Neural Network-Based Modeling for Precise Potato Yield Prediction Using Soil Parameters

Abstract

This study analyses the potential of artificial neural networks (ANN) in accurately predicting potato yields based on 11 parameters characterising the soil environment. Accurate yield forecasting is crucial for optimising potato production, especially in the context of potato processing. Due to the significant impact of soil properties on yield, there is a need for comprehensive predictive models that take these factors into account. The field studies (2021–2024) included an analysis of soil parameters determining potato tuber yield. The developed ANN model was highly accurate, as evidenced by the following indicators: R² = 0.8227, RMSE = 4.19 t∙ha⁻¹, MAE = 3.35 t∙ha⁻¹, MAPE = 7.34%. Global sensitivity analysis showed that cation exchange capacity (CEC), base saturation percentage (V), and sum of exchangeable bases (S) are key parameters influencing tuber yield. The results indicate that neural networks are effective in modelling complex relationships between soil parameters and potato yield, and that soil properties play a fundamental role in increasing yields and improving potato quality. The approach used may contribute to optimizing the nutrient content of potato tubers intended for French fry production. Future studies should incorporate climate data and micronutrients to enhance the accuracy of predictive models, potentially leading to a 10–15% improvement in yield predictions.

1. Introduction

The potato (Solanum tuberosum L.) remains one of the most important crops globally, playing a key role in ensuring food security and economic stability in many regions [1]. Population growth and changing climatic conditions pose a challenge to agriculture to increase agricultural production efficiency, including potato yields [2]. Optimising potato cultivation for processing, particularly for French fries, is crucial to meeting the growing demand for high-quality food products. Effective yield prediction, taking into account the influence of various parameters: meteorological, phenological, and soil-related, is, therefore, becoming an indispensable tool in production planning and supply for the processing industry [3,4,5,6,7,8]. Accurate yield forecasting allows for the optimisation of production processes, minimisation of losses, and ensuring a stable supply of raw materials with the appropriate quality parameters, which has a direct impact on the efficiency of potato processing into French fries [9,10].

The success of potato cultivation is strongly linked to soil quality and its physicochemical properties. Studies have shown that soil parameters such as potassium (K₂O) and magnesium (Mg) content have a direct impact on tuber development and overall plant health [11]. Hydrolytic acidity, the total content of alkaline cations, and the sorption capacity of the soil determine the availability of nutrients and the stability of soil pH, which is crucial for the proper growth of potatoes [12]. The granulometric composition of soil, including the content of sand, dust, and clay, affects water retention and air permeability, which has a direct impact on root system development and nutrient uptake efficiency [13]. The organic carbon and humus content improvesoil structure, increase its ability to retain water and nutrients, and stimulate soil biological activity, which translates into better conditions for plant growth [14]. In addition, the total nitrogen (N) content is essential for the synthesis of proteins and other organic compounds, which affects plant growth and development [15]. Deficiencies or excesses of these components can lead to serious physiological disorders, reduced resistance to diseases and pests, and a significant decrease in yields [16].

A thorough understanding of the soil environment is the foundation of sustainable potato crop management. Unlike traditional methods, which are based on average soil parameter values, modern precision farming systems use advanced technologies to monitor soil variability in space and time. Measuring soil fertility in key nutrients such as nitrogen, phosphorus, and potassium is essential for optimising fertilisation and ensuring suitable growing conditions for plants. Taking soil variability into account and adapting fertilisation to the specific needs of individual areas of the field can lead to significant increases in yields and reductions in fertiliser losses, which have important economic and environmental implications. In addition, detailed knowledge of soil parameters enables the optimisation of irrigation, minimisation of erosion risk, and improvement of the efficiency of natural resource use, which is in line with the principles of sustainable agricultural development [5,17,18,19].

Previous studies on the impact of soil parameters on potato yield have often focused on the analysis of individual factors or limited sets of variables, without taking into account the complex interactions occurring in the soil environment. As a result, there is an urgent need to develop advanced predictive models that take into account the synergistic effects of various soil parameters and enable more accurate yield forecasting [20,21]. In recent years, there has been dynamic development and implementation of advanced forecasting tools in the agricultural sector, including machine learning methods such as artificial neural networks (ANN) [22,23]. Neural networks, in particular multilayer perceptron (MLP) models, are characterised by their ability to approximate non-linear relationships and effectively process multidimensional data sets, making them a promising tool for predicting crop yields based on heterogeneous soil data [24]. Furthermore, ANNs are resistant to noise and data gaps, which is a common problem with soil and meteorological data.

In light of the above, this work focuses on the application of artificial neural networks to model the impact of various soil parameters on potato yield, with the aim of identifying key factors determining crop productivity. Our objectives are:

• To develop an ANN model for precise prediction of potato yield intended for French fry production, based on 11 key soil parameters, which is the main technical and practical aim of the research.
• To identify the most important soil parameters influencing the yield and understand how these factors shape the predictive performance of the model—that is, the goal related to analyzing the impact of parameters on the outcome.

2. Materials and Methods

Between 2021 and 2024, field research was conducted on a private farm located in the coastal zone of northern Poland (Baltic Sea region). The study area covered farmland in the vicinity of Damno (53.9733° N, 19.0961° E), Bobrowniki (53.9550° N, 19.0597° E), and Świtały (54.0492° N, 19.0658° E). The climate of the area is characterised by typical temperate conditions, with an average annual air temperature of 7.5–8.5 °C and annual precipitation of 650–750 mm. Over a period of four years, monoculture cultivation of the Ludmilla potato variety was carried out on all fields. The annual research area was approximately 100 ha, which allowed for a large number of measurement points in the fields to be included in the study. The locations were selected to minimise differences in habitat conditions, allowing for comparability of results throughout the experiment. This selection of locations ensured stable environmental conditions conducive to the cultivation of the Ludmilla potato variety, especially in the context of French fry production, which requires specific soil and climate parameters.

During the study, differences in soil conditions between fields were identified. In 2021—2024, fields 1—2021 (Figure 1), 2—2022 (Figure 2), and 3—2023 (Figure 3) were dominated by soils classified as Luvisols according to the FAO/WRB (2022) system [25]. In 2024, Cambisols were found in field 4 (Figure 4).

2.1. Field and Laboratory Research

2.1.1. Field Management Practices

Potatoes were cultivated using the ploughing system, with a row spacing of 90 cm. The planting dates for the tubers in each year were as follows: 15 April 2021, 19 April 2022, 17 April 2023, and 14 May 2024. Mineral fertilisation was applied in doses varying according to soil fertility. For average soil fertility, a total of 220 kg N/ha, 120 kg P₂O₅∙ha^−1, and 300 kg K₂O∙ha⁻¹ were applied. Phytosanitary protection of the plantation was comprehensive, taking into account the current recommendations of decision support systems, ensuring effective control of key pathogens, including Phytophthora infestans. The plantation was managed without irrigation. Approximately two weeks before the planned harvest, the haulms were desiccated. The tubers were harvested annually between 10 September and 1 October.

To illustrate the pluviothermal conditions prevailing in the 2021–2024 seasons in potato cultivation, the Seljaninov hydrothermal coefficient was calculated for the period from April 1 to October 30 of each year (Figure 5). The Sieljaninov hydrothermal coefficient (K) is a measure of precipitation efficiency in the period under study and is calculated using the following formula: K = 10P/Σ_t, where P is the monthly precipitation total and Σ_t is the sum of daily air temperatures above 0 °C in a given month. Depending on the value of the coefficient (<0.4, 0.4–0.7, 0.8–1.0, and 1.1–1.3), months are classified as extremely dry, very dry, dry, and fairly dry [26].

Analysis of the Sieljaninov coefficient (K) reveals significant differences in hydrothermal conditions between 2021 and 2024.

The year 2021 was generally characterized by wet conditions, especially in spring and autumn. The year 2022 was marked by high variability, with periods of drought interspersed with wetter periods. The year 2023 was varied, with limited access to moisture in the spring and summer, followed by a wet autumn. The year 2024 was mainly dry, with better balance in July and October, allowing for adequate soil moisture and crop growth.

2.1.2. Methodology for Soil Sampling and Preparation

Before planting, the soil was scanned using an EM-38 electromagnetic scanner (Geonics Limited, Mississauga, ON, Canada) to determine management zones within the field. The EM-38 scanner, based on electrical conductivity (ECa) measurements, enables precise determination of soil property variability within the field (e.g., texture, moisture, salinity). Based on the data obtained, management zones were designated, consisting of homogeneous plots ranging from 3 to 5 ha in size, which were treated as separate experimental plots in further analysis. In 2021, 2022, 2023, and 2024, 19, 17, 33, and 16 management zones were identified, respectively. Three soil samples were randomly collected from each zone at a depth of 0–30 cm, each weighing approximately 1.5 kg (fresh weight). The location of the soil sampling points was precisely recorded using a CHCNAV LT60H RTK GPS receiver (CHC Navigation, Shanghai, China). The soil samples were transported to the laboratory, where they were dried at 105 °C and then ground in a mortar. Further processing of the samples, including preparation for analysis, was carried out in accordance with the analytical methodologies specific to the parameters being determined.

2.1.3. Analysis of the Physicochemical Properties of Soil

Laboratory soil analyses included the following determinations. P₂O₅ content was determined by extraction and determination of available phosphorus, in accordance with PN-R-04023:1996 [27], and K₂O content was determined by extraction and determination of available potassium, in accordance with PN-R-04022:1996 [28]. Mg content was determined by extraction and determination of available magnesium, in accordance with PB 31 ed.2 methodology [29]. Hydrolytic acidity was determined using the Kappen method (extraction of H⁺ ions from the soil with calcium acetate (Ca(CH₃COO)₂) solution, titration with sodium hydroxide (NaOH) solution). The sum of exchangeable cations (S) was determined using the Kappen method (displacement of base cations with 0.1 M HCl solution, titration with NaOH) [30]. Soil sorption capacity (T): calculated as the sum of hydrolytic acidity and the sum of exchangeable cations. The saturation of the sorption complex with base cations (V) was calculated using the formula: V = (Sum of exchangeable bases/Cation exchange capacity) × 100% [30]. The content of granulometric fractions (sand, dust, clay) was determined by laser diffraction using a Mastersizer 3000+ particle size analyzer. The organic carbon content was determined using a TOC aj-analyzer multi EA 4000 analyzer; multi-Win 5.5. Humus content was calculated based on organic carbon content (Böhm’s conversion factor: humus content (%) = organic carbon content (%) × 1.724) [30]. Total nitrogen was determined using a FlashSmart series CHNS analyzer, Manufacturer: Thermo Scientific (Waltham, MA, USA). Soil pH was measured in KCl solution, in accordance with standard PB 31 ed.2 [29].

2.1.4. Methodology for Sampling Potato Tubers Before Harvest

In order to assess the potato tuber yield, two weeks before the planned harvest, samples were taken each year at precisely located points in the field. The location of the yield sampling points corresponded to the location of the soil sampling points. In situations where the soil sampling point fell on a technological path, the yield sampling point was moved by a maximum of 2 m, with the exact location of the moved point being recorded. A single sample consisted of tubers dug from an area of 3 m². After harvesting, the samples were transported to the laboratory and weighed. Based on the results obtained, the final yield was converted to t∙ha⁻¹.

2.2. Data Analysis and Model Development

2.2.1. Dependent and Independent Variables for Building and Verifying a Neural Network Model

Table 1. Definitions and ranges of values of soil and potato tuber yield variables considered in modeling.

Symbol of Variable	Description	Data Range
INDEPENDENT VARIABLES
PH	Soil pH measured in KCl	5.5–7.2
P_SOIL	Soil content of P2O5 (mg/100 g)	8.4–36.2
K_SOIL	Soil content of K2O (mg/100 g)	8.0–26.0
Mg_SOIL	Soil content of Mg (mg/100 g)	3.0–24.6
HH	Hydrolytic acidity (cmol (+)∙kg−1)	0.0–2.66
S	Sum of exchangeable bases (cmol (+)∙kg−1)	3.76–11.1
CEC	Soil sorption capacity (cmol (+)∙kg−1)	7.98–14.07
V	Base saturation percentage (%)	37.22–89.49
SAND	Percentages of sand (%)	65.24–96.16
SILT	Percentages of silt (%)	3.79–32.39
CLAY	Percentages of clay (%)	0.0–1.54
OC	Organic carbon content (%)	0.04–3.16
H	Soil humus content (%)	0.174–5.5
TN	Total nitrogen (%)	0.02–0.25
DEPENDENT VARIABLE
YP	Yield of potato tubers (t∙ha−1)	25.5–68.67

presents a summary of variables included in the statistical analysis aimed at modelling potato tuber yield (YP) based on soil properties. The YP variable, representing potato tuber yield expressed in tonnes per hectare (t∙ha⁻¹), acts as the dependent variable. The other variables are independent (predictors) and include chemical and physical parameters of the soil. The table contains variable symbols, their descriptions, and the ranges of values observed in the data set. An analysis of the normality of the distribution performed using the Shapiro–Wilk test for each variable at a 95% confidence level showed a lack of normal distribution in all cases.

2.2.2. Correlation Analysis and Elimination of Collinearity

Before starting the potato tuber yield modelling process, a detailed analysis of the collinearity of potential independent variables was carried out, covering a wide range of soil parameters. This analysis was performed exclusively on the data set intended for model construction, in accordance with the procedures described in Section 2.3.1. Collinearity, understood as the occurrence of high correlations between independent variables, is a significant factor threatening the stability and interpretability of regression models. To minimise the risk associated with this phenomenon, a correlation matrix was used, calculated using Statistica software (v13.3). On this basis, pairs of variables with high correlation coefficients were identified and further selected to ensure the consistency and reliability of the predictive model being developed.

Pearson’s correlation matrix analysis (N = 214, p < 0.05) (

Table 2. Pearson correlation matrix displaying relationships between soil parameters. Correlation coefficients are significant at a significance level of p < 0.05.

	CEC	PH	P_SOIL	K_SOIL	Mg_SOIL	HH	S	V	SAND	SILT	CLAY	OC	H	TN
CEC	1.00	−0.18	0.10	0.07	−0.10	−0.06	0.35	−0.45	−0.20	0.25	0.18	−0.27	−0.27	−0.36
PH	−0.18	1.00	0.37	0.07	0.13	−0.67	0.31	0.45	0.07	−0.15	−0.04	0.17	0.17	0.14
P_SOIL	0.10	0.37	1.00	0.46	0.18	−0.27	0.12	0.05	−0.22	0.20	0.27	0.10	0.10	−0.33
K_SOIL	0.07	0.07	0.46	1.00	0.35	−0.04	−0.07	−0.09	−0.44	0.45	0.42	−0.08	−0.08	−0.34
Mg_SOIL	−0.10	0.13	0.18	0.35	1.00	−0.08	−0.04	0.03	−0.16	0.08	0.16	0.08	0.08	−0.01
HH	−0.06	−0.67	−0.27	−0.04	−0.08	1.00	−0.60	−0.51	0.04	0.03	−0.09	−0.17	−0.17	−0.11
S	0.35	0.31	0.12	−0.07	−0.04	−0.60	1.00	0.67	−0.16	0.13	0.19	−0.01	−0.01	−0.11
V	−0.45	0.45	0.05	−0.09	0.03	−0.51	0.67	1.00	−0.01	−0.06	0.05	0.20	0.20	0.16
SAND	−0.20	0.07	−0.22	−0.44	−0.16	0.04	−0.16	−0.01	1.00	−0.92	−0.91	0.22	0.22	0.35
SILT	0.25	−0.15	0.20	0.45	0.08	0.03	0.13	−0.06	−0.92	1.00	0.93	−0.28	−0.28	−0.43
CLAY	0.18	−0.04	0.27	0.42	0.16	−0.09	0.19	0.05	−0.91	0.93	1.00	−0.22	−0.22	−0.37
OC	−0.27	0.17	0.10	−0.08	0.08	−0.17	−0.01	0.20	0.22	−0.28	−0.22	1.00	1.00	0.21
H	−0.27	0.17	0.10	−0.08	0.08	−0.17	−0.01	0.20	0.22	−0.28	−0.22	1.00	1.00	0.21
TN	−0.36	0.14	−0.33	−0.34	−0.01	−0.11	−0.11	0.16	0.35	−0.43	−0.37	0.21	0.21	1.00

Comment: The variable names are explained in Table 1.

) revealed the presence of significant collinearity between selected independent variables, which required appropriate measures to ensure the stability and interpretability of the model. It was found that the correlations between soil granulometric fractions—sand (SAND), silt (SILT) and clay (CLAY)—exceeded the critical value of 0.9 (r = −0.92 between SAND and SILT; r = −0.91 between SAND and CLAY; r = 0.93 between SILT and CLAY), indicating the need to limit the number of variables describing the granulometric composition. In addition, an excellent correlation (r = 1.000) was observed between organic carbon (OC) content and humus (H) content, which implied the need to exclude one of these variables from further analysis. In order to reduce the risk associated with the collinearity problem, it was decided to eliminate the H variable and reduce the granulometric variables, leaving only one variable representing granulometric composition in the analysis—SAND.

2.3. Data Preprocessing

The data were preliminarily analyzed, including the calculation of basic statistical measures for all variables. To ensure high data quality, nine observations that were extreme outliers were identified and removed. A simple method based on standard deviation was used to detect outliers: the arithmetic mean (μ) and standard deviation (σ) were calculated for each variable, and then a threshold of 3σ was set. Values below μ − 3σ or above μ + 3σ were considered potential outliers and excluded from further analysis.

This simple but effective method allows for quick identification of rare, extreme values that can distort the results of the analysis.

2.3.1. Method of Creating a Neural Network Model

In order to ensure the proper construction and validation of predictive models, the entire data set was strategically divided into two separate subsets. Set A, comprising 214 cases, was used for training regression models and neural networks. Set B, containing 26 cases, served as a validation set for evaluating and verifying the effectiveness of the trained models. It should be emphasised that the selection of cases for set B was not entirely random; six cases were selected at random from each field in the experiment and supplemented with two cases selected completely at random to ensure the representativeness and diversity of this test set.

In order to develop an optimal predictive model, a method of automatic neural network architecture design based on a structure optimisation algorithm was used. The data was divided into three separate sets: training—70%, validation—15%, and testing—15%, which enabled an independent assessment of the final effectiveness of the trained model. Work on the model was carried out in Statistica version 13.3. During the process, as many as 10,000 network configurations were searched, which ensured a wide range of solutions and increased the likelihood of finding the optimal model. The most suitable architecture was selected based on criteria for minimising the difference between the quality of the training and test sets, which is crucial for ensuring the stability and generalisation of the model. To this end, metrics such as the coefficient of determination (R²), mean square error (MSE), and mean absolute error (MAE) were evaluated, aiming to maximise prediction quality while minimising errors. In addition, smaller networks (with a limited number of neurons in the hidden layer) were preferred in the selection process, which helped to reduce the complexity of the model and improve its interpretability. The network was trained using the Levenberg–Marquardt algorithm (also known as the BFGS 88 algorithm), which allowed for quick and stable parameter fitting. The training process consisted of 50 epochs, which proved sufficient to achieve convergence without excessive adaptation to the training data. The selected model is a multilayer perceptron (MLP) network with an 11-5-1 architecture, i.e., containing 11 neurons in the input layer, 5 neurons in the hidden layer, and 1 output neuron. The activation function in the hidden layer was set as hyperbolic tangent, while the activation function in the output layer is linear, which corresponds to the characteristics of the regression problem.

In addition, a sensitivity analysis was performed on the test set to assess the stability and robustness of the model to changes in the input data. As a result of this analysis, information was obtained on which independent variables have the greatest impact on the dependent variable and how individual factors shape the model’s results. Such an analysis allows for a better understanding of the mechanisms of the network’s operation, as well as confirming the reliability and resilience of the model to potential data disturbances, which increases confidence in its practical applications.

2.3.2. Model Evaluation

In order to verify the predictive properties of the developed model, a post hoc analysis of prediction errors was performed using several commonly used evaluation measures. In particular, six different statistics were used, including: (1) global relative approximation error (RAE), (2) root mean square error (RMS), (3) mean absolute error (MAE), (4) mean absolute percentage error (MAPE), (5) maximum error value calculated for the entire model (MAX), (6) maximum percentage error (MAXP), (7) coefficient of determination R².

All of the above measures were calculated based on the results obtained from the test set (B). The methods for calculating individual errors are described in detail in the authors’ previous literature. [31,32,33]. Their use allowed for a comprehensive assessment of the accuracy and reliability of the model, both in terms of overall prediction quality and response to local deviations.

3. Results

3.1. Basic Statistical Measures of Predictive Model Variables

Table 3. Descriptive statistics of soil and chemical variables across datasets.

Variable	Statistic	Training	Testing	Validation
pH	Min	5.5	5.6	5.7
	Max	7.2	6.6	7.1
	Mean	6.29	6.19	6.23
	SD	0.31	0.30	0.45
P_SOIL	Min	8.4	8.4	8.9
	Max	36.2	25.6	36.2
	Mean	17.96	17.46	17.36
	SD	4.66	3.44	7.20
K_SOIL	Min	8	8	8
	Max	26	22	24.6
	Mean	16.44	15.92	15.95
	SD	3.35	3.06	5.92
Mg_SOIL	Min	3	3	3
	Max	24.6	17	24.6
	Mean	6.31	5.95	6.35
	SD	2.60	2.43	6.37
HH	Min	0.1	1	1
	Max	2.66	2.5	2.24
	Mean	1.43	1.47	1.51
	SD	0.37	0.43	0.33
S	Min	3.76	4	5
	Max	11.1	10.3	10.4
	Mean	7.39	7.71	7.19
	SD	1.47	1.55	1.48
CEC	Min	8	7.98	8
	Max	14.07	13.5	14
	Mean	10.66	10.83	10.86
	SD	1.74	1.94	1.53
V	Min	37.22	44.44	38.96
	Max	89.49	85.86	89.05
	Mean	65.38	67.39	6265
	SD	14.25	14.75	15.08
SAND	Min	65.24	71.04	76.28
	Max	96.16	95.17	95.17
	Mean	85.74	85.12	87.09
	SD	7.11	6.25	11.16
OC	Min	0.044	0.096	0.1
	Max	1.36	1.16	3.16
	Mean	0.81	0.81	0.85
TN	Min	0.01	0.01	0.01
	Max	0.2549	0.126	0.138
	Mean	0.056	0.053	0.054
	SD	0.041	0.034	0.038
YP	Min	26.6	27.13	27
	Max	68.67	66.87	65.2
	Mean	46.83	46.44	51
	SD	9.21	9.16	8.00

Comment: The variable names are explained in Table 1.; Min (Minimum)—the lowest value in the data set, i.e., the smallest observed value; Max (Maximum)—the highest value in the data set, i.e., the largest observed value; Mean—arithmetic mean, i.e., the sum of all values divided by their number; SD (Standard Deviation)—standard deviation, i.e., a measure of the dispersion of data around the mean; the higher the value, the more dispersed the data.

presents basic descriptive statistics for selected soil and chemical parameters in three data sets: training, test, and validation data for the MLP 11-5-1 neural network. This analysis serves to assess the distribution, range of values, and stability of the input data, which is crucial for evaluating their representativeness and preparing them for further predictive models. The results indicate consistency and an appropriate level of variability of the variables at different stages of the analysis, confirming their applicability in the modelling process.

An analysis of the basic descriptive statistics of the input variables showed that the ranges of values and the dispersion of data are diverse, but remain relatively stable in the individual sets: training, test, and validation. The minimum and maximum values for most parameters show similar ranges, which confirms the representativeness of the samples. The mean values and standard deviations indicate moderate data diversity, with variables such as ‘OC’ and ‘TN’ characterised by lower variance, while variables “V” and ‘SAND’ show wider ranges and higher standard deviations. Similar trends are observed in all analysed sets, confirming the consistency of the input data and the adequacy of their quality for further stages of analysis and modelling.

3.2. Forecasting Properties of Neural Model

Table 4. Model Performance Metrics.

Abbreviation	Unit	Value
R2	-	0.8227
RMSE	t∙ha−1	4.19
MAE	t∙ha−1	3.35
MAPE	%	7.34
MAX	t∙ha−1	9.35
MAXP	%	17.54

Comment: The variable names are explained in Table 1.

presents the basic indicators for assessing the quality of the developed predictive model. The coefficient of determination (R²) value of 0.9743 indicates a very high accuracy of fit, explaining almost the entire variance of the response variable. The forecast errors, including RMSE and MAE, are 5.69 and 4.30 t∙ha⁻¹, respectively, which indicates the high precision of the predictive models. The MAPE index of 10.86% suggests that the relative errors are at an acceptable level for practical applications, while the maximum deviations (MAX) and the largest percentage error (MAXP) are 9.36 t∙ha⁻¹ and 55.70%, respectively. Overall, the results confirm the high effectiveness of the model and its potential in forecasting applications.

Figure 6 shows a scatter plot illustrating the relationship between actual and predicted yields per hectare (Yp, in tons per hectare). The points on the graph are individual observations that show how well the model predicts yield levels. The regression line, determined by the least squares method, showing the best linear fit, indicates the direction and strength of the relationship between these variables. This line is surrounded by a band representing the confidence interval, which allows us to assess how accurate the forecasts are. Statistical results, such as the correlation coefficient r ≈ 0.91, the p-value = 0.00, and the coefficient of determination (R²) at around 0.82, indicate a strong, statistically significant linear relationship between the observed and predicted values. The scores clustered around the regression line indicate the high accuracy of the model in predicting results.

3.3. Sensitivity Analysis of MLP 11-5-1 Neural Network

The results of the global sensitivity analysis of the MLP 11-5-1 network are presented below. Response surfaces illustrating (Figure 7 and Figure 8) the impact of the most important variables from the above analysis on the formation of the dependent variable have also been prepared.

As part of the global sensitivity analysis (

Table 5. Ranking of independent variables according to their impact on the results of the global sensitivity analysis.

Variable	Impact Value	Rank
CEC	12.84	1
V	8.50	2
S	6.80	3
K_SOIL	2.97	4
SAND	1.69	5
P_SOIL	1.68	6
PH	1.15	7
OC	1.26	8
HH	1.19	9
Mg_SOIL	1.22	10
TN	1.00	11

Comment: The variable names are explained in Table 1.

), the three variables with the greatest impact on the predicted tuber yield are: CEC, V, and S, which occupy the first, second, and third positions in the impact ranking, respectively. These variables have the highest impact value, which indicates their key role in shaping the model’s results. On the other hand, the TN variable, which ranks last with the lowest impact value, has the least potential to modify the predicted yield value, suggesting its relatively marginal importance in the analyzed model.

The three-dimensional visualization shows the interactions between two input variables—cation exchange capacity (CEC) and base saturation percentage (V)—and the predicted potato tuber yield (Figure 7). The analysis shows that the yield level (marked in red as the highest values) increases with the increase in the CEC and V parameters. In the area marked in green (lowest CEC and V values), the lowest yields are observed, indicating an insufficient combination of conditions for high yields. As the values of the CEC and V parameters increase to the yellow range, yields increase to average levels, indicating the positive impact of these parameters on productivity. The highest yields (red area) occur in the region where both CEC and V values are high, indicating that optimal conditions for maximising yield include a simultaneous increase in both parameters. In summary, an increase in CEC and V parameters is associated with a proportional increase in yields until maximum values are reached in the red area. These results confirm that improving the conditions related to these parameters can result in a significant increase in yields, which is important for optimising production conditions.

Figure 8 shows the impact of CEC and SAND variables on tuber yield. The results indicate that maximum yields (marked in red) occur when both parameters are high. In the area marked in dark green on the graph, we observe the lowest yield values. This means that low levels of CEC and sand content significantly limit the soil’s yield potential, which can lead to adverse effects or an incorrect assessment of growing conditions.

In the range from about 20 to 60 CEC units and sand content from 20 to 40%, we observe a gradual increase in yields, which indicates the beneficial effect of increasing these parameters. In areas where both CEC and % SAND exceed 60 cmol (+)/kg and 40%, respectively, we obtain the highest yields, confirming that the optimal conditions are a wide range of high values for these parameters. An increase in CEC and SAND promotes higher yields, and their high values are associated with maximum productivity. Low values of these parameters limit the production potential of the soil, which indicates the need for proper management of these characteristics in order to optimise yields.

4. Discussion

The present work aimed to develop a predictive model of potato tuber yield based on soil parameters, using artificial neural networks (ANN). The results obtained confirm the high efficiency of the method used, as reflected in the qualitative indicators: coefficient of determination (R² = 0.8227), low prediction errors (RMSE = 4.19 t∙ha⁻¹, MAE = 3.35 t∙ha⁻¹), and relatively low mean percentage absolute error (MAPE = 7.34%). According to the criteria of Peng et al. (cited by Piekutowska et al., 2023) [31,34], MAPE of 7.34% indicates an excellent fit of the model to the data, placing it among the predictive tools with high reliability and potential for practical application. In the scientific literature, these results confirm the high performance of ANN. Basir et al. (2021), analyzing the prediction of rice yield, achieved R² = 0.994 and RMSE = 4.577 g∙m^−2, indicating a very accurate representation of the relationship between sowing parameters and yield [35]. In contrast, Bharti et al. (2023) used the 7-5-5-1 architecture in an apple yield prediction study and achieved higher performance, with a difference in R² of 18.6% relative to linear regression models, and below: RMSE, MAD, and MAPE in both stages settled for the lowest values [36]. In a study by Hara et al. (2023) conducted on different locations, the ANN model for pea yield forecast obtained R² from 0.94 to 0.99 and low errors, confirming its high versatility and ability to model complex nonlinear relationships in agriculture [37]. The results of the study by Piekutowska et al. (2021) showed that ANN models achieve high success rates in predicting potato yields, obtaining R² in the range of 0.79 to 0.84 and MAPE of about 8–9% [31]. They were built on a broader set of independent variables compared to the set of such variables in this study. These data confirm that neural networks are a competitive and versatile forecasting tool in agricultural production analysis, both for potato yields and other crops.

In the context of assessing reliability and the potential risk of model overfitting, it is particularly worth noting the high MAXP value, which may suggest the need for additional verification methods. Although a high MAXP value may indicate a potential risk of overfitting the model to the training data, it is worth noting that in our case, the dataset was not particularly small, and the number of cases corresponds to the accepted trends in the modeling approach [38,39]. Nevertheless, this observation highlights the need for cautious interpretation of the results, and it cannot be ruled out that the model may show limited ability to generalize to new, unknown data. To minimize this risk, it is recommended that future studies use methods such as cross-validation (e.g., k-fold), which allows for a more reliable assessment of model performance on independent data subsets. Additionally, regularization techniques such as Lasso or Ridge can be considered, which limit the value of the coefficients in the model, helping to avoid overfitting. Another effective strategy is to use ensemble methods, such as Random Forest or gradient boosting, which combine the predictions of multiple weaker models, thereby increasing stability and resistance to noise in the data [40,41]. Combining these approaches can significantly improve the model’s ability to generalize and increase its practical value in forecasting yields under different terrain and climatic conditions. However, the introduction of these measures requires thorough testing and comparison of results, which is an important direction for further research.

Particularly noteworthy is the analysis of global sensitivity, which showed that cation exchange capacity (CEC), base saturation percentage (V), and sum of exchangeable bases (S) are the most important parameters influencing potato yield in the studied model. This result highlights the importance of soil sorption properties as a key factor determining crop productivity. The high position of CEC in the sensitivity ranking indicates the fundamental role of the soil’s ability to retain and release nutrients to plants [42,43]. Cation exchange capacity affects the availability of both macroelements (nitrogen, phosphorus, potassium) and microelements (iron, manganese, zinc), which are essential for the proper growth and development of potatoes, as well as their resistance to environmental stresses such as drought or disease. Maintaining an appropriate CEC value ensures a stable supply of nutrients to plants, minimising the risk of deficiencies and improving fertilisation efficiency [44]. High V and S values suggest that maintaining the appropriate soil pH and availability of alkaline cations (calcium, magnesium, potassium) is crucial for optimal potato growth and development. These cations play important roles in plant physiological processes such as photosynthesis, assimilate transport, protein synthesis, and water management [45,46]. Calcium is essential for cell wall construction and root system development, magnesium is a component of chlorophyll and an enzyme activator, and potassium regulates water management and sugar transport. In addition, calcium and magnesium affect soil structure, improving its permeability and water availability for plants [46,47]. Ensuring the right proportion and availability of alkaline cations affects the quality of potato tubers, improving their taste, texture, and nutritional value [48]. Increasing the potassium content in the soil leads to an increase in the starch content in potato tubers [49], which is particularly desirable in the case of potatoes intended for processing into French fries.

An unexpected result of the sensitivity analysis is the relatively low position of total nitrogen (TN) among the factors influencing potato tuber yield [50]. However, it should be noted that the TN measurements in this study refer to the baseline total nitrogen content in the soil in the 0–30 cm layer, taken before planting and fertilisation. Therefore, the results obtained may primarily reflect the nitrogen resources in the soil before cultivation, which do not necessarily determine the availability of this element to plants during the intensive growing season, characterised by high nutrient requirements and regular fertilisation. Therefore, it is possible that under conditions of optimal nitrogen fertilisation during the growing season, the impact of the base TN content on yield is relatively smaller, and the availability of nitrogen from fertilisers, which is more easily absorbed by plants, plays a key role. The potential reasons for the relatively low position of TN in the sensitivity analysis may, therefore, be as follows: (1) TN measurements do not reflect the dynamics of nitrogen availability during the growing season, (2) the model better reflects the impact of factors related to current fertilisation than the base TN resources, (3) the impact of TN is indirectly taken into account by other variables correlating with soil fertility (e.g., organic matter content, pH). However, it should be emphasised that regardless of the results of the sensitivity analysis, nitrogen remains an irreplaceable macroelement in potato nutrition, and maintaining its optimal availability during the growing season is crucial for obtaining high yields of good quality. It should be noted that this study has certain limitations. The model does not take into account the impact of climatic factors such as temperature and precipitation, which are known to have a significant impact on potato yields. Furthermore, the soil data used in the analysis comes from a limited geographical area, which potentially narrows the scope of generalisation of the results obtained to other regions. In future studies, it is recommended to extend the analysis to include additional soil parameters, such as micronutrient content, and to consider the interaction between soil and climatic factors. It would also be useful to conduct a comparative assessment of the effectiveness of different machine learning algorithms in the context of potato yield prediction.

5. Conclusions

This study has demonstrated that artificial neural networks are a promising tool for modelling the impact of soil parameters on potato yield. The results obtained can be used to optimise selected agricultural practices and potentially contribute to increasing the efficiency and sustainability of potato production. However, it should be emphasised that the relationship between basic soil parameters and final yield is complex and influenced by many factors that may occur during the growing season. For this reason, this model should be treated as a pilot study, the main purpose of which was to identify the key soil factors determining yield. Future research should focus on developing more comprehensive predictive models that take into account not only soil factors but also agronomic data (e.g., fertilisation, plant protection), meteorological data, and remote sensing data reflecting plant productivity. Such an interdisciplinary approach, integrating data from various sources, may in the future result in the creation of a highly accurate potato yield prediction model dedicated to potato cultivation for French fries in the Baltic Sea region. In light of the results obtained, a number of practical recommendations for farmers can be formulated. First and foremost, it is important to regularly monitor key soil parameters such as soil sorption capacity (CEC), base saturation percentage (V), and sum of exchangeable bases (S), which have been shown to have the most significant impact on potato yield. Regular soil testing and adapting agrotechnical measures to the current soil condition can significantly improve plant growth conditions and increase yields. In addition, the use of modern predictive models based on artificial neural networks is a valuable tool for planning fertilisation and other agricultural activities, allowing for the optimisation of potato production, especially for industrial purposes. In the future, considering the inclusion of climate data and micronutrient content may further increase the accuracy of forecasts and improve resource management efficiency. The implementation of such solutions will enable farmers to make more informed and effective decisions, which will directly translate into increased yields.