.

To address these limitations, recent studies have increasingly explored the integration of fuzzy logic into machine learning and deep learning pipelines. Fuzzy logic has been applied across various machine learning domains [19-21] and has consistently improved model performance by enhancing robustness, convergence, and generalization. Fuzzy logic offers a mathematically grounded framework for handling uncertainty, ambiguity, and data imprecision—characteristics that are highly prevalent in biological datasets. For instance, hybrid neuro-fuzzy systems have demonstrated improved transparency and interpretability in complex models through rule-based reasoning mechanisms without sacrificing predictive performance [22]. Likewise, the intuitionistic fuzzy broad learning system (IF-BLS) has shown superior diagnostic accuracy in noisy datasets such as Alzheimer’s disease classification [23]. Among recent contributions, reference [24] provided a systematic review of predictive uncertainty estimation techniques in machine learning, highlighting the value of probabilistic frameworks such as fuzzy logic for improving both the reliability and robustness of model outcomes under uncertain or imbalanced conditions.

The practical utility of fuzzy logic is further exemplified in clinical applications, where real-time interpretability and robustness are paramount. In stroke rehabilitation systems, fuzzy logic has been instrumental in enabling responsive feedback control, thereby enhancing therapy outcomes in environments characterized by patient variability and uncertainty. For instance, Das et al. [25] introduced a hybrid model combining fuzzy logic with machine learning to monitor lower limb exercises in stroke patients, facilitating real-time feedback and progress tracking without human intervention [25].

In the realm of medical imaging, fuzzy-augmented deep learning frameworks have demonstrated superior performance over traditional architectures by effectively managing ambiguous pixel-level data and improving diagnostic reliability. A comprehensive survey by Zheng et al. [26] highlighted the efficacy of fuzzy deep learning models in handling uncertain medical data, emphasizing their advantages in enhancing model interpretability and generalization across various clinical scenarios. Additionally, a novel ensemble fuzzy deep learning approach was proposed for brain MRI analysis, integrating volumetric fuzzy pooling and attention mechanisms to improve the segmentation of brain tissues and abnormalities, thereby advancing diagnostic accuracy [27].

Motivated by these challenges and limitations, we focus our methodological innovation on the development of a novel fuzzy logic-augmented optimizer, named FuzzyAdam, specifically tailored for RNA–protein binding site prediction in noisy, imbalanced, and temporally dynamic biological datasets. Unlike conventional optimizers such as Adam, FuzzyAdam introduces an adaptive learning mechanism driven by fuzzy inference rules that respond to fluctuations in loss dynamics and gradient behaviour. This approach enhances training stability, improves generalization, and promotes interpretability, without requiring architectural changes or manual reweighting.

Our contribution is centered on the design and implementation of FuzzyAdam, which enhances convergence stability without requiring any modification to the underlying model architecture or data representation. This approach offers a principled mechanism for handling uncertainty during training—a characteristic often presents in biological datasets—even when explicit noise or imbalance is not modeled.

To the best of our knowledge, this is the first application of fuzzy-enhanced optimization within the context of RBP interaction modeling, offering a new dimension of control and interpretability in biological deep learning frameworks.

3.1 Conceptual overview

The FuzzyAdam optimizer is a novel extension of the standard Adam optimizer, incorporating fuzzy logic-based adaptation into the learning rate dynamics. While Adam utilizes first and second moment estimates of gradients to perform parameter updates with adaptive learning rates, FuzzyAdam introduces a fuzzy inference mechanism that dynamically adjusts the effective learning rate scaling at each iteration based on recent training dynamics.

Let $\theta_t$ denote the model parameters at iteration t, and let $\mathcal{L}\left(\theta_t\right)$ be the corresponding loss. The update rule for FuzzyAdam is defined as:

$\theta_{t+1}=\theta_t-\eta \cdot \lambda_t \cdot \frac{\widehat{m}_t}{\sqrt{\hat{v}_t}+\varepsilon}$                (5)

where:

• $\eta>0$ is the base learning rate,
• $\hat{m}_t$ and $\hat{v}_t$ are the bias-corrected first and second moment estimates, respectively,
• $\varepsilon$ is a small constant to ensure numerical stability.

$\lambda_t \in \mathbb{R}^{+}$ is a fuzzy scaling factor adaptively determined at each step through a fuzzy inference system. The fuzzy factor $\lambda_t$ modulates the update magnitude in a context-aware manner by evaluating features extracted from training behavior, such as: Change in loss: $\Delta \mathcal{L}_t=\mathcal{L}_t-\mathcal{L}_{t-1}$ and Gradient norm: $\left\|g_t\right\|$.

The full step-by-step procedure of FuzzyAdam is outlined in Algorithm 1.

FuzzyAdam extends the standard Adam algorithm by embedding fuzzy reasoning to improve adaptivity and robustness. Unlike traditional optimizers that rely on fixed or heuristically scheduled learning rates, FuzzyAdam dynamically adjusts the effective step size in each iteration. This adjustment is guided by fuzzy inference over signals such as gradient magnitude, loss trajectory, and momentum variance—enabling smoother updates and improved convergence.

By integrating fuzzy logic, FuzzyAdam captures vague and nonlinear dependencies in training dynamics that are otherwise difficult to encode using classical heuristics. As a result, it offers several desirable properties:

• Adaptivity: it responds in real-time to changes in training dynamics;
• Interpretability: its decisions are governed by human-readable fuzzy rules;
• Stability: it maintains smoother convergence under noisy or chaotic training conditions.

In summary, FuzzyAdam can be viewed as a cognitively augmented optimizer that combines the statistical strength of Adam with the flexible reasoning capabilities of fuzzy logic. This synergy makes it particularly effective for scenarios where manual learning rate tuning is impractical, and training stability is critical.

3.2 Convergence analysis

Theorem: Let $\mathcal{L}(\theta)$ be a continuously differentiable loss function that is bounded below and has Lipschitz continuous gradients with constant $\mathcal{L}_g>0$, i.e.,

$\left\|\nabla \mathcal{L}(\theta)-\nabla \mathcal{L}\left(\theta^{\prime}\right)\right\| \leq \mathcal{L}_g\left\|\theta-\theta^{\prime}\right\|$ forall $\theta, \theta^{\prime}$.

Assume:

• The gradients are bounded: $\left\|g_t\right\| \leq G<\infty$.
• The gradient norms are square-summable: $\sum_{t=1}^{\infty}\left\|g_t\right\|^2<\infty$.
• The fuzzy scaling factor $\lambda_t \in\left[\lambda_{\min }, \lambda_{\max }\right]$, where $0<\lambda_{\min } \leq \lambda_{\max }<\infty$.

Then, the sequence $\left\{\theta_t\right\}$ generated by the FuzzyAdam update rule:

$\theta_{t+1}=\theta_t-\eta \cdot \lambda_t \cdot \frac{\widehat{m}_t}{\left(\sqrt{\hat{v}_t}+\varepsilon\right)}$

Converges to a stationary point $\theta^*$, i.e., $\lim _{t \rightarrow \infty}\left\|\nabla \mathcal{L}\left(\theta_t\right)\right\|=0$ and $\sum_{t=1}^{\infty}\left\|\theta_{t+1}-\theta_t\right\|^2<\infty$.

Proof: Define the update step as:

$\Delta_t=\eta \cdot \lambda_t \cdot \frac{\widehat{m}_t}{\sqrt{\hat{v}_t}+\varepsilon}$,

So the update rule becomes:

$\theta_{t+1}=\theta_t-\Delta_t$.

By the standard descent lemma for functions with Lipschitz continuous gradients [29], we have:

$\mathcal{L}\left(\theta_{t+1}\right) \leq \mathcal{L}\left(\theta_t\right)+\nabla \mathcal{L}\left(\theta_t\right)^{\top}\left(\theta_{t+1}-\theta_t\right)+\left(\frac{\mathcal{L}_g}{2}\right)\left\|\theta_{t+1}-\theta_t\right\|^2$.

Substituting $\theta_{t+1}-\theta_t=-\Delta_t$ gives:

$\mathcal{L}\left(\theta_{t+1}\right) \leq \mathcal{L}\left(\theta_t\right)-\nabla \mathcal{L}\left(\theta_t\right)^{\top} \Delta_t+\left(\frac{L_g}{2}\right)\left\|\Delta_t\right\|^2$

Note that: $\Delta_t=\eta \cdot \lambda_t \cdot \frac{\widehat{m}_t}{\sqrt{\hat{v}_t}+\varepsilon}$, and since $\lambda_t \in\left[\lambda_{\min }, \lambda_{\max }\right]$, we can bound the update norm: $\left\|\Delta_t\right\| \leq \eta \cdot \lambda_{\max } \cdot\left\|\frac{\widehat{m}_t}{\sqrt{\hat{v}_t}+\varepsilon}\right\| \leq C$, for some constant $C>0$.

Assuming $\widehat{m}_t$ aligns with $g_t$ (as is standard under bias correction), then:

$\nabla \mathcal{L}\left(\theta_t\right)^{\top} \Delta_t \geq \eta \cdot \lambda_{\min } \cdot \gamma \cdot\left\|\nabla \mathcal{L}\left(\theta_t\right)\right\|^2$,

for some $\gamma>0$ due to moment alignment.

Thus:

$\mathcal{L}\left(\theta_{t+1}\right) \leq \mathcal{L}\left(\theta_t\right)-\eta \cdot \lambda_{\text {min }} \cdot \gamma \cdot\left\|\nabla \mathcal{L}\left(\theta_t\right)\right\|^2+\left(\frac{\mathcal{L}_g}{2}\right) \cdot C^2$

Summing from $t=1$ to T:

$\mathcal{L}\left(\theta_1\right)-\mathcal{L}\left(\theta_{T+1}\right) \geq \eta \cdot \lambda_{\min } \cdot \gamma \sum_{t=1}^T\left\|\nabla \mathcal{L}\left(\theta_{-} t\right)\right\|^2-T \cdot\left(\frac{\mathcal{L}_g}{2}\right) \cdot C^2$.

Because $\mathcal{L}$ is bounded below, the left-hand side is finite. Hence:

$\sum_{t=1}^{\infty}\left\|\nabla \mathcal{L}\left(\theta_t\right)\right\|^2<\infty$

which implies:

$\lim _{t \rightarrow \infty}\left\|\nabla \mathcal{L}\left(\theta_t\right)\right\|=0$.

Furthermore, since: $\left\|\theta_{t+1}-\theta_t\right\|^2=\left\|\Delta_t\right\|^2 \leq\left(\eta \cdot \lambda_{\max } \cdot C\right)^2$ and $\sum_{t=1}^{\infty}\left\|g_t\right\|^2<\infty$, we have:

$\sum_{t=1}^{\infty}\left\|\theta_{t+1}-\theta_t\right\|^2<\infty$.

The convergence of the FuzzyAdam optimizer is guaranteed under the same conditions that ensure the convergence of traditional Adam. The introduction of the fuzzy scaling factor does not interfere with convergence but instead provides a dynamic adjustment to the learning rate that can enhance optimization stability. The fuzzy scaling factor $\lambda_t$, driven by loss differences and gradient norms, adapts the learning process based on the changing optimization landscape, potentially leading to faster convergence in scenarios where traditional methods struggle with noisy gradients or fluctuating loss landscapes.

Thus, FuzzyAdam maintains the theoretical guarantees of Adam while introducing a flexible mechanism to handle uncertainty in the optimization process, making it particularly well-suited for complex, non-stationary environments, such as those encountered in deep neural networks.

3.3 Protein sequence retrieval from UniProt

To construct an image-based representation of protein sequences suitable for CNN input, we developed a systematic pipeline that retrieves curated RBP and non-RBP sequences from UniProt, computes 2-mer (dipeptide) frequency distributions, and renders them into 2D heatmaps. This approach leverages the physicochemical and contextual information embedded in short amino acid motifs, while maintaining compatibility with image-based deep learning architectures.

Protein identifiers were programmatically retrieved from the UniProt Knowledgebase (UniProtKB) [30] using a RESTful API. We utilized the controlled vocabulary keyword KW-0694 to isolate RBPs, while non-RBP sequences were obtained using the negation NOT keyword:KW-0694. The API was queried in batches (batch size = 500), and all primary accession numbers were stored locally to ensure experimental reproducibility. query_rbp = "keyword:KW-0694" and query_nonrbp = "NOT keyword:KW-0694".

For each accession ID, the corresponding FASTA sequence was downloaded via:

https://rest.uniprot.org/uniprotkb/{accession}.fasta.

We excluded sequences shorter than 50 amino acids to prevent sparse or degenerate feature encodings.

To convert sequences into a biologically informed numerical representation, we employed dipeptide frequency encoding, where each sequence is mapped to a 20×20 matrix based on the normalized frequency of overlapping 2-mers (subsequences of length 2) constructed from the 20 canonical amino acids:

Let:

• $A=\{A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y\}$ be the amino acid alphabet,
• $K=2$ the k-mer length,
• $S=s_1, s_2, \ldots, s_n$ be a protein sequence of length n,
• K be the set of all possible 2-mers, i.e., $|K|=400$.

We slide a window of size 2 across S to extract all overlapping 2-mers $w_i=s_i s_{i+1}$, for $i=1, \ldots, n-1$, then compute the frequency of each k-mer as:

$f_k=\frac{\sum_{i=1}^{|n-K+1|} \delta\left(w_i=k\right)}{\sum_{j=i}^{|K|} \sum_{i=1}^{|n-K+1|} \delta\left(w_i=j\right)}=\frac{c_k}{\sum_{j=i}^{|K|} c_j}$

where:

• $f_k$ is the normalized frequency of k-mer $k \in K$,
• $c_k$ is the raw count of k-mer kkk in sequence SSS,
• $\delta\left(w_i=k\right)$ is the indicator function that returns 1 if the i-th window equals k, 0 otherwise.

The final 400-dimensional vector $f=\left[f_1, f_2, \ldots, f_{400}\right]$ is reshaped into a 2D matrix of shape 20×20, forming a structured feature map that preserves residue co-occurrence patterns.

To transform the frequency matrices into image-like representations for CNN processing, we employed seaborn.heatmap() using the perceptually uniform viridis colormap. Each heatmap was saved in .png format with fixed resolution (3×3 inches, 300 dpi), no axis ticks, and no colorbar to ensure that the model focuses solely on data-intrinsic patterns. Each protein sequence thus yields a single heatmap representing its 2-mer structural signature. All output images were organized into class-specific directories: images/RBP/ and images/nonRBP/. This representation enables effective convolutional pattern extraction while grounding model input in domain-specific priors.

3.4 Feature extraction using EfficientNet

To extract high-dimensional semantic representations of protein images, we utilized a pre-trained EfficientNet-B3 model as the backbone feature extractor. The original classification head was removed, and an adaptive average pooling layer was employed to produce fixed-length feature vectors of size 1536. The feature extraction was performed in evaluation mode to ensure deterministic behaviour during inference. All feature vectors were computed in batches using a GPU-enabled environment to accelerate processing.

3.5 Graph construction

A similarity-based graph was constructed to model pairwise relations among protein representations. We computed the cosine similarity matrix $S \in R^{N \times N}$ from the extracted feature matrix $F \in R^{N \times 1536}$, where $N$ denotes the number of samples. Self-similarity entries along the diagonal of $S$ were set to zero to eliminate self-loops.

To build the graph structure dynamically and avoid overconnectivity, we adopted an adaptive top- $k$ thresholding strategy. For each node, only the top- 10 most similar neighbors were retained to form directed edges, resulting in a sparse adjacency matrix $A$. The edge list $E$ was then derived by collecting non-zero indices from $A$, and converted into a PyTorch Geometric format edge index.

3.6 Graph representation dataset

The final graph dataset was formulated as a torch_geometric.data.Data object, where each node represents a protein sample, its features are the extracted embeddings from EfficientNet-B3, and its label corresponds to either RNA-binding or non-binding class. Formally, the graph G=(V,E) is defined as follows:

• V: set of nodes with $x_i \in R 1536$,
• E: set of directed edges based on cosine similarity,
• $y_i \in\{0,1\}$: label indicating class membership.

3.7 Graph convolutional network architecture

To perform classification over the constructed protein image graph, we designed a deep GCN composed of multiple convolutional blocks with skip connections and a gating mechanism. The architecture is optimized for learning rich, spatially-aware representations from graph-structured data.

The model accepts node-level features of size 1536, which originate from the EfficientNet-B3 feature extractor described in Section 3.4. The GCN consists of three stacked GCNConv layers with hidden dimensionality of 128. Each layer is followed by batch normalization and Parametric ReLU (PReLU) activation to improve convergence stability and address vanishing gradients.

To maintain information from the initial representation, a residual connection is implemented via a linear transformation of the input features. These residual features are combined with the output of the final GCN layer using a gated fusion mechanism, which adaptively weights both pathways. This gating is implemented as a sigmoid-activated feedforward layer over the concatenation of the learned and residual embeddings.

Formally, given an input feature matrix $\boldsymbol{X} \in \mathbb{R}^{N \times d}$ and edge index $\mathcal{E}$, the intermediate representations through the network are computed as:

$\begin{gathered}\mathbf{H}^{(1)}=\operatorname{PReLU}_1\left(\operatorname{BN}_1\left(\operatorname{GCNConv}_1(\mathbf{X}, \mathcal{E})\right)\right) \\ \mathbf{H}^{(2)}=\operatorname{PReLU}_2\left(\operatorname{BN}_2\left(\operatorname{GCNConv}_2\left(\mathbf{H}^{(1)}, \mathcal{E}\right)\right)\right) \\ \mathbf{H}^{(3)}=\operatorname{PReLU}_3\left(\operatorname{BN}_3\left(\operatorname{GCNConv}_1\left(\mathbf{H}^{(2)}, \mathcal{E}\right)\right)\right)\end{gathered}$

The residual projection $\mathbf{R}=\mathbf{X W}_r$ is combined with the output $\mathbf{H}^{(3)}$ using a learned gating vector $g \in[0,1]^{N \times h}$, computed as:

$\boldsymbol{g}=\sigma\left(\right. \text {Linear} \left.\left(\left[\mathbf{H}^{(3)} \| \mathbf{R}\right]\right)\right)$

The final node representation is given by the gated combination:

$\mathbf{Z}=\boldsymbol{g} \odot \mathbf{H}^{(3)}+(1-\boldsymbol{g}) \odot \mathbf{R}$

If a graph-level prediction is required, the node-level features Z are aggregated using global mean pooling. The resulting pooled embedding is passed through a multi-layer perceptron (MLP) classifier consisting of a hidden ReLU-activated dense layer with dropout and a final softmax layer for binary classification.

3.8 Training strategy

The GCN model was optimized using both the standard Adam optimizer and the proposed FuzzyAdam optimizer for comparative evaluation. For both optimizers, the initial learning rate was set to 0.01 and weight decay of $5 \times 10^{-4}$ mitigate overfitting. The cross-entropy loss function was employed, which is appropriate for binary classification tasks such as RNA binding site prediction.

Training was conducted in a GPU-enabled environment using Google Colab, with all computations executed on an NVIDIA Tesla T4 GPU. Both the model and graph data were deployed on the same device to ensure efficient training. The experiments were run for 500 epochs to allow for full convergence, with performance metrics recorded at each iteration for both optimizers.

In this study, we introduced FuzzyAdam, a fuzzy logic-enhanced variant of the Adam optimizer, tailored to improve the performance of deep learning models in RNA-binding site prediction. By integrating fuzzy rules into the adaptive learning rate mechanism, FuzzyAdam enables more nuanced weight updates, particularly in regions with uncertain or borderline patterns. Experimental results on balanced RBP and non-RBP datasets demonstrate that FuzzyAdam consistently achieves lower false negative rates and competitive performance across key metrics such as accuracy, F1-score, MCC, and balanced accuracy.

Ablation studies further reveal that the fuzzy rule configuration plays a significant role in optimizing sensitivity without sacrificing overall performance. This suggests that FuzzyAdam may offer an advantage in biological classification tasks where minimizing false negatives is crucial. While FuzzyAdam is not explicitly informed by biological principles, its ability to handle ambiguous features and imprecise patterns may be particularly suitable for noisy biological data.

Overall, FuzzyAdam offers a computationally efficient and generalizable optimization strategy that improves sensitivity and robustness without increasing model complexity. Its potential applicability to other bioinformatics tasks warrants further investigation, especially in settings involving multi-omics data or limited supervision.