Manta-Ray Foraging Optimization (MRFO)

• MRFO is a bio-inspired metaheuristic that mimics manta ray foraging using chain, cyclone, and somersault operators to iteratively refine candidate solutions in diverse search spaces.
• It is applied to optimize feature selection in speech emotion recognition, refine deep features in multiview imaging, search neural architectures in federated diagnosis, and tune hidden-layer weights in extreme learning machines.
• The effectiveness of MRFO hinges on careful solution encoding and fitness definition, with empirical studies reporting improved accuracy, reduced RMSE, and more compact feature selection relative to other metaheuristics.

Manta-Ray Foraging Optimization (MRFO) is a bio-inspired, population-based metaheuristic that mimics manta rays’ foraging for plankton through three search operators: chain foraging, cyclone foraging, and somersault foraging. In recent arXiv-described applications, MRFO appears as a general continuous optimizer, a wrapper feature selector, a deep-feature refinement module, a structural optimizer for lightweight neural architectures, and a trainer for hidden-layer parameters in extreme learning machines. These uses span speech emotion recognition, multiview scleral imaging for glycemic assessment, federated MRI diagnosis, and ab-initio energy-landscape prediction, illustrating that MRFO is better understood as a search framework whose practical role depends on how candidate solutions, fitness, and downstream models are defined (Chattopadhyay et al., 2020, Khan et al., 13 Mar 2026, Khan et al., 9 Jul 2025, Rubio-Solis, 16 May 2026).

1. Biological inspiration and canonical search model

Subsequent summaries describe MRFO as having been introduced by Zhao et al. in 2020 and grounded in three observed behaviors of manta rays: moving in chains, forming cyclone-like spirals near food-rich regions, and performing somersaults around the best food spot. In algorithmic terms, these become three update operators applied to a population of candidate solutions, with each agent occupying a position in a bounded search space and the current best solution acting as a recurrent attractor (Chattopadhyay et al., 2020, Rubio-Solis, 16 May 2026).

In the standard formulation, a population of $N$ manta rays is initialized uniformly within bounds, and each individual position $x_i(t) \in \mathbb{R}^D$ is iteratively updated. The initialization rule is written as

$$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$

This representation is continuous, but later applications adapt it to discrete or mixed spaces by mapping positions to feature masks, architecture choices, or neural-network parameters. That pattern suggests that MRFO is not tied to a single problem class; its identity lies in its update dynamics rather than in any fixed representation (Rubio-Solis, 16 May 2026).

The canonical division of labor across the three operators is consistent across the papers. Chain foraging is primarily exploitative, because each agent follows either the current best or its predecessor in the chain. Cyclone foraging mixes exploration and exploitation through spiral motion toward either the best point or a random point. Somersault foraging perturbs solutions around the current best and functions as a diversification step that can also intensify local search. Later variants preserve this basic interpretation even when they rename or omit operators (Chattopadhyay et al., 2020, Khan et al., 9 Jul 2025).

2. Search operators and governing equations

A standard chain-foraging update is given in component form as

$P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$

with

$b = 2a\sqrt{\lvert \ln(a) \rvert}.$

Here the leader moves directly toward the global best, whereas all other agents combine local following behavior with global guidance. This is the core exploitation mechanism in the classical formulation (Chattopadhyay et al., 2020).

Cyclone foraging introduces spiral-like motion. A standard $N$-dimensional update is

$P_j^k(n+1) = \begin{cases} P_{\text{best}}^k + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + \gamma \left(P_{\text{best}}^k(n) - P_j^k(n) \right), & j=1,\[4pt] P_{\text{best}}^k + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + \gamma \left(P_{\text{best}}^k(n) - P_j^k(n) \right), & j=2,3,\dots,N, \end{cases}$

where

$\gamma = 2 e^{c \frac{I - n + 1}{I} \sin(2\pi c)}.$

The factor $\gamma$ decays with iteration index $n$, so the operator is more exploratory earlier and more exploitative later. Classical MRFO also permits occasional random reinitialization within bounds, written as

$x_i(t) \in \mathbb{R}^D$0

which serves as an escape mechanism from local optima (Chattopadhyay et al., 2020).

Somersault foraging is written as

$x_i(t) \in \mathbb{R}^D$1

where $x_i(t) \in \mathbb{R}^D$2 is the somersault factor and $x_i(t) \in \mathbb{R}^D$3 are random variables. This step treats the current best as a pivot and repositions agents around it. In one implementation-oriented description, the full algorithm alternates between cyclone and chain foraging with probability $x_i(t) \in \mathbb{R}^D$4, then applies somersault foraging to all agents before updating the best solution again (Rubio-Solis, 16 May 2026).

Not all later MRFO-based systems preserve the full three-phase structure. In FOLC-Net, the authors decompose their customized version into “Cyclone Aging” and “Chain Foraging,” denoted CYA and CHF, and do not explicitly implement or name a somersault phase. That paper therefore uses MRFO mechanisms as a tailored structural search procedure rather than as a verbatim reproduction of the classical algorithm (Khan et al., 9 Jul 2025).

3. Solution encodings and objective functions

The most consequential design choice in MRFO is not the update rule alone but the encoding of a candidate solution and the definition of its fitness. Across the reported applications, MRFO is used with four distinct encodings: binary-like feature masks over hand-crafted speech features, feature-selection indicators over deep multiview embeddings, discrete architectural settings for a shallow federated CNN, and continuous hidden-layer weight vectors for an extreme learning machine.

Work	Candidate representation	Optimization target
Speech emotion recognition (Chattopadhyay et al., 2020)	Subset of 959 concatenated MFCC+LPC features	Maximize classification performance
ScleraGluNet (Khan et al., 13 Mar 2026)	Feature-selection indicators over CNN branch embeddings	Refine features before transformer fusion
FOLC-Net (Khan et al., 9 Jul 2025)	$x_i(t) \in \mathbb{R}^D$5	Maximize validation accuracy
EELM-MRFO-LF (Rubio-Solis, 16 May 2026)	Hidden-layer input weights in $x_i(t) \in \mathbb{R}^D$6	Minimize RMSE

In speech emotion recognition, MRFO is applied as a wrapper feature selector over a 959-dimensional feature vector formed by concatenating 216 MFCC features and 743 LPC features. The paper does not explicitly formalize the binarization step, but it states that MRFO is used for optimized feature selection and reports compact selected subsets of 43 and 61 features. This suggests a continuous-to-binary mapping from agent positions to select-or-discard decisions at evaluation time, although the threshold is not specified. Fitness is effectively cross-validated classifier performance, with accuracy as the central optimization criterion and precision, recall, and F1 used for analysis (Chattopadhyay et al., 2020).

In ScleraGluNet, MRFO is placed after five parallel CNN branches and before transformer-based cross-view attention. Its purpose is to mitigate redundancy from multiview feature concatenation and enhance focus on relevant vascular features. The paper is explicit that MRFO operates on feature vectors, not on images or network weights, and that it identifies an optimal subset of features by eliminating those deemed irrelevant or highly correlated. It is less explicit about whether refinement is performed per view or on a concatenated multiview embedding, and it does not print an explicit objective function. The text states only that MRFO is a feature subset optimizer in relation to downstream classification and/or regression performance (Khan et al., 13 Mar 2026).

FOLC-Net uses MRFO as a structural optimizer for a shallow CNN called ShallowFed. The search space has five dimensions—filters, kernel sizes, activation functions, dropout, and number of neurons—and each candidate architecture is represented as

$x_i(t) \in \mathbb{R}^D$7

The optimized settings are drawn from discrete sets: filters $x_i(t) \in \mathbb{R}^D$8, kernel sizes $x_i(t) \in \mathbb{R}^D$9, activation functions $$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$0, dropout $$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$1, and neurons $$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$2. The objective is single-objective and aims to maximize validation accuracy (Khan et al., 9 Jul 2025).

In EELM-MRFO-LF, each manta ray encodes the full set of hidden-layer input weights of a single-layer feedforward network. Output weights are not optimized by MRFO; they are solved analytically via the Moore–Penrose pseudoinverse. The fitness function is the training RMSE,

$$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$3

subject to bound constraints $$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$4 with $$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$5 and $$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$6 (Rubio-Solis, 16 May 2026).

4. Variants, customizations, and hybrid forms

The applications show that MRFO is rarely used as an immutable template. Instead, recent work either embeds it into larger deep-learning systems or modifies one of its phases to address perceived search deficiencies.

ScleraGluNet uses MRFO as a post-CNN feature refinement stage in a multiview pipeline: image acquisition across five gaze directions, preprocessing and vessel enhancement, five parallel CNN branches, MRFO-based feature refinement, transformer-based cross-view fusion, and multitask heads for three-class classification and fasting plasma glucose estimation. The paper explicitly states that MRFO is not used for CNN weight optimization, end-to-end hyperparameter tuning, optimizer replacement for gradient descent, or architecture search. In that system, MRFO is a black-box, population-based metaheuristic inserted between representation learning and attention-based fusion (Khan et al., 13 Mar 2026).

FOLC-Net adopts “MRFO mechanisms” for efficient model structure generation, but the mechanism is customized. Its CYA equations introduce random and best-guided cyclone updates with a time-varying factor

$$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$7

and its CHF equations encode leader-follower motion toward both predecessor and best structure. The paper states that the optimization begins with an initial population of 10 candidate solutions, uses eight iterative cycles each lasting 40 epochs, imposes a maximum of ten iterations, and adds a patience mechanism with value 10. MRFO is therefore adapted to discrete neural architecture search rather than used as a generic continuous solver (Khan et al., 9 Jul 2025).

The most explicit algorithmic extension is MRFO with Lévy Flight. In EELM-MRFO-LF, Lévy Flight is inserted during somersault foraging to increase diversity and reduce premature convergence. After the classical somersault update, each agent is perturbed as

$$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$8

where the Lévy steps are generated by Mantegna’s algorithm,

$$x_i(0) = x_l + \text{rand} \cdot (x_u - x_l), \quad i=1,\dots,N.$$9

with $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$0. The implementation reported in that paper uses population size $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$1, maximum iterations $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$2, somersault factor $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$3, bounds $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$4, a sigmoid activation function, and 300 hidden nodes (Rubio-Solis, 16 May 2026).

A plausible implication of these implementations is that “MRFO” in current applied literature often denotes a family of search patterns rather than a uniquely fixed algorithmic object. Some works use the classical three-phase form, some omit phases, and some hybridize it with external stochastic processes such as Lévy Flight (Khan et al., 9 Jul 2025, Rubio-Solis, 16 May 2026).

5. Reported applications and empirical performance

In speech emotion recognition, MRFO-based feature selection is reported on the SAVEE and Emo-DB datasets. Using concatenated MFCC and LPC features followed by MRFO and MLP classification, the method achieved classification accuracies of 97.06% and 97.68%, respectively, while selecting 43 features for SAVEE and 61 for Emo-DB out of the original 959. The same study reports direct comparisons against GA, GWO, and PSO. On SAVEE, GA used 325 features for 72.23% accuracy, GWO used 89 features for 84.31%, PSO used 35 features for 81.32%, and MRFO used 43 features for 97.06%. On Emo-DB, GA used 230 features for 85.91%, GWO used 76 for 82.21%, PSO used 26 for 87.29%, and MRFO used 61 for 97.68% (Chattopadhyay et al., 2020).

In ScleraGluNet, MRFO is one component of a multiview deep-learning framework for three-class metabolic status classification and continuous fasting plasma glucose estimation from multidirectional scleral vessel images. The dataset comprised 445 participants and 2,225 anterior-segment images acquired from five gaze directions per participant. An ablation study compared four variants: a single-view CNN baseline, a multiview CNN without MRFO or transformer fusion, a multiview CNN with MRFO feature refinement only, and the full ScleraGluNet model with both MRFO and transformer cross-view fusion. The text states that the performance enhancement brought by multiview learning, optimization of features, and MRFO-based transformer fusion is incremental. For the full model, the reported performance is 93.8% overall accuracy, one-vs-rest AUCs of 0.971, 0.956, and 0.982 for normal, controlled diabetes, and high-glucose diabetes, respectively, and for fasting plasma glucose estimation MAE $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$5 mg/dL, RMSE $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$6 mg/dL, $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$7, and $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$8, with Bland–Altman mean bias $P_j^k(n+1)= \begin{cases} P_j^k(n) + a\left(P_{\text{best}}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=1,\[4pt] P_j^k(n) + a\left(P_{j-1}^k(n) - P_j^k(n)\right) + b\left(P_{\text{best}}^k(n) - P_j^k(n)\right), & j=2,3,\dots,N, \end{cases}$9 mg/dL and 95% limits of agreement from $b = 2a\sqrt{\lvert \ln(a) \rvert}.$0 to $b = 2a\sqrt{\lvert \ln(a) \rvert}.$1 mg/dL (Khan et al., 13 Mar 2026).

In federated MRI diagnosis, FOLC-Net incorporates MRFO mechanisms, global model cloning, and ConvNeXt. The framework is described as lightweight, with approximately 1.217 million parameters and a storage requirement of only 0.9 MB. Its reported accuracies are 98.01% on all views combined, 99.44% on axial, 98.27% on coronal, and 92.44% on sagittal data. The paper highlights the sagittal result by comparing 92.44% against 88.37% for a study method based on DL + Residual Learning and 88.95% for DL models. These are framework-level outcomes rather than an isolated ablation of MRFO alone, since the paper does not include a direct “with MRFO vs without MRFO” comparison (Khan et al., 9 Jul 2025).

In crystal-structure-related energy prediction, EELM-MRFO-LF is compared with ELM-GA, ELM-WOA, ELM-PSO, ELM-MRFO, and basic ELM under the same single-hidden-layer feedforward architecture. For predicting unrelaxed formation energies, the reported RMSE of EELM-MRFO-LF is approximately 9.3 meV/atom, compared with approximately 10.52 meV/atom for traditional EELM-MRFO. The paper also reports better $b = 2a\sqrt{\lvert \ln(a) \rvert}.$2, closer alignment with DFT calculations, and fewer outliers in unseen-data predictions for the Lévy-flight-enhanced variant (Rubio-Solis, 16 May 2026).

6. Reproducibility, limitations, and methodological cautions

Several methodological cautions recur across the applications. First, MRFO is not used in a single uniform way. In speech emotion recognition it is a wrapper feature selector; in ScleraGluNet it refines deep features; in FOLC-Net it searches architecture settings; and in EELM-MRFO-LF it optimizes hidden-layer weights while output weights are solved analytically. A common misconception is therefore to treat MRFO as if it were always a replacement for gradient descent or always a feature selector. The reported literature does not support that simplification (Chattopadhyay et al., 2020, Khan et al., 13 Mar 2026, Khan et al., 9 Jul 2025, Rubio-Solis, 16 May 2026).

Second, direct claims of superiority over other metaheuristics are strongly application-dependent. The speech paper provides explicit comparisons against GA, GWO, and PSO and reports large gains for MRFO. By contrast, ScleraGluNet does not directly compare MRFO with PSO, GA, WOA, or GWO on the scleral dataset, and FOLC-Net likewise does not include an in-paper MRFO-versus-random-search or MRFO-versus-other-metaheuristic ablation. Those papers support the claim that MRFO-based designs were effective in their respective frameworks, but they do not establish global best-in-class status across optimizers (Chattopadhyay et al., 2020, Khan et al., 13 Mar 2026, Khan et al., 9 Jul 2025).

Third, reproducibility is uneven. ScleraGluNet does not report MRFO population size, maximum number of iterations, pseudocode, stopping criteria, or internal control parameters. The speech paper also does not explicitly list numerical MRFO hyperparameters. FOLC-Net is more explicit about population size, iteration cap, and training cycles, and EELM-MRFO-LF reports a comparatively complete parameterization. This suggests that, in the current literature, “MRFO-based” may describe a clear functional role without fully exposing the optimizer configuration required for exact replication (Khan et al., 13 Mar 2026, Chattopadhyay et al., 2020, Khan et al., 9 Jul 2025, Rubio-Solis, 16 May 2026).

Finally, the papers repeatedly imply a familiar trade-off of population-based wrapper optimization: flexibility versus cost. ScleraGluNet notes that wrapper feature selection requires many model evaluations and can be significantly more expensive than simple filter methods, although it does not quantify this cost. The speech and architecture-search applications likewise evaluate many candidate subsets or structures, and EELM-MRFO-LF introduces additional complexity through Lévy-flight generation and parameter tuning. The cumulative picture is that MRFO is most attractive when the search space is high-dimensional, multimodal, derivative-free, or discretized, and when the additional search cost is acceptable relative to the expected gain in feature compactness, architectural adequacy, or predictive accuracy (Khan et al., 13 Mar 2026, Rubio-Solis, 16 May 2026).