Zone of inhibition prediction in silver nanoparticle antimicrobial activity via physics-guided liquid state machine

Highlights

• •This study proposes a physics-guided liquid state machine predictive model.
• •The model predicts antimicrobial activity from the silver nanoparticles synthesis.
• •The prediction model achieves a test coefficient of determination of 0.956.
• •Shapley additive explanations analysis reveals the importance of features.

Abstract

Silver nanoparticles are widely applied in medicine, packaging, and environmental remediation because of their potent antimicrobial properties. Nevertheless, predictive modeling of their zone of inhibition (ZOI) remains a longstanding challenge due to limited experimental datasets and the lack of physical consistency in machine learning approaches. In this study, a physics-guided liquid state machine (PG-LSM) was developed, integrating key physics-informed features of nanoparticle formation with experimental ZOI data through a series of reservoirs to predict antimicrobial efficacy. Three pretrained reservoirs of particle shape, particle core size, and ultraviolet-visible peak are transferred as encoders into a terminal ZOI predictor. Across nine baseline models, PG-LSM achieved the strongest test performance with a coefficient of determination (R²) of 0.956, a root mean square error (RMSE) of 1.151, and a mean absolute error (MAE) of 0.486 with close agreement between validation and test scores, indicating reliable generalization under small-sample conditions. Ablation studies confirmed the additive value from each reservoir; removing any intermediate stage degraded accuracy and stability. SHAP analysis revealed that exposure dose concentration and duration dominated the antimicrobial activity of AgNPs, whereas microbial species, capping agent, and reducing agent had a lower impact. PG-LSM advances materials informatics for AgNP antimicrobial assessment through physics-guided embeddings integrated with reservoir computing. The framework delivers accurate, stable, and interpretable ZOI predictions and remains practical for limited datasets.

Nomenclature

Abbreviation

AE

Autoencoder

AgNPs

Silver nanoparticles

ANN

Artificial neural network

CNN

Convolutional neural network

DT

Decision tree

FNN

Feedforward neural network

GPR

Gaussian process regression

LGBM

Light gradient boosting machine

LIF

Leaky integrate-and-fire

LM

Levenberg–Marquardt

LSM

Liquid state machine

LSPR

Local surface plasmon resonance

LSTM

Long short-term memory

MAE

Mean absolute error

MSE

Mean squared error

PG-LSM

Physics-guided liquid state machine

PSO

Particle swarm optimization

R²

Coefficient of determination

RF

Random forest

RMSE

Root mean square error

ROS

Reactive oxygen species

SHAP

Shapley additive explanations

UV–Vis

Ultraviolet–visible

XGBoost

Extreme gradient boosting

ZOI

Zone of inhibition

Symbol

$x_{ij}$

True value of the feature $j$ for the sample $i$

$z_{ij}$

Standardized value of the feature $j$ for the sample $i$

$N$

Number of samples

${\mu }_j$

Average value of the feature $j$

${\sigma }_j$

Standard deviation of the feature $j$

$V_l\left(t\right)$

Membrane potential of the neuron $l$ at time $t$

$V_{rest}$

Resting potential

${\tau }_m$

Membrane time constant

$R_m$

Membrane resistance

$I_i\left(t\right)$

Total synaptic input current to the neuron $l$

$\alpha$

Leak rate

$w_{lm}^E$

Synaptic weights of an excitatory neuron

$w_{ln}^I$

Synaptic weights of an inhibitory neuron

$w_l^S$

Synaptic weights of a self-feedback neuron

${\kappa }_E$

Synaptic kernels of an excitatory neuron

${\kappa }_I$

Synaptic kernels of an inhibitory neuron

${\kappa }_I$

Synaptic kernels of a self-feedback neuron

$s_m\left(t\right)$

Spike trains of an excitatory neuron $m$

$s_n\left(t\right)$

Spike trains of an inhibitory neuron $n$

$s_l\left(t\right)$

Spike trains of a neuron $l$

$V_{thres}$

Membrane threshold potential

$S_l\left(t\right)$

Spike indicator of the neuron $l$ at time $t$

${\widehat{S}}_l\left(n\right)$

Smoothed spike train of a neuron $l$

$\gamma *$

Exponential filter constant

$z\left(n\right)$

Reservoir readout output at step $n$

${\varphi }_j$

SHAP value of the feature $j$

$P$

Feature index of the predictor

$S$

Coalition feature without the considered feature $j$

$\left|S\right|$

Coalition size of the feature

$f\left(s\right)$

Model outcomes of coalition features $S$

$f(S\cup \{j\left\}\right)$

Model outcomes of the coalition feature $S$ with the feature $j$

$y_i$

Observed values

${\widehat{y}}_i$

Predicted values

$\overline{y}$

Average value of observed data

1. Introduction

Nanoparticles are typically defined within the 1–100 nm size range. They exhibit physicochemical properties that differ from bulk materials due to their high surface-to-volume ratio. Among various nanoparticles, silver nanoparticles (AgNPs) have attracted exceptional attention because of their antimicrobial activity, local surface plasmon resonance (LSPR), catalytic activity, and electrical properties, which support applications in medicine, sensing, electronics, packaging, and environmental remediation [1]. The antimicrobial activity of AgNPs is particularly valuable, enabling their use in wound dressings, medical device coatings, textiles, food preservation, and water purification [2].

The antimicrobial activity of AgNPs arises from multiple synergistic mechanisms. At the nanoscale, AgNPs can penetrate and disrupt the bacterial membrane [3], generate reactive oxygen species (ROS) that oxidize biomolecules, and release silver ions (Ag⁺) that bind to thiol groups in enzymes and nucleic acids, therefore disrupting metabolic and replication processes in microbial cells [4]. Together, these mechanisms account for the broad-spectrum activity of AgNPs against a range of microorganisms, including Gram-positive and Gram-negative bacteria, and fungi [5]. The antimicrobial activity of AgNPs is directly related to their physicochemical properties [6]. Smaller nanoparticles exhibit faster dissolution of AgNPs and stronger interactions with microbial membranes [7]. AgNPs with higher specific surface area increase the interaction with cell membranes, while shapes with more exposure of reactive crystal facets promote silver ion release, both of which contribute to enhanced antimicrobial activity [8]. Surface capping and functionalization stabilize colloidal AgNPs while modifying charge and ligand availability at the biological interface [9]. Aggregation state and dispersion stability control effective surface exposure and cellular penetration [10].

The AgNPs can be synthesized through physical, chemical, and green methods. Although physical approaches such as laser ablation, evaporation–condensation, and ball milling yield high-purity nanoparticles, particle stability and agglomeration can be challenging due to the absence of stabilizers or capping agents [11]. Chemical synthesis using reducing agents such as sodium borohydride, along with capping agents, offers better control over nanoparticle characteristics [12]. However, the impact on environmental safety and biomedical compatibility is a concern. Green synthesis methods employing plant extracts, microorganisms, and biomolecules as reducing agents have emerged as highly attractive strategies for producing AgNPs due to their intrinsic biocompatibility, environmental sustainability, and cost-effectiveness. However, these biologically mediated processes often exhibit substantial variability resulting from the variable composition of biological sources used, which strongly influences nanoparticle physicochemical properties [13]. In addition to the choice of reductants, parameters such as precursor concentration, type of capping agent, reaction temperature, and time also directly determine AgNPs characteristics that govern antimicrobial activities. The interdependent nature of physicochemical characteristics of AgNPs and their synthesis parameters poses a fundamental challenge for predictive modeling of AgNPs antimicrobial efficacy.

Despite recent developments in synthesis methods and progress in antimicrobial research, the systematic design of AgNPs with reliable antimicrobial effectiveness remains challenging. The challenge highlights the importance of approaches that integrate synthesis conditions and antimicrobial activity with predictive modeling. Recent studies have applied various machine learning approaches to predict particle size and antibacterial performance of AgNPs, with antibacterial activity often expressed in terms of zone of inhibition (ZOI) or related biological outcomes. Appendix A (Table A1) summarizes representative contributions, highlighting algorithms, dataset sizes, predictive targets, performance, and key limitations. A combination of a hybrid artificial neural network and particle swarm optimization (ANN-PSO) algorithm [14] and an ANN trained with the Lavenberg-Marquardt (LM) backpropagation algorithm [15] were used to predict the size of AgNPs synthesized using opium syrup and Coleus aromaticus leaf extract as reducing agent, respectively. Both studies demonstrated high R² values of 0.99, which suggests that the developed models are highly accurate in predicting and optimizing nanoparticle size under specific experiments, even when trained on relatively small datasets. Mirzaei et al [16] collected literature data on nanoparticle characteristics and microbial exposure conditions to model the antibacterial activity of Ag, Fe₃O₄, and ZnO nanoparticles, with Random Forest (RF) achieving the best performance (R² = 0.78). Saadat et al [17] predicted the antimicrobial activity of AgNPs using compiled data on nanoparticle size and bacterial exposure conditions, along with detailed synthesis parameters, identifying XGBoost as the best model with an R² of 0.84. Nathanael et al [18] optimized microfluidic synthesis conditions and applied decision tree (DT), XGBoost and RF algorithms to predict particle size, achieving modest R² values between 0.61 and 0.66 with predictive specificity limited to microfluidic settings. Furxhi et al [19] have employed machine learning models to predict both core size and antimicrobial activity of AgNPs. Core size prediction based on synthesis variables yielded an R² of 0.70 using RF and XGBoost algorithms, while antimicrobial activity prediction incorporated additional AgNPs physicochemical properties, such as shape, core size, and UV–Vis peak, as well as exposure conditions, reached R² of 0.72 by using a light gradient boosting machine (LGBM). These studies have consistently shown that without explicit modeling of fundamental physicochemical processes, the models are prone to overfitting and offer limited interpretability regarding why particular features dominate the biological activity of nanomaterials. While some prediction models may achieve high accuracy in narrowly constrained datasets, their performance typically deteriorates when extended to diverse synthesis conditions, microbial strains, or exposure environments.

Although recent studies demonstrate that predictive models can offer valuable insights into AgNPs synthesis and the ZOI test, the reported accuracies are often encouraging, but closer examination reveals essential shortcomings in terms of data scope, robustness, and explainability. Consequently, several critical gaps remain to be addressed as follows. Predictive models of ZOI performance often rely on narrowly scoped datasets, which restrict their applicability across broader synthesis and microbial test conditions. Small datasets are common in nanomaterials research due to the high cost and complexity of experiments. However, conventional algorithms frequently struggle to extract meaningful patterns without overfitting. As a result, models that achieve high accuracy in controlled laboratory settings often lose reliability when applied to heterogeneous synthesis routes, microbial strains, and exposure environments. These challenges emphasize the need for frameworks capable of operating effectively under limited data conditions while embedding physics-guided principles that stabilize predictions and enhance generalization.

Physics-guided approaches have demonstrated potential in domains such as materials optimization [20] but remain underutilized in the antimicrobial modeling of AgNPs. The absence of embedded physical principles limits both the robustness and explainability of existing predictive frameworks. While reservoir computing has advanced considerably in recent years, its application within the field of nanomaterials remains largely unexplored. Liquid state machine (LSM) offers efficient nonlinear representations in small-sample regimes [21]. The aim of the present study is to introduce a physics-guided liquid state machine (PG-LSM) to predict ZOI. By embedding physics-guided principles of nanoparticle formation and antimicrobial activity into a data-driven framework, the model remains robust and interpretable even under small-sample conditions. The main contributions of this work are summarized as follows. First, a physics-guided multi-reservoir architecture is introduced, wherein particle shape, particle core size, and UV–Vis peak intermediate features are encoded through a series of reservoirs and subsequently transferred to a terminal ZOI prediction stage. Second, the framework demonstrates superior predictive accuracy in limited data regimes by reducing overfitting and maintaining stability across diverse synthesis and antimicrobial testing conditions. Finally, the integration of Shapley Additive Explanations (SHAP) analysis provides physics-guided explainability by aligning computational predictions with experimentally established nanoparticle processes, which directly connects predictive modeling with physics-based understanding.

The remainder of this paper is organized into six sections. Section 2 presents data collection, data preprocessing, and details of the proposed PG-LSM framework. Section 3 reports the comparative evaluation against machine learning and deep learning baselines, as well as ablation studies and SHAP-based feature attribution for interpretability. Finally, Section 4 is the conclusion of this study.

2. Methodology

This section summarizes the overall workflow of the study through the proposed PG-LSM framework. It details the data collection and preparation process, physics-guided LSM model development, hyperparameter tuning, SHAP analysis, and model performance evaluation.

2.1. Data collection and data preparation

The dataset used in this study was obtained from the studies of nanoparticle-related antimicrobial experiments reported by Furxhi et al [19]. Each record contained experimental features of nanoparticles, synthesis conditions, and corresponding biological outcomes. For the present work, only entries reporting antimicrobial activity in the form of ZOI measurements were retained, as these directly aligned with the scope of this study. The gathered data consisted of two feature types, which are numerical and categorical. For the numerical features, the data cleaning process involves excluding samples with particle core sizes larger than 100 nm. Additionally, groups of repeated synthesis conditions that yielded different experimental outcomes for particle size, UV–Vis peak, and ZOI were consolidated by computing their average values. After filtering out categorical features, a new variable of microorganism type was introduced as an additional experimental input feature. Through these preparation steps, the dataset was systematically examined to eliminate incomplete, inconsistent, and non-representative data in the column. Finally, the data cleaning and preparation process resulted in a refined subset of 199 samples. The datasets obtained are presented in Appendix A (Table A2).

The prepared dataset included measurements from two complementary experiments, which are AgNPs synthesis and the ZOI test. The first experiment focused on systematically varying synthesis parameters to achieve the desired particle size, while the second experiment aimed to evaluate antimicrobial performance. A comprehensive summary of the experimental features is provided in Table 1, which details the experimental feature names, codes, data types, measurement units, and domain sizes. Furthermore, Appendix A (Table A5) describes these inputs in detail and explains their relationship to the synthesis and ZOI test tasks. Numerical variables are represented by their measurement ranges, whereas categorical variables are described by the number of distinct classes. The structured representation ensures that both quantitative and qualitative features are explicitly defined before model integration.

Table 1.

Summary of experimental features and domains after data preprocessing.

Experimental	Feature name	Code	Type	Domain size	Unit
AgNPs synthesis feature	Synthesis duration	x1	Numerical	0.16–72	h
	Capping agent concentration	x2	Numerical	0.02–100	mg·mL⁻¹
	Temperature	x3	Numerical	25–100	oC
	Scale synthesis	x4	Numerical	11–510	mL
	Precursor concentration	x5	Numerical	1–100	mM
	Reducing agent quantity	x6	Numerical	1–500	mL
	Synthesis steps	x7	Categorical	2 classes	-
	External energy	x8	Categorical	2 classes	-
	Stirring	x9	Categorical	6 classes	-
	Capping agent class	x10	Categorical	4 classes	-
	Capping agent	x11	Categorical	19 classes	-
	Order of reagents	x12	Categorical	3 classes	-
	Treatment	x13	Categorical	3 classes	-
	Reducing agent source	x14	Categorical	2 classes	-
	Reducing agent class	x15	Categorical	4 classes	-
	Reducing agent	x16	Categorical	19 classes	-
ZOI test	Exposure dose concentration	x17	Numerical	0.0098–6	mg·mL⁻¹
	Exposure duration	x18	Numerical	24–72	h
	Microbial species	x19	Categorical	22 classes	-
	Microbial culture medium	x20	Categorical	6 classes	-
	Microorganism type	x21	Categorical	3 classes	-
AgNPs synthesisi ntermediate output	Shape	L1	Categorical	3 classes	-
	Particle core size	L2	Numerical	4.9–57.4	nm
	UV–Vis peak	L3	Numerical	350–780	nm
ZOI test output	Zone of inhibition	y	Numerical	0–25	mm

Note: The domain size represents the experimental range and the number of distinct classes for numerical features and categorical features, respectively.

The preprocessed dataset consisted of both numerical and categorical features; each was treated with distinct strategies for model compatibility. Numerical features were standardized using z-score scaling as expressed in Eq. (1). The standardization procedure transformed values to have zero mean and unit variance, reducing discrepancies due to different scales. Thus, categorical features were transformed into integer indices and subsequently represented using one-hot encoding, which was treated as discrete identifiers.

$$z_{ij}=\frac{x_{ij}-{\mu }_j}{{\sigma }_j}$$ (1)

where $z_{ij}$ represents the standardized value of the feature $j$ for sample $i$, $x_{ij}$ represents the true value of the feature $j$ for sample $i$, ${\mu }_j$ represents the standard deviation of the feature $j$, and ${\sigma }_j$ represents the standard deviation of the feature $j$. Then, the final dataset was partitioned into three subsets, which are 70 % for training, 15 % for validation, and 15 % for testing.

2.2. Physics-guided liquid state machine (PG-LSM)

The dataset presented multiple challenges for deep learning models due to variability across synthesis conditions, categorical feature types, and limited data problems. Standard modeling frameworks, such as recurrent and convolutional networks, require large sequential datasets and perform poorly under such constraints. To overcome the limitation of standard LSM, the PG-LSM was designed as a hybrid framework that integrates physics-guided principles into intermediate reservoir computing. The experimental features from AgNPs synthesis and characterization were combined with physics-guided LSM reservoirs for ZOI prediction. The connection between input to intermediate variables is expected to enhance prediction accuracy.

2.2.1. Model development of PG-LSM

The proposed PG-LSM framework integrates synthesis conditions of AgNPs into three intermediate reservoirs, such as shape, core size, and UV–Vis peak, as listed in Table 1. While the shape and size of AgNPs are directly affected by synthesis conditions, the UV–Vis peak corresponds to their optical response induced by LSPR, which is related to the morphology and stability of nanoparticles [22]. The framework combined three reservoirs to enhance the prediction of antimicrobial performance via the ZOI reservoir, which integrates microbial exposure features such as exposure concentration, duration, bacterial species, and culture medium.

An LSM processes time-varying signals by projecting inputs into a fixed, high-dimensional network of spiking neurons while restricting learning to a simple readout layer [23]. Unlike traditional recurrent networks trained through backpropagation through time, the LSM fixes reservoir synapses and neuron parameters at an initial and the readout is learned. Multiple input channels deliver spike trains through static input-to-reservoir synapses, after which the readout layer linearly combines membrane potentials and spike events over time. A schematic of the LSM architecture consists of an input layer, a recurrent reservoir of spiking neurons, and a readout layer.

The behavior of LSM is governed by four hyperparameters. The reservoir size sets the dimensionality of the state space, improving nonlinear separability up to the memory demand. The spectral radius controls dynamic gain and fading memory stability, the leak rate determines the neuron time constant, which trades responsiveness against temporal integration, and the input-scaling coefficient multiplies the static input-to-reservoir synapses. In deep LSM configuration, these hyperparameters define the initialization and stability constraints, while internal parameters are optimized jointly with the readout under a spectral radius constraint.

The reservoir neurons follow the standard leaky integrate-and-fire (LIF) formulation, widely adopted in liquid state machine studies from [24]. To provide a more structured representation, the extended canonical leaky LIF is formulated by decomposing the synaptic input into mechanistically interpretable components as expressed in Eq. (2). The dynamics combine multiple interacting components. The leak term drives exponential decay toward the resting potential, controlling the fading-memory horizon. External inputs emerge through static synapses as spike-encoded features, while recurrent excitation reinforces temporal patterns and stabilized recurrent inhibition. An autaptic self-feedback term further adjusts neuronal responsiveness based on the firing history of the neurons.

$$\frac{d{V}_l\left(t\right)}{dt}=-\alpha (V_l\left(t\right)-V_{\mathrm{r}})+\frac{R_m}{{\tau }_m}\left[I_l^{ext}\left(t\right)+\sum _{m\in E}{w}_{lm}^E(s_m*{\kappa }_E)\left(t\right)-\sum _{n\in I}{w}_{ln}^I(s_n*{\kappa }_I)\left(t\right)+w_l^S(s_l*{\kappa }_s)\left(t\right)\right]$$ (2)

where $V_l\left(t\right)$ denotes the membrane potential of a neuron $l$, $V_r$ denotes the resting potential, ${\tau }_m$ denotes the membrane time constant, $R_m$ denotes the membrane resistance, $I_i\left(t\right)$ denotes the total synaptic input current, $\alpha$ denotes the leak rate, and $\frac{R_m}{{\tau }_m}$ denotes a factor that scales synaptic input current into voltage changes according to the membrane resistance. $w_{lm}^E$, $w_{ln}^I$, and $w_l^S$ denote the recurrent excitation, recurrent inhibition, and autaptic self-feedback weight. $s_m\left(t\right)$, $s_n\left(t\right)$, and $s_l\left(t\right)$ denote the spike train of excitatory, inhibitory, and current neurons. ${\kappa }_E$, ${\kappa }_I$, and ${\kappa }_s$ denote the synaptic kernel of excitatory, inhibitory, and autaptic self-feedback neurons.

After each membrane update, the neuron $l$ applied a hard threshold. If the updated potential exceeds the threshold as described in Eq. (3). The neuron emits a spike, and its membrane is reset to the resting potential; the neuron remains inactive for a time constant of the membrane. Otherwise, no spikes are produced. Combined with leak and balanced recurrence, the threshold–reset mechanism yields a high-dimensional and fading-memory state that uniquely encodes each input sequence, even when constructed from short spike encodings of tabular features [25].

$$S_l(t+1)=\{\begin{array}{cc}1,& V_l(t+1)\ge V_{thres}\\ 0,& \text{Otherwise}\end{array}$$ (3)

where $S_l(t+1)$ denotes the spike indicator of a neuron $l$ at the next time step, $V_{thres}$ denotes the membrane threshold.

In the baseline configuration, reservoir parameters remain fixed after initialization and only the readout is trained. In the deep LSM, input weights, leak rates, biases, and recurrent matrices are optimized together with the readout, while fading-memory stability is maintained by constraining the spectral radius and input-scaling factor. To improve conditioning, spike trains are smoothed using an exponential filter, and the readout combines the filtered responses with weights estimated by minimizing a regularized mean squared error as expressed in Eq. (4)-(5).

$${\widehat{S}}_l\left(n\right)=(1-{\gamma }^{*}){\widehat{S}}_l(n-1)+{\gamma }^{*}{S}_l\left(n\right)$$ (4)

$$z\left(n\right)=\sum _{n=1}^N{V}_l{\widehat{S}}_l\left(n\right)$$ (5)

where ${\gamma }^{*}$ is the filter constant controls the balance between recent and past spike, ${\widehat{S}}_l$ is the smooth spike trains of the neuron $l$, and $z\left(n\right)$ is the readout output of the reservoir.

2.2.2. Transfer learning of physics-guided reservoir

The overall framework of the PG-LSM is presented in Fig. 1. Fig. 1a outlines the complete workflow, which includes data collection, cleaning, partitioning, model development, and hyperparameter optimization. Within this workflow, the prediction network is implemented as a series of multi-reservoir LSM, as detailed in Fig. 1b, where each stage transforms physics-guided embeddings that contribute to ZOI prediction.

Fig. 1.

The proposed PG-LSM framework: a) overall data processing workflow and b) multi-reservoir architecture.

A transfer learning strategy is applied in which the particle shape, core size, and UV–Vis peak reservoirs are pretrained on their respective tasks and subsequently frozen to stabilize learning and prevent overfitting. The input features are first processed through input weight selection before being passed to the pretrained reservoirs. These three reservoirs were operated independently, and their frozen embeddings are transferred directly to the terminal ZOI reservoir, where the influence of particle shape, core size, and UV–Vis peak is correlated with ZOI performance. During the transfer learning process, the learnable parameters of the upstream reservoirs (reservoirs that predict three intermediate variables) remain fixed after pretraining, ensuring that their physics-informed representations are preserved while reducing computational cost [26]. Only the projection layers linking the frozen reservoirs and input features to the ZOI reservoir (x₁₇-x₂₁) are updated during fine-tuning. The framework utilizes prior physics-guided information while focusing training capacity on the ZOI prediction task.

2.2.3. Hyperparameter tuning

According to the model architecture, a hyperparameter tuning process was undertaken to calibrate both reservoir dynamics and the readout head of PG-LSM. Bayesian optimization employed a Gaussian process surrogate model to approximate prediction loss over the multidimensional hyperparameter space [27]. Candidate configurations were proposed by an acquisition function that balances exploration of under-sampled regions with exploitation of areas likely to yield performance gains. During each iteration, the PG-LSM was trained on the chosen partition, resulting in mean squared error (MSE) for informed updates to the posterior surrogate model, which guides subsequent proposals toward regions with the highest expected improvement [28]. Compared to grid search and unguided random search, Bayesian optimization markedly reduces the number of required training runs and uncertainty in parameter estimation to avoid suboptimal parameter regions [29]. The hyperparameters and their searching domains during Bayesian optimization were selected to balance representation complexity, computational stability, and generalization in a small-sample regime. The full range of these hyperparameters is listed in Appendix A (Table A6).

Reservoir size governs the dimensionality of the LSM, enabling interactions between physics-guided inputs while controlling variance and computational cost. To maintain stability and reduce the risk of overfitting, the reservoir size was constrained by a rule-of-thumb principle, where the number of training samples exceeds the number of reservoir nodes. The node count was also aligned to powers of two to enhance computational efficiency. Alongside dimensionality, dropout was applied to moderate reservoir activity and prevent overfitting by suppressing overly dominant liquid dynamics. However, excessively high probabilities tend to eliminate too much information flow and destabilize representations [30].

The learning rate regulates the trade-off between convergence speed and stability of the optimizer, with small values 10^–5 and large values 10^–2. In parallel, the leak rate shapes the effective neural memory, allowing the reservoir to capture both short- and long-term dependencies by sweeping across low to high values. Long-term stability further depends on the recurrent gain; the spectral radius was constrained to be less than 0.95 to respect the fading-memory heuristic, while the input-scaling factor was applied to avoid under-exciting or saturating units. Finally, a regularization factor applied at the readout improves conditioning during training and further mitigates the overfitting problem.

The optimization objective was defined as minimizing the MSE between predicted and experimental ZOI values using the validation dataset, expressed as Eq. (6).

$$L_{MSE}=\frac{1}{N}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2$$ (6)

where $y_i$ denoted the observed value and ${\widehat{y}}_i$ denoted the predicted value.

2.2.4. SHAP analysis

SHAP assigns an additive contribution to each input feature such that contributions sum to the prediction values of the model [31], which provides two complementary views of model behavior. For per-sample analysis, local SHAP attributes a signed contribution to each feature for a single observation, indicating both direction and strength of influence on prediction relative to a reference expectation for diagnosing case-level sensitivity. While global SHAP aggregates absolute contributions across many samples to quantify overall importance and rank features by average impact. The global SHAP distribution demonstrates systematic tendencies, stability across folds, and interaction-sensitive patterns that illustrate nonlinear responses and possible thresholds. The SHAP value equation is expressed as follows using Eq. (7).

$${\varphi }_j=\frac{1}{N}\sum _{S\subseteq P\setminus \left\{j\right\}}\frac{\left|S\right|!(N-\left|S\right|-1)!}{(N-1)!}\left[f(S\cup \{j\left\}\right)-f\left(S\right)\right]$$ (7)

where $P$ denotes the feature index of the predictor, $S$ denotes a coalition not containing the feature $j$, $\left|S\right|$ denotes the coalition size of the feature, $f\left(S\right)$ denotes the model outcome when only the feature in $S$ are presented, and $f(S\cup \{j\left\}\right)$ denotes a predicted outcome when the feature $j$ is combined with $S$.

2.2.5. Model performance evaluation

Following hyperparameter optimization, generalization performance was quantified on the test dataset using three complementary metrics of coefficients of determination (R²), root mean square error (RMSE), and mean absolute error (MAE) as defined in Eqs. (8)–(10). R² measures the proportion of variance in the observed values explained by the model. RMSE computes the square root of the average squared residuals, providing an interpretable scale for typical prediction uncertainty. MAE summarizes the average absolute deviation and is less sensitive to extreme errors. Interpreting indicators altogether provides an overall comprehension of model performance. A high R² together with low RMSE and MAE indicates accurate and well-calibrated predictions.

$$R^2=1-\frac{\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{\sum _{i=1}^n{(y_i-\overline{y})}^2}$$ (8)

$$RMSE=\sqrt{\frac{1}{n}\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}$$ (9)

$$MAE=\frac{1}{n}\sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|$$ (10)

where $y_i$ denoted observed value, ${\widehat{y}}_i$ denoted predicted value, $\overline{y}$ denoted the average value of the observed dataset, and $n$ denoted the total number of observations.

3. Results

3.1. Comparative performance with baseline models for zone of inhibition prediction

The predictive performance of PG-LSM was benchmarked against eight representative models from both deep learning and machine learning algorithms. The deep learning group comprised five architectures of feedforward neural network (FNN) as a baseline for nonlinear regression, an autoencoder network (AE) and a convolutional neural network (CNN) for intermediate feature extraction, a long short-term memory network (LSTM) for sequential dependency modeling, and a simple liquid state machine (LSM) representing recurrent spiking dynamics without physical guidance. Three machine learning baselines are Gaussian process regression (GPR) as a kernel-based nonparametric method, random forest (RF), and XGBoost as an ensemble model capturing nonlinear interactions.

Fig. 2 demonstrates the comparison of predicted and observed zone of inhibition (ZOI) values across three datasets. Fig. 2a corresponds to the training dataset, Fig. 2b to the validation dataset, and Fig. 2c to the test dataset. Visually, the issue of overfitting in GPR is evident; while the GPR data align perfectly along the diagonal in the training set (Fig. 2a), they disperse considerably in the test set (Fig. 2c). Similarly, the FNN exhibits a high degree of scatter below the identity line in the test set, confirming its inability to generalize. In contrast, PG-LSM predictions cluster tightly along the identity line even at larger ZOI diameters and maintain consistency in the dense lower-range region, whereas ensemble models like RF and XGBoost exhibit greater dispersion.

Fig. 2.

Performance comparison across three datasets of a) training dataset, b) validation dataset, and c) testing dataset for ZOI prediction.

According to the results (detailed results in Appendix A, Table A8) and Fig. 2, the PG-LSM achieved the best overall performance among the deep learning groups on the test dataset. PG-LSM achieved R² of 0.956, RMSE of 1.151, and an MAE of 0.486. The alignment between training, validation, and test results shows stable generalization with consistently low error magnitudes. In contrast, the simple LSM achieves lower accuracy on the test data with an R² of 0.930, RMSE of 1.450, and an MAE of 1.055. Among the deep learning architectures, the LSTM network ranked second with an R² of 0.898, RMSE of 1.750, and an MAE of 1.281, followed by CNN, AE, and FNN, which exhibited progressively weaker test performance. Within the machine learning group, XGBoost achieved the highest test accuracy with an R² of 0.912, an RMSE of 1.624, and an MAE of 1.148, while RF produced comparable but slightly weaker results. GPR achieved the most accurate prediction on the training and validation datasets but deteriorated significantly on the test dataset with an R² of 0.758, RMSE of 2.694, and an MAE of 1.135.

3.2. Feature attribution and explainability analysis

Fig. 3a presents the global SHAP summary of the PG-LSM model for ZOI prediction. The plot simultaneously encodes three aspects of feature attribution, which consist of the horizontal axis that denotes the SHAP values (where positive values indicate an increase in predicted ZOI and negative values indicate a decrease), the vertical axis that ranks features by overall importance (mean absolute SHAP scores), and the color gradient that represents the original feature value. This visualization allows for the assessment of how low (blue) versus high (pink) input values influence the model output. As observed in Fig. 3a, exposure dose concentration (x₁₇), exposure duration (x₁₈), and microbial species (x₁₉) emerge as the top three predictors governing the antimicrobial activity. These are followed by the capping agent (x₁₁) and reducing agent (x₁₆) species or chemicals used in AgNPs synthesis. In contrast, descriptors such as synthesis duration (x₁)_, microorganism type (x₂₁), precursor concentration (x₅), capping agent class (x₁₀)_, and stirring (x₉) exhibited lower attribution scores in the present analysis, reflecting a comparatively modest influence relative to the dominant synthesis and microbial test features. While the predicted ZOI depends on various factors, SHAP analysis allows the identification of general contribution trends of individual parameters to the ZOI outcome. Regarding exposure dose concentration (x₁₇), increased exposure concentrations are associated with positive SHAP values, indicating a greater contribution to ZOI prediction. Conversely, for exposure duration (x₁₈), shorter exposure times predominantly yield positive SHAP values, whereas extended exposure durations correlate with negative SHAP values and diminished predicted ZOI. The variation in 22 categorized microbial species for the ZOI test (x₁₉) shows heterogeneous distribution of both positive and negative SHAP values, suggesting species-specific responses to AgNPs exposure. Similarly, the capping agent (x₁₁) and reducing agent (x₁₆) demonstrate widely distributed SHAP values that depend on the sources used, including specific plant species, algae extract, biomolecules, and chemicals.

Fig. 3.

SHAP-based interpretability of the PG-LSM framework: (a) global SHAP summary for ZOI prediction, (b–c) species-specific absolute SHAP rankings for Aeromonas and Staphylococcus, respectively, (d) global SHAP for the core-size reservoir; (e) global SHAP for the UV–Vis reservoir.

Within the proposed PG-LSM framework, the synthesis parameters of AgNPs are initially processed through intermediate reservoirs, including particle core size, shape, and UV–Vis absorption peak, prior to their contribution to the final ZOI prediction. Therefore, separate global SHAP analyses for core size and UV–Vis absorption peak intermediate output were performed to evaluate the relative importance of synthesis parameters in determining these intermediates and their influence on the predicted antibacterial response (Fig. 3d and Fig. 3e, respectively). SHAP analysis of particle shape was not included because shape is a categorical feature, and the interpretation is not straightforward due to the categorical encoding. The five most influential synthesis variables for core size prediction of AgNPs were the reducing agent (x₁₆), the capping agent (x₁₁), the capping agent concentration (x₂), stirring (x₉), and the reducing agent quantity (x₆). Similarly, for UV–Vis peak prediction, the two most important contributors were the same as those affecting core size, which are the capping agent (x₁₁) and the reducing agent (x₁₆), followed by the scale of synthesis (x₄), the synthesis step (x₇), and stirring (x₉). When compared with the features contributing to ZOI prediction, the highly SHAP-ranked synthesis-related variables consistently shared with both core size and UV–Vis peak predictions include the capping agent (x₁₁) and reducing agent (x₁₆), demonstrating their influence across AgNPs synthesis, intermediate nanoparticle properties, and ZOI outcome.

To understand how these feature dependencies shift across different bacterial species, Fig. 3b and Fig. 3c present a local SHAP comparison for Aeromonas and Staphylococcus, respectively. The comparison illustrates the relative influence of synthesis and ZOI test features when the bacterial species changes. In the Aeromonas case (Fig. 3b), exposure dose concentration (x₁₇) acts as the overwhelming determinant with a mean absolute SHAP value of 1.295, followed by exposure duration x₁₈ (0.909) and microbial species x₁₉ (0.641). Microorganism type (x₂₁) and reducing agent class (x₁₅) also appear in the top five, suggesting specific sensitivity of Aeromonas to these parameters. However, the Staphylococcus case (Fig. 3c) reveals a distinct rearrangement of feature importance. While x₁₇ remains the top feature, its dominance decreases (SHAP value drops to 1.071), and the influence of the microbial species feature (x₁₉) notably increases to 0.786. Furthermore, the feature set shifts: reducing agent (x₁₆) and precursor concentration (x₅) emerge as top determinants for Staphylococcus, replacing microorganism type (x₂₁) and reducing agent class (x₁₅). Compared with Aeromonas, the influence of microbial species (x₁₉) is enhanced, demonstrating that species-specific characteristics affect the antimicrobial activity of AgNPs against Staphylococcus.

3.3. Ablation study

An ablation study was performed to isolate the contribution of each physically guided reservoir stage and to benchmark the PG-LSM against baseline variants. Table 2 summarizes the five experimental designs by specifying which reservoirs remained active in each variant. Experiment 1 represents the baseline simple LSM, where all inputs are collapsed into a single recurrent reservoir without physical guidance. Experiments 2–4 represent partial-ablation variants, where one specific physical reservoir (shape, core size, or UV–Vis) is omitted to isolate its individual impact. Experiment 5 represents the proposed model, which is the fully integrated PG-LSM with all liquid reservoirs.

Table 2.

Experimental design and performance results of the ablation study for PG-LSM.

Experiment	Single reservoir	Multi-reservoir			Purpose	Testing performance
		Shape	Core size	UV–Vis		R2	RMSE	MAE
Experiment 1	✓	—	—	—	Baseline LSM model	0.930	1.450	1.055
Experiment 2	—	—	✓	✓	Effect of excluding shape	0.926	1.489	0.799
Experiment 3	—	✓	—	✓	Effect of excluding core size	0.910	1.640	0.783
Experiment 4	—	✓	✓	—	Effect of excluding UV–Vis	0.933	1.413	0.826
Experiment 5	—	✓	✓	✓	Effect of the proposed design	0.956	1.151	0.486

According to the results in Table 2, Experiment 5 (the fully integrated PG-LSM) achieved the best generalization performance with a testing R² of 0.956 and the lowest RMSE (1.151) and MAE (0.486). In contrast, Experiment 1, consisting of a single unguided reservoir (unguided baseline), yielded a lower R² of 0.930 and substantially higher RMSE and MAE values of 1.450 and 1.055, respectively. Regarding the partial-ablation variants (Experiments 2–4), Experiment 3 (excluding core size) resulted in the lowest performance among all configurations, with an R² of 0.910 and the highest RMSE of 1.640. Experiment 2 (excluding shape) produced an R² of 0.926 and an RMSE of 1.489. Experiment 4 (excluding UV–Vis) produced an R² of 0.933 and RMSE of 1.413. Experiment 5 produced a higher R² and lower error than Experiment 4, indicating that the physics-guided framework is most effective when the UV–Vis reservoir is utilized together with shape and core size reservoirs.

Prior to the integration into the PG-LSM, the intermediate reservoirs were validated independently as detailed in Appendix B (Fig. B2). The shape reservoir showed low misclassification rates across classes. The core-size reservoir generated predictions with absolute errors below 0.20 nm relative to experimental measurements. The UV–Vis reservoir produced absolute prediction errors below 5 nm compared to observed peaks.

4. Discussion

4.1. Enhanced generalization through physics-guided reservoir design

The comparative analysis highlights the advantage of embedding physics-guided into reservoir computing. The superior performance of PG-LSM arises from the integration of physics-guided constraints into the reservoir dynamics. In contrast, deep learning models such as CNN, AE, and FNN rely on generic representation learning, which is less effective under a limited dataset. For the machine learning group, although ensemble methods such as RF and XGBoost captured important nonlinear relationships in the limited dataset, they lacked physical constraints to incorporate physicochemical principles, which limits their ability to generalize the observed data. Furthermore, the discrepancy between RMSE and MAE in GPR further indicates that a small subset of outlier samples disproportionately affected its predictions.

Under the limited-data regime of Fig. 2, embedding physical consistency during training reduces dispersion, particularly in scenarios with limited data and when calculating extreme values. For the proposed model, the predictions across the training, validation, and testing datasets closely align with the identity line, indicating stable learning with minimal variance. Validation and test predictions remain tightly clustered along the diagonal, confirming reliable generalization under distributional shifts. The PG-LSM maintains alignment with physics-guided consistency and achieves higher accuracy with lower errors on testing data. The results confirm robust generalization under limited-sample conditions.

4.2. Mechanistic interpretation of feature contributions

The global SHAP analysis of the PG-LSM model shown in Fig. 3a clearly reveals that ZOI prediction is primarily driven by variables in the ZOI test, followed by AgNPs synthesis-related parameters. The most impactful features are exposure dose concentration (x₁₇) and exposure duration (x₁₈), indicating that ZOI prediction depends largely on the interactions between AgNPs and microbials within the culture medium. Positive SHAP values associated with higher exposure doses or AgNPs concentrations indicate that higher silver ion release enhances diffusion, leading to larger predicted ZOI. This can be explained by the diffusion-based ZOI assays, in which the inhibition zone formation is primarily determined by the diffusion of silver ions released from AgNPs acting as silver ion reservoirs, rather than by the nanoparticle diffusion [32]. Further inspection of the SHAP values relative to the corresponding exposure concentrations suggests a notable change around 0.5 mg⋅mL^-1, above which AgNPs concentration generally contributes positively and enhances the predicted ZOI. Exposure duration (x₁₈) exhibits positive SHAP values at 24 h, whereas longer exposures (48 h and 72 h) are associated with negative SHAP contributions, indicating a decrease in predicted ZOI. The negative contribution of the longer contact time to ZOI may be due to reduced availability of diffusible silver ions over time, which may be limited by diffusion in bioassays [33], as well as by binding of silver ions to microbial cellular components [3], and chemical interactions with constituents of the culture medium [34]. Microbial species (x₁₉), a category feature, exhibit varying SHAP values across species, implying different species-specific characteristics and their sensitivity to AgNPs. Distinctly high positive SHAP values are associated with Gram-negative bacteria, such as Acinetobacter and Pseudomonas, suggesting that these microbial species are predicted to be more sensitive to AgNPs exposure and tend to exhibit higher predicted ZOI values. These results may be due to the thin peptidoglycan layer of the Gram-negative cell wall, which facilitates the easier penetration of AgNPs, and to the negative charge of lipopolysaccharide (LPS) in the cell membrane, which promotes adhesion of AgNPs to the outer membrane [4]. Ranking after ZOI test variables, AgNPs synthesis parameters, particularly capping agent (x₁₁) and reducing agent (x₁₆), show the dominant but non-uniform SHAP trends, suggesting the variety of sources, including specific plant species, algae extracts, biomolecules, and chemicals, contribute differently to characteristics of AgNPs and thus antimicrobial activity. The capping agent governs nanoparticle stabilization and surface properties, influencing reactivity with microbial membranes [9], while reducing agents of different strengths affect the size and shape of AgNPs [12]. Employed as both capping agent and reducing agent, C. hirta, E. hirta, and bark extract of Alstonia scholaris tend to exhibit negative SHAP values, whereas Ziziphus nummularia leaf extract, Rumex hymenosepalus extract, and Citrus limon zest extract are associated with highly positive SHAP values, indicating their outstanding positive contribution to predicted ZOI.

The capping agent (x₁₁) and reducing agent (x₁₆) consistently appear as the most influential predictors of core size and UV–Vis absorption peak, indicating their shared impact on physicochemical characteristics and antimicrobial performance of AgNPs. Negative SHAP values associated with particular capping and reducing agents correspond to a decrease in predicted AgNPs core sizes and blue-shifted UV–Vis absorption peaks, suggesting that strong reducing agents facilitate rapid reduction of silver ions and limit core size, while effective capping agents provide surface stabilization that suppresses subsequent particle growth and aggregation [12]. For example, Ziziphus nummularia leaf extract exhibits high negative SHAP values in both core size and UV–Vis peak predictions, corresponding to a small core size (5 nm) and a UV–Vis absorption peak at 415 nm. These physicochemical characteristics are consistent with its high positive SHAP value in ZOI prediction, indicating enhanced antimicrobial activity. The higher surface area of smaller nanoparticles promotes greater silver ion release that leads to more efficient interaction with microbial cells, contributing to an enlarged inhibition zone [35].

Together, based on the local SHAP, these features (x₁₇ and x₁₈) reflect the strong dependence of Gram-negative inhibition on nanoparticle dosage and exposure time. x₁₉ and x₂₁ also show the variations in microbial species and types of microorganism, which directly affect sensitivity to AgNPs. Among synthesis factors, x₁₅ demonstrates substantial effects, showing the importance of reducing agent classes such as chemicals, plant extracts, algae extracts, and biomolecules. While x₁₇ and x₁₈ remain dominant (for Staphylococcus), their absolute SHAP values are lower than those of Aeromonas, indicating that antibacterial potency in Gram-positive bacteria is less dependent on AgNPs dose and exposure time. Furthermore, x₁₆ and x₅ gain importance relative to Aeromonas, suggesting that the choice of reducing agent and precursor concentration, which affect particle morphology and stability of AgNPs, play a more decisive role in overcoming the thicker peptidoglycan barrier of Gram-positive cells.

4.3. Implications for nanoparticle engineering and predictive modeling

Ablation studies of Experiments 2–4 provide further insights into the contributions of individual reservoirs. The removal of the shape reservoir in Experiment 2 resulted in reduced test performance (R² = 0.926, RMSE = 1.489, MAE = 0.799). Elimination of the core-size reservoir in Experiment 3 produced a slightly higher decline (R² = 0.910, RMSE = 1.640, MAE = 0.783), which reflects the importance of particle size descriptors. The ablation of either size or shape emphasizes the role of particle morphology in determining antimicrobial activity.

A smaller size results in a higher surface area of AgNPs, which enhances antimicrobial performance by promoting silver ion release and increasing interactions with microbial cell membranes [35]. While the effect of particle size on antimicrobial activity of AgNPs can be directly determined, the effect of shape is more complex. Particle geometries with higher specific surface area, sharp edges that facilitate membrane penetration, and more reactive exposed facets tend to increase the antimicrobial effect of AgNPs [36]. Due to these influences, the prediction of microbial inhibition of AgNPs may rely more on size than on shape. Excluding the UV–Vis reservoir (Experiment 4) yielded an R² of 0.933 with an RMSE of 1.413 and an MAE of 0.826, indicating that optical descriptors contribute implicitly, but their absence is less detrimental than the removal of morphological or size-related reservoirs. The results suggest that the UV–Vis peak influences model performance; however, its effect is less than that of size and shape. A possible reason is that UV–vis spectroscopy provides optical information related to particle characteristics, but it is not a direct factor in antimicrobial activity.

4.4. Limitations and future directions

Despite the robust performance of the PG-LSM in handling small-sample regimes, it is crucial to recognize the boundaries of the presented approach. Our reliance on a dataset of 199 samples that are consolidated from specific literature sources. While the framework excels at predicting outcomes for specific synthesis routes using chemical and green methods, future studies must expand the training datasets to include other methodologies to ensure true universal applicability. Another challenge lies in the granularity of chemical features. By treating ingredients, such as reducing agents, as categorical variables, the model effectively memorizes labels rather than learning the underlying molecular properties that drive the reaction. This prevents the framework from extrapolating to novel reagents it has not encountered before. To evolve from an optimization tool into a discovery engine, future work will replace these discrete labels with quantitative molecular descriptors, allowing the reservoirs to interpret the chemical logic of new candidates. Finally, the transition of the PG-LSM model from analyzing historical data to guiding real-time discovery is also an interesting goal. By embedding this model into a closed-loop active learning cycle, we can move beyond prediction to the generative design of targeted antimicrobial nanotherapeutics.

5. Conclusions

The present study introduced a physics-guided liquid state machine (PG-LSM) for the prediction of the zone of inhibition (ZOI) of AgNPs under limited-data conditions. A multi-reservoir design of pretrained particle shape, core size, and UV–Vis intermediates transferred as frozen encoders into the final ZOI stage delivered high accuracy. PG-LSM outperformed eight deep and classical baselines on the test set (R² = 0.956, root mean square error = 1.151, and mean absolute error = 0.486) and maintained close agreement between validation and test metrics, indicating stable generalization in a small-sample regime. Ablation results confirmed that removing any intermediate reservoir weakened performance, emphasizing the benefit of incorporating particle shape, core size, and UV–Vis peak into the antimicrobial efficacy prediction. Global SHAP analysis ranks exposure dose concentration and exposure duration as the primary drivers of predicted ZOI, followed by microbial species and synthesis variables such as capping and reducing agents. SHAP patterns suggest a response change around 0.5 mg·mL⁻¹ for exposure dose, where higher concentrations more often associate with positive contributions to predicted ZOI. Exposure duration shows positive contributions at 24 hours, while 48 and 72 hours are more often associated with negative contributions. Microbial species show heterogeneous contributions across categories, consistent with species-dependent sensitivity. Intermediate-reservoir checks support consistent representation transfer, with low misclassification for shape, core-size absolute errors below 0.20 nm, and UV–Vis peak absolute errors below 5 nm. Separated SHAP analyses for intermediate reservoirs identify dominant synthesis drivers of nanoparticle properties. Core-size and UV–Vis peak predictions share the same top two drivers, which are reducing agent and capping agent, while the third-ranked driver differs between capping agent concentration for core size and scale of synthesis for the UV–Vis peak. Shared importance of capping and reducing agents across ZOI, core size, and UV–Vis supports a coherent mechanistic link from synthesis chemistry to intermediate particle properties and antimicrobial activity.

CRediT authorship contribution statement

Tanawadee Dechakupt: Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Formal analysis, Conceptualization. Chinakrit Akkawong: Writing – review & editing, Writing – original draft, Visualization, Software, Resources, Investigation, Data curation. Santi Bardeeniz: Writing – review & editing, Writing – original draft, Visualization, Software, Methodology, Investigation, Conceptualization. Kulpavee Jitapunkul: Writing – review & editing, Supervision. Chanin Panjapornpon: Writing – review & editing, Writing – original draft, Validation, Supervision, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

The data used in this study is provided in the Supplementary materials.