Temperature-Compensated Multi-Objective Framework for Core Loss Prediction and Optimization: Integrating Data-Driven Modeling and Evolutionary Strategies

Abstract

Magnetic components serve as critical energy conversion elements in power conversion systems, with their performance directly determining overall system efficiency and long-term operational reliability. The development of accurate core loss frameworks and multi-objective optimization strategies has emerged as a pivotal technical bottleneck in power electronics research. This study develops an integrated framework combining physics-informed modeling and multi-objective optimization. Key findings include the following: (1) a square-root temperature correction model (exponent = 0.5) derived via nonlinear least squares outperforms six alternatives for Steinmetz equation enhancement; (2) a hybrid Bi-LSTM-Bayes-ISE model achieves industry-leading predictive accuracy (R² = 96.22%) through Bayesian hyperparameter optimization; and (3) coupled with NSGA-II, the framework optimizes core loss minimization and magnetic energy transmission, yielding Pareto-optimal solutions. Eight decision-making strategies are compared to refine trade-offs, while a crow search algorithm (CSA) improves NSGA-II’s initial population diversity. UFM, as the optimal decision strategy, achieves minimal core loss (659,555 W/m³) and maximal energy transmission (41,201.9 T·Hz) under 90 °C, 489.7 kHz, and 0.0841 T conditions. Experimental results validate the approach’s superiority in balancing performance and multi-objective efficiency under thermal variations.

1. Introduction

Amidst the rapid progression of power electronics toward higher frequencies and power densities, magnetic components serve as critical energy conversion elements in power conversion systems, with their performance directly determining overall system efficiency and long-term operational reliability [1,2,3]. The proliferation of wide-bandgap semiconductor devices has pushed switching frequencies into the MHz regime [4], imposing significant challenges in thermal management [5], electromagnetic interference (EMI) mitigation [6,7,8], and efficiency optimization [9,10]. Against this backdrop, the development of accurate core loss modeling frameworks and multi-objective optimization strategies has emerged as a pivotal technical bottleneck in power electronics research.

Power losses in magnetic components comprise two primary components: winding losses and core losses [11,12,13]. Winding losses, originating from Joule heating in conductive materials, can be quantitatively assessed through finite element analysis (FEA) incorporating high-frequency skin effect corrections [14,15]. In contrast, core loss mechanisms exhibit greater complexity, arising from irreversible energy dissipation caused by magnetic domain wall motion, eddy currents, and residual losses under alternating magnetic fields [16,17,18]. According to Bertotti’s seminal theory, total core loss can be decomposed into three distinct contributions: hysteresis loss (linearly proportional to magnetization frequency), eddy current loss (quadratic dependency on frequency), and residual loss (encompassing relaxation phenomena and other complex mechanisms) [13,19,20]. This intrinsic nonlinearity positions core loss as a critical limiting factor for system energy efficiency: (1) Joule heating from losses accelerates material degradation and induces thermal runaway risks [21]. (2) Nonlinear loss characteristics may deteriorate EMI spectral distributions, compromising electromagnetic compatibility (EMC) [6]. Consequently, the establishment of high-precision core loss prediction models holds strategic importance for achieving the tripartite objectives of “high efficiency—high density—high reliability”.

The existing representative core loss analysis methods in recent years are shown in

Table 1. Summary of existing approaches for core loss analysis.

Approach Category	References	Variables Considered	Applicable Scenarios	Limitations
Traditional Physical Models	[22,23]	Material microstructures, electromagnetic parameters (permeability, coercivity)	Fundamental loss mechanism analysis	Neglects coupling effects between multiple factors
Classical Empirical Models	[31,32]	Frequency, flux density amplitude	Sinusoidal excitation, isothermal conditions	Poor accuracy under non-sinusoidal waveforms or thermally dynamic environments
Data-Driven Methods	[33]	Operational parameters (frequency, temperature)	PV power forecasting	Requires large datasets; limited interpretability
	[34]	Waveform temporal dependencies	Small-sample loss prediction	Computational complexity for high-dimensional data
	[36]	Multi-objective loss-heat coupling	Core loss optimization	GAN-based methods may suffer from instability in training
	[38]	Multi-objective loss	Kolmogorov-Arnold (FS-KAN) network	Time-consuming; Low modeling efficiency
	[35]	High-frequency HSV driving strategies	Energy loss minimization in electromagnetic systems	Focus on single-objective optimization; lacks temperature-awareness

. The physical mechanisms governing core losses involve profound coupling between material microstructures and macroscopic electromagnetic parameters [22,23]. Empirical studies reveal that loss characteristics depend not only on intrinsic material properties such as permeability and coercivity but also exhibit complex nonlinear relationships with operational parameters including frequency [24], flux density amplitude [25], ambient temperature [26,27,28], and excitation waveforms [29,30]. While classical models like the Steinmetz Equation provide a theoretical foundation for loss calculation, their assumptions of sinusoidal excitation and isothermal conditions limit practical applicability, particularly under non-sinusoidal or thermally varying conditions [31,32]. Recent data-driven approaches demonstrate superior adaptability. Oumiguil and Nejmi [33] employed extreme gradient boosting (XGBoost) to daily PV power forecasting, while Liu and Liang [34] implemented a CNN-Bi-LSTM architecture for small-sample loss prediction. Yu et al. [35] explored the energy loss and heat generated by high-speed solenoid valves (HSV) under high-frequency operating conditions and utilized NSGA-II to provide efficient HSV driving strategy optimization. Shen et al. [36] combined the enhancement method of GAN with NSGA-II to calculate the multi-objective optimization of core loss. Core loss prediction and optimization is an issue that is receiving increasing attention [30,37,38].

However, there are still the following three challenges at present: (1) The traditional core loss model (Stanmetz equation) has relatively limited applicability, as it is primarily derived for sinusoidal excitations and temperature-invariant scenarios, resulting in significant errors in practical engineering applications when applied to non-sinusoidal waveforms or thermally dynamic conditions. (2) Traditional empirical models are limited in accuracy when dealing with complex waveforms (such as other triangular and trapezoidal waves), nonlinear temperature rise effects (abnormal changes in core losses at different temperatures), and the coupling effects of multiple factors (magnetic energy, temperature, waveforms, etc.). These limitations necessitate data-driven deep learning approaches to enhance predictive capability. (3) In engineering applications, the design of magnetic cores should not only focus on minimizing losses but also take into account the efficiency of magnetic energy transmission simultaneously to ensure multi-objective optimization.

This study proposes a multi-dimensional core loss analysis framework. First, a temperature-compensated Steinmetz Equation is developed by introducing six nonlinear correction strategies (linear, exponential, logarithmic, quadratic, square root, and multiplicative models), with comparative analysis of prediction accuracy under variable thermal conditions. Second, a Bayesian-optimized BiLSTM (BiLSTM-Bayes) deep learning model is constructed to capture temporal dependencies in excitation waveforms, demonstrating superior performance in modeling nonlinear loss behaviors. Finally, a multi-objective optimization paradigm is established combining the following: (1) a BiLSTM-Bayes-ISE core loss predictor as the objective function; and (2) magnetic energy transfer efficiency as an additional optimization target. The NSGA-II is employed to generate Pareto front, with crow search algorithm (CSA)-enhanced initial population initialization improving convergence characteristics. Through comparative analysis of eight decision-making strategies (weighted sum, utility function, etc.), optimal operational conditions achieving simultaneous core loss minimization and transmitted magnetic energy maximization are identified.

2. Methodology

2.1. Classical Core Loss Equation (SE) and Improved Strategy (ISE)

2.1.1. Classical Core Loss Equation (SE)

The Steinmetz Equation (SE), as a classic core loss prediction model, is widely applied in fields such as power electronic transformers and magnetic components [39]. SE is expressed as follows:

$$P=k\cdot f^{\alpha }\cdot B_m^{\beta }$$

(1)

where P is the core loss; B_m is the peak (maximum) of magnetic flux density; f is the frequency; and k, α, and β are the coefficients related to the material properties fitted from experimental data. The effect evaluation of coefficient fitting adopts the following five indicators.

(1)

Max Error

Max Error reflects the prediction error of the model in the worst case. The smaller the Max Error, the better the prediction of the model.

$$MaxError=\max \left(\left|True-Pre\right|\right)$$

(2)

(2)

Mean squared error (MSE)

MSE is the average of the squared errors and is highly sensitive to large errors and outliers. The smaller the MSE, the closer the predicted value is to the true value.

$$MSE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(True-Pre\right)}^2}$$

(3)

(3)

Root mean squared error (RMSE)

RMSE is the square root of the mean squared error and can visually reflect the prediction error. The smaller the RMSE, the higher the prediction accuracy of the model.

$$RMSE={\displaystyle \frac{1}{n}}\sqrt{{\displaystyle \sum _{i=1}^n{\left(True-Pre\right)}^2}}$$

(4)

(4)

Mean Absolute Error (MAE)

MAE is the average value of the absolute values of errors, reflecting the average degree of error. The smaller the MAE, the smaller the average prediction error of the model.

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|True-Pre\right|}$$

(5)

(5)

Coefficient of determination (R2)

R² reflects the degree of fit between the predicted value and the true value, with a value range of [0, 1]. The closer R² is to 1, the higher the accuracy of the model.

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(True-Pre\right)}^2}}{{\displaystyle \sum _{i=1}^n{\left(True-Ave\right)}^2}}}$$

(6)

2.1.2. Improved Core Loss Equation (ISE)

Recognizing the limitations of the classical Steinmetz Equation (SE) under non-sinusoidal excitations and thermally varying conditions, this study proposes a systematic temperature-compensated modeling approach. The original SE exhibits significant prediction errors under temperature deviations from the reference condition (T₀ = 25 °C). To address this, six temperature correction strategies were formulated by introducing a thermal modification factor Φ(T) into the original SE framework as follows:

$$P=k\cdot f^{\alpha }\cdot B_m^{\beta }\cdot \Phi \left(T\right)$$

(7)

We selected six different improved equations, hoping to compare their accuracy to determine the final improvement method. The functions were chosen to represent plausible temperature-loss relationships observed in magnetic materials (e.g., linear thermal expansion at low temperatures, nonlinear saturation effects at elevated temperatures). The set includes linear, exponential, logarithmic, polynomial, radical, and multiplicative forms to comprehensively capture potential thermal dependencies. All functions contain a minimal number of parameters (k, α, β, C) to avoid overfitting while retaining sufficient flexibility for curve fitting. Mathematical formulations are detailed below.

(1)

Linear Correction:

$$\Phi \left(T\right)=1+C\cdot (T-T_0)$$

(8)

(2)

Exponential Correction:

$$\Phi \left(T\right)=e^{C\cdot \left(T-T_0\right)}$$

(9)

(3)

Logarithmic Correction:

$$\Phi \left(T\right)=1+C\cdot \log (1+\left|T-T_0\right|)$$

(10)

(4)

Quadratic Correction:

$$\Phi \left(T\right)=1+C\cdot {(T-T_0)}^2$$

(11)

(5)

Square Root Correction:

$$\Phi \left(T\right)=1+C\cdot \sqrt{\left|T-T_0\right|}$$

(12)

(6)

Multiplicative Correction:

$$\Phi \left(T\right)={\left(1+T-T_0\right)}^C$$

(13)

where T is the current working temperature, and T₀ is the normal laboratory temperature (25 °C); k, α, β and C are the coefficients related to the material properties fitted from experimental data.

2.2. Core Loss Prediction Model Based on Bi-LSTM-Bayes-ISE

2.2.1. Bi-LSTM Method

The accuracy of traditional empirical models is limited when dealing with complex waveforms, nonlinear temperature rise effects, and the coupling of multiple factors. Therefore, it is necessary to rely on data-driven deep learning methods to enhance the predictive ability. To select the optimal model, we conducted a comparative study of three machine learning models (SVR, Decision Tree, Linear Regression) and two deep learning models (LSTM and GRU). To this end, this study proposes a core loss prediction framework based on Bi-LSTM (Figure 1). Results showed that Bi-LSTM outperformed other models in capturing bidirectional contextual information (forward/backward waveform history) and handling multi-factor interactions. This model can effectively capture the nonlinear influence of historical information of the excitation waveform on the current loss.

2.2.2. Bayesian Optimization Algorithm

Bayesian Optimization employs Gaussian Processes to iteratively refine prior distributions by observing objective function outputs at sampled input points [40]. This method demonstrates superior computational efficiency, achieving rapid convergence with significantly fewer iterations compared to traditional approaches. As illustrated in Figure 2, the framework jointly optimizes multiple hyperparameters in a model-driven paradigm, capturing synergistic interactions through advanced acquisition strategies. By integrating techniques such as Tree-structured Parzen Estimator (TPE), Adaptive TPE (ATPE), and Gaussian Process (GP), the algorithm systematically identifies optimal hyperparameters for core loss prediction tasks.

2.2.3. Bi-LSTM-Bayes-ISE Framework

The Bi-LSTM-Bayes-ISE framework constitutes an integrated predictive methodology for core loss estimation. The workflow commences with data preprocessing encompassing temperature, excitation waveform, material type, and core loss measurements, followed by feature encoding of categorical variables while designating specific core loss per unit volume as the target variable. A data-driven predictive model is initially developed using Bi-LSTM. Subsequently, Bayesian hyperparameter optimization is employed to globally tune critical architectural parameters including network depth, hidden neuron count, and learning rate, resulting in the Bi-LSTM-Bayes configuration. Building upon this, the framework integrates a physics-informed Steinmetz Equation (SE) correction module to refine predictions through domain-specific knowledge, culminating in the final Bi-LSTM-Bayes-ISE model. Quantitative evaluation through comparative analysis of prediction errors before and after optimization demonstrates the framework’s capacity to systematically enhance accuracy while maintaining computational efficiency.

2.3. Core Loss Optimization Model Based on NSGA-II-CSA

2.3.1. NSGA-II

Multi-objective optimization problems (MOOs) are prevalent in engineering design, characterized by conflicting objective functions that preclude resolution through conventional single-objective optimization methods [41]. The NSGA-II represents a classical and computationally efficient evolutionary algorithm framework, widely implemented in complex engineering optimization domains including power electronics, electric machine design, structural optimization, and path planning [42].

Formally, the MOO formulation seeks decision variables x∈χ that simultaneously optimize multiple objective functions f_i(x) under the constraint that no solution pair exhibits mutual dominance. The set of objective functions for NSGA-II is defined as follows:

$$\min \mathrm{F}\left(x\right)={\left[f_1\left(x\right),f_2\left(x\right),\dots ,f_m\left(x\right)\right]}^{\mathrm{T}},x\in \chi$$

(14)

Define two solutions x₁ and x₂ that satisfy the following conditions:

$$\forall i,f_i\left(x_1\right)\le f_i\left(x_2\right)\wedge \exists j,f_j\left(x_1\right)\le f_j\left(x_2\right)$$

(15)

Then it is said that x₁ governs x₂. In engineering applications, the design of magnetic cores should not only focus on minimizing losses but also take into account the efficiency of magnetic energy transmission simultaneously. Therefore, in this paper, a multi-objective optimization model is further constructed based on the Bi-LSTM-Bayes-ISE prediction model to simultaneously optimize the two major objectives of core loss and magnetic energy transmission.

2.3.2. Crow Search Algorithm (CSA)

The CSA is employed to dynamically calibrate NSGA-II’s operational parameters as pragmatic tools for specific applications (addressing hyperparameter-induced variability in multi-objective electromagnetic optimization) [43,44,45]. Rooted in a probabilistic model of crows’ caching behavior, CSA introduces stochastic perturbation mechanisms to balance exploration and exploitation [46]. In this framework, CSA adapts NSGA-II’s population size, crossover probability, and mutation rate based on solution diversity metrics and convergence progress. The algorithm initializes a population of candidate parameter sets, iteratively refining them through the following: (1) global search via Levy flight-inspired jumps to escape local optima, and (2) local refinement through neighborhood-based adjustments. Empirical validation on benchmark functions demonstrates CSA’s capacity to maintain stable convergence trajectories while preserving Pareto front diversity [47,48]. This integration specifically targets the non-convex optimization landscape of core loss minimization, where traditional parameter tuning often struggles to balance computational efficiency and solution quality [45,49].

2.3.3. Pareto Front and Solution Strategies

NSGA-II finds the Pareto Optimal Set, which together constitute the optimal frontier curve or surface in the target space [49,50]. The Pareto solution set is a group of non-dominated solutions in multi-objective optimization, meaning that these solutions cannot improve another objective without sacrificing the performance of one. In fact, choosing an optimal solution (or the most representative one) from the relevant Pareto solution set requires the adoption of certain decision-making methods.

It may not be possible to find a global optimum by only one method. Eight different decision-making methods were adopted to obtain the optimal solution, namely, the Weigiht Sum Method (WSM), Ideal Point Method (IPM), Entropy Weight Method (EWM), Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) Method, Utility Function Method (UFM), Ranking-Based Selection Method (RBSM), Interactive Method (IM), and Hierarchical Optimization Method (HOM). The selection of eight decision-making methods was driven by the need to comprehensively evaluate trade-offs between core loss minimization and magnetic energy transmission efficiency under diverse operational scenarios. These methods were chosen to represent distinct decision-making philosophies. This selection strategy ensures robustness against method-specific limitations while covering all major multi-criteria decision-making (MCDM) paradigms. By incorporating this diversity, we ensure robustness against method-specific biases while capturing variations in engineering priorities (e.g., cost-sensitive vs. performance-critical applications).

(1)

Weight Sum Method (WSM)

$$J={\displaystyle \sum _{i=1}^n{w}_i\cdot f_i\left(x\right)}$$

(16)

where w_i is normalized weights satisfying Σw_i = 1, and f_i is objective function.

(2)

Ideal Point Method (IPM)

$$d_j=\sqrt{{\displaystyle \sum _{i-1}^n{\left(\frac{f_i\left(x_j\right)-f_i^{*}}{f_i^{*}-f_i^{nad}}\right)}^2}}$$

(17)

where f_i^* is the ideal value, f_i^nad is the nadir value, and x_j is Pareto solution.

(3)

Entropy Weight Method (EWM)

$$e_i=-k{\displaystyle \sum _{j=1}^m{p}_{ij}}\ln p_{ij},k>0$$

(18)

$$p_{ij}={\displaystyle \frac{f_{ij}}{{\displaystyle \sum _{j=1}^m{f}_{ij}}}}$$

(19)

where f_ij is the normalized objective value.

(4)

TOPSIS Method

$$C_i={\displaystyle \frac{d_i^{-}}{d_i^{-}+d_i^{+}}}$$

(20)

where d_i⁺ and d_i^- are the distance to idea and anti-ideal solutions, respectively.

(5)

Utility Function Method (UFM)

$$U\left(x\right)={\displaystyle \sum _{i=1}^n{\varphi }_i\left(f_i\left(x\right)\right)},{\varphi }_i\ge 0$$

(21)

where ${\varphi }_i$ is the monotonic utility function reflecting decision-maker preferences.

(6)

Ranking-Based Selection Method (RBSM)

$$R_j={\displaystyle \sum _{i=1}^n\frac{1}{r_{ij}}}$$

(22)

where r_ij is the rank of solution j for objective i (1 = best).

(7)

Interactive Method (IM)

$$x^{\left(k+1\right)}=\arg {\min }_{x\in PF}\left\{{\displaystyle \sum _{i=1}^n{\lambda }_i^{\left(k\right)}{f}_i\left(x\right)}\right\}$$

(23)

where λ_i^(k) are the preference weights updated in iteration k.

(8)

Hierarchical Optimization Method (HOM)

$$\begin{array}{l}{\min }_{x\in PF}{f}_1\left(x\right)\\ \mathrm{s}.\mathrm{t}.f_2\left(x\right)\le {\epsilon }_2,\dots ,f_n\left(x\right)\le {\epsilon }_n\end{array}$$

(24)

where ε_i are the predefined thresholds for lower-priority objectives.

2.4. The Proposed Prediction and Optimization Framework

This study proposes a comprehensive prediction and optimization framework (Figure 3) to explore the optimal trade-off point between minimum loss and maximum energy efficiency, and to guide the selection of magnetic materials and the adjustment of working parameters in different application scenarios. Firstly, the SE equation and the optimization of its improved equation considering temperature were discussed. Then, based on the Bi-LSTM-Bayes-ISE core loss prediction model and combined with the measurement index of transmission magnetic energy, a comprehensive optimization model for core loss prediction optimization based on the NSGA-II-CAS algorithm is established. By analyzing the influence of multiple variables, the optimal conditions are identified.

2.5. Materials and Data Preprocessing

Core loss characterization commonly employs the AC power method, with the experimental setup depicted in Figure 4. The toroidal core (mean magnetic path length l_e, cross-sectional area A_e) is equipped with dual symmetrically wound coils (N₁ = N₂ turns). A signal generator produces sinusoidal/arbitrary waveforms at frequency f (period T = 1/f), amplified by a high-frequency power amplifier before driving the excitation coil.

Governed by Ampère’s Law, the excitation current I(t) generates magnetic field strength H(t) = N₁I(t)/l_e (magnetomotive force per unit length). Concurrently, time-varying B(t) (magnetic flux density) is induced per Faraday’s Law, producing measurable voltage in the sensing coil. Simultaneous acquisition of I(t) and sensing voltage u(t) enables H(t)/B(t) waveform reconstruction for core loss density calculation (Equation (25)), with B(t) characteristics serving as operational state indicators.

$$P={\displaystyle \frac{1}{T}}{\displaystyle {\int }_0^T\frac{u\left(t\right)\cdot I\left(t\right)dt}{A_e\cdot l_e}}={\displaystyle \frac{1}{T}}{\displaystyle {\int }_{B\left(0\right)}^{B\left(T\right)}HdB}$$

(25)

The dataset adopted in this study (

Table 2. Statistical values of the dataset in this study.

Materials	Parameters	Qualitative Data	Quantitative Data
			Minimum	Median	Maximum	Mean	Standard
Material 1	Temperature (°C)	25, 50, 70, and 90
	Waveform	sine, triangle, and trapezoid
	Frequency f (Hz)		50,020	158,500	446,410	174,017.8294	101,221.4147
	Core Loss P (W/m3)		684.0462	44,323.2844	3,616,132.5360	179,886.9442	339,525.6526
	Peak Magnetic Flux Density Bm (T)		0.0108	0.0614	0.2790	0.0831	0.0671
Material 2	Temperature (°C)	25, 50, 70, and 90
	Waveform	sine, triangle, and trapezoid
	Frequency f (Hz)		49,990	158,750	501,180	210,265.51	134,207.3594
	Core Loss P (W/m3)		415.6131	55,545.2364	2,750,045.7730	234,317.1443	409,095.6793
	Peak Magnetic Flux Density Bm (T)		0.0096	0.0615	0.3133	0.0826	0.0722
Material 3	Temperature (°C)	25, 50, 70, and 90
	Waveform	sine, triangle, and trapezoid
	Frequency f (Hz)		49,990	158,750	501,180	212,495.4344	135,724.3408
	Core Loss (W/m3)		739.3341	61,055.7401	3,525,389.2960	264,453.0732	465,459.5626
	Peak Magnetic Flux Density Bm (T)		0.0097	0.0614	0.3133	0.0830	0.0733
Material 4	Temperature (°C)	25, 50, 70, and 90
	Waveform	sine, triangle, and trapezoid
	Frequency f (Hz)		50,010	125,930	446,690	170,944.9929	110,310.508
	Core Loss P (W/m3)		452.2277	25,284.3844	2,322,456.1470	109,469.2491	213,889.8311
	Peak Magnetic Flux Density Bm (T)		0.0108	0.0393	0.2776	0.0599	0.0541

) includes material, temperature, frequency, core loss, excitation waveform, and magnetic flux density time series of 1024 sampling points in each group. The dimensions of the four different material datasets are 3400 × 1028, 3000 × 1028, 3200 × 1028, and 2800 × 1028 respectively, among which the materials (four types) and excitation waveforms (sine, triangle, and trapezoid) are label data. Source of data set for the 21st session of Chinese graduate students mathematical contest in modeling problem C public data sets (https://cpipc.acge.org.cn/cw/detail/4/2c90801791c6c0a80191f9a6b0366533, accessed on 21 September 2024). The original source of this dataset is the MagNet open-source database [36,51]. The database was jointly developed and established by Princeton University and Dartmouth College. The excitation waveforms have been classified according to the waveform characteristics of the magnetic flux density time series. Considering the transparency of the data, all data can be directly obtained from the official links. In this study, only the statistical data characteristics of all four different material types are presented as shown in Table 2. While the dataset provides comprehensive multi-material coverage, two limitations warrant attention: The excitation waveform classification relies on predefined flux density characteristics, which may not capture all real-world variations.

The collection of this dataset was carried out using the double-line method (Figure 4), and the quality of the dataset was strictly controlled by the official and authoritative organizing committee of the China Postgraduate Mathematical Modeling Contest. At the same time, considering the occurrence of errors, while modeling and predicting, we adopted the 3sigma (3σ) method to handle outliers on the data in the dataset. We excluded a very small number of core loss data samples that did not meet 3σ. According to the calculation of sigma based on Equation (26), it can be known that the probability of the data distribution within the 3sigma interval (μ − 3σ, μ + 3σ) is 99.73%.

$$\begin{array}{l}\mu ={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{x}_i}\\ \sigma =\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{\left(x_i-\mu \right)}^2}}{n}}}\end{array}$$

(26)

where n is the sample quantity, x_i is the core loss, μ is the average core loss of all samples, and σ is the standard deviation of the core loss of all samples.

3. Results and Discussion

3.1. Core Loss Coefficient Fitting and Temperature Correction Equation

3.1.1. Coefficient Fitting of the SE Equation

In order to study the temperature correction scheme of the SE equation under the condition that the magnetic waveform is sinusoidal, the coefficients are fitted first. By analyzing the existing experimental data, without considering the influence of temperature, k, α, and β of the SE equation are fitted using f and B_m. The fitting results are shown in Figure 5.

The four fitting methods adopted are linear fitting, nonlinear least square method, annealing algorithm, and genetic algorithm. Among them, the linear fitting method is simple and has a small computational load, but it has poor adaptability to nonlinear relationships. The nonlinear least square method is suitable for complex nonlinear relationships and performs parameter fitting by minimizing the error square. Annealing algorithms and genetic algorithms are applicable to multimodal optimization problems, which can avoid local optima, but they have relatively high computational complexity. For the four proposed methods, f and B_m are used as inputs to fit the core loss and find the optimal coefficient. In Figure 5, the fitting results of the four methods are relatively close. The predictions for the first 500 or so data points are all smaller, while the predicted values after 500 are larger than the true values. The core losses obtained after fitting by four methods were evaluated and the errors were analyzed. The error results of core loss calculated by four methods are shown in

Table 3. Performance of coefficient fitting for sine wave SE equation (unit: W/m³).

Fitting Method	MaxError	MSE	RMSE	MAE	R2
Linear fitting	283,722.71	1,791,962,523.43	42,331.57	19,692.29	0.9396
Nonlinear least square method	240,008.01	1,616,232,322.95	40,202.39	20,464.07	0.9455
Annealing algorithm	305,695.37	2,440,533,358.32	49,401.75	25,342.42	0.9177
Genetic algorithm	243,344.87	1,617,941,105.55	40,223.63	20,651.16	0.9454

.

The MaxError, MSE, RMSE, and MAE of the nonlinear least squares method are all the smallest (Table 3), while it has the largest R² (0.9455). The calculation results of the annealing algorithm are similar to those of the nonlinear least square method, but the local errors are relatively large. Therefore, among the four methods, the nonlinear least square method has the best fitting effect. The nonlinear least squares fitting process adopts the Levenberg–Marquardt algorithm and the Gauss–Newton method and has better convergence performance for nonlinear problems.

3.1.2. ISE Equation and Verification

Based on the nonlinear least squares method for optimal coefficient fitting of the core loss model, the accuracy of core loss prediction is further improved. Different temperature correction forms are adopted, and the temperature factor is taken into account in the SE equation to enhance the calculation accuracy of core loss. Taking the sinusoidal waveform data in Material 1 as the analysis object, the modified equation considering temperature constructed was applied to the same data set to calculate the modified predicted value of core loss. Compare the errors between the SE and ISE, and evaluate the improvement effect of the modified equation in predicted value. According to the six modified equations proposed in Section 2.1.2, the equation coefficients were re-fitted, and the fitting results are shown in Figure 6.

The results of core loss calculated by various correction methods after introducing temperature as a variable for correction are all closer to the true value (Figure 6). Among them, the overall visual effect of square root correction and multiplication correction is better. Error analysis was conducted on six correction methods to screen out the one with the best correction effect.

Table 4. Core loss calculation performance of different correction methods (unit: W/m³).

Correction Methods	MaxError	MSE	RMSE	MAE	R2
Linear	106,838.69	264,281,458.96	16,256.73	9564.32	0.9911
Exponential	100,395.71	203,849,338.53	14,277.58	8331.75	0.9931
Logarithmic	91,338.985	166,977,784.68	12,921.98	7249.31	0.9944
Quadratic	127,806.45	607,657,689.76	24,650.71	14,154.62	0.9795
Square Root	85,248.85	136,153,072.040	11,668.46	6776.91	0.9954
Multiplicative	83,406.21	185,980,517.43	13,637.46	7629.69	0.9937

presents the performance analysis results of core loss calculation for different correction methods.

The error generated by linear correction is much higher than that of the other five methods, and the effect is average (Table 4). The errors of exponential correction and logarithmic correction are relatively low, and the R² values are high, with better effects. The square correction error is relatively large, and the R² value is also relatively low, making it the method with the worst correction effect among all methods. Among all the square root correction error indicators, it is the lowest, and the R² value (0.9954) is the highest, which is the best effect among all the methods. The effect of multiplication correction is similar to that of square root correction. The R² value (0.9937) is slightly lower than that of the square root correction method, and its effect is second only to that of the square root correction. To analyze the effects of the six correction methods more intuitively, the results were visualized using kernel density plots. The kernel density plots of the six methods are shown in Figure 7.

It can be seen from Figure 7 that the fitting accuracy of the square correction method is the worst, and the data has shown discrete bifurcations. The core loss calculated by the square root correction method has a higher fitting accuracy with the real core loss curve, which proves that the core loss temperature correction model using the square root correction method has the best effect. The square root obtained so far is 1/2 of the temperature, which is not necessarily the optimal parameter. Therefore, the correction effects of both linear correction and quadratic correction methods are weaker than those of the square root correction method.

Based on the proposed core loss temperature correction model using the square root correction method, the exponential power of the temperature is further optimized. Let the exponential power of the temperature be the coefficient γ (ranging from 0 to 1) for parameter optimization (Figure 8 and

Table 5. Performance of the square root correction optimization algorithm (unit: W/m³).

Algorithm Type	MaxError	MSE	RMSE	MAE	R2
Optimization	86,582.60	134,966,429.94	11,617.50	6736.64	0.9954
Non-optimized	85,248.85	136,153,072.040	11,668.46	6776.91	0.9954

).

It can be seen from the result analysis in Table 5 that optimizing the value of the exponential power can slightly improve the model accuracy, but the optimized equation is not conducive to calculation (with an exponent of 0.46465). Considering the actual situation, we need to reduce the computational complexity. The square root (with an exponent of 0.5) correction model is still adopted. The core loss after fitting the SE equation by the nonlinear least square method, the core loss after fitting the SE equation by square root correction considering the temperature factor (ISE), and the actual core loss value are compared as shown in Figure 9. When the temperature factor is not considered, the core loss fitted by the nonlinear least square method still deviates significantly from the true value. After temperature correction by the SE equation, the calculated value of the core loss is significantly closer to the true value, which proves the correctness of the core loss temperature correction model adopted. It can adapt to different temperature changes and make the prediction effect of core loss better.

3.2. Core Loss Prediction Based on Bi-LSTM-Bayes-ISE

3.2.1. Bi-LSTM

The establishment of a core loss prediction model based on Bi-LSTM deep learning is a process involving the processing of complex data sequences. Core loss is closely related to factors such as material type, temperature, and excitation waveform, and these data usually exhibit significant time series characteristics. Therefore, Bi-LSTM is highly suitable for predicting core losses. Figure 10a presents the comparison results between the predicted values and the true values of core loss by the Bi-LSTM deep learning algorithm. Figure 10b shows the statistics of the error values of core loss. Figure 10c presents the visualization processing results of the kernel density of the predicted values.

From the comparison results of the true values and predicted values of the core loss prediction model in Figure 10a, it can be seen that there is a certain error especially for some data points between 1500 and 2500. The error values is relatively large. Meanwhile, according to the kernel density graph of the predicted core loss values, it can be seen that the data points are relatively discrete and the fitting accuracy is poor. Figure 11 presents the analysis results of core loss performance and error based on Bi-LSTM and three basic machine learning models. The prediction error of Bi-LSTM is smaller than that of Linear, SVR, and Decision Tree (Figure 11a). It can be seen that the values of RMSE, MSE, MAE, and SMAPE are relatively large. Although Bi-LSTM performs better than other basic models, there is still room for improvement, which inspires us to expand further optimizations (such as Section 3.2.2 and Section 3.2.3).

Furthermore, we also conducted a comparative analysis of the Bi-LSTM model with other classic deep learning models (LSTM, GRU), as shown in Figure 12. Bi-LSTM is closest to the 1:1 error line in both Figure 11b and Figure 12b, indicating that the error gap is the smallest. The performance results of all the basic models are presented in

Table 6. Core loss prediction performance based on four basic models.

Model	RMSE	MSE	MAE	MAPE	SMAPE	R2
Bi-LSTM	70,129.68	4.92 × 109	44,975.22	459.14	87.81	0.9023
LSTM	85,106	7.24 × 109	57,879	553.37	83.223	0.8562
GRU	76,115	5.79 × 109	40,982	315.93	72.656	0.8850
SVR	2.55 × 105	6.53 × 1010	2.38 × 105	3536.6	137.05	−0.2955
Decision Tree	72,423	5.25 × 109	53,946	488.3	42.45	0.8859
Linear	1.45 × 105	2.11 × 1010	1.13 × 105	1885.9	121.12	0.5803

. Based on the comprehensive performance results, Bi-LSTM has the largest R² and the best performance, and can be considered as the basic model for core loss prediction.

3.2.2. Bi-LSTM-Bayes

Because the predicted values obtained by the core loss prediction model based on Bi-LSTM still have a certain error compared with the true values, the Bayesian method is adopted to optimize the model. Bayesian hyperparameter optimization aims to automatically adjust the hyperparameters of the model to improve its performance.

Table 7. Parameter range and optimal hyperparameters for Bayesian hyperparameter optimization.

Hyperparameter	Variable Name	Range	Optimal Hyperparameters
Hidden units	NumHiddenUnits	[20, 100]	50
Learning rate	LearnRate	[1 × 10−4, 1 × 10−2]	0.009965
Maximum training number	MaxEpochs	[50, 150]	56
Batch size	MiniBatchSize	[16, 128]	28

presents the parameter range and the optimal hyperparameters for Bayesian hyperparameter optimization, including the number of hidden units, learning rate, maximum training number, and batch size.

Based on the optimal hyperparameters obtained from the optimization of Bayes hyperparameters, the Bi-LSTM-Bayes core loss prediction model is constructed below and compared with the real values. Figure 13 presents the comparison result between the predicted value and the true value of core loss. The accuracy of the prediction model corrected with the optimal hyperparameters has been significantly improved (compared with Figure 10). In particular, the error values of some data points from 1500 to 2500 have been significantly reduced.

Table 8. Core loss prediction performance based on Bayes.

Model	RMSE	MSE	MAE	MAPE	SMAPE	R2
Bi-LSTM	46,653.23	2.18 × 109	26,622.94	165.04	63.79	0.9568
LSTM	66,107.28	4.37 × 109	37,681.16	193.03	74.98	0.9134
GRU	62,423.30	3.89 × 109	35,581.31	181.26	70.11	0.9228
SVR	120,707.60	1.46 × 1010	44,131.70	348.29	113.56	0.7112
Decision Tree	70,850.72	5.02 × 109	40,384.90	201.02	73.54	0.9005
Linear	104,403.05	1.09 × 1010	68,803.32	304.54	101.22	0.7840

presents the error analysis results based on the Bi-LSTM-Bayes core loss prediction model. Compared with Table 6, the RMSE, MSE, MAE, and SMAPE values have generally decreased, and the R² value of the Bi-LSTM-Bayes model has also increased (reaching 0.9568, an increase of approximately 6%). It can be seen that after Bayesian hyperparameter optimization, the accuracy of the core loss model has been significantly improved. Compared with LSTM, GRU, SVR, Decision Tree, and Linear, Bi-LSTM shows relatively better performance whether it is optimized by Bayes or not.

3.2.3. Bi-LSTM-Bayes-ISE

Although the accuracy of the core loss model based on Bi-LSTM has been improved after Bayesian hyperparameter optimization, the physical model based on the aforementioned SE modified equation also has high prediction accuracy and a high-precision function model. Therefore, based on the Bi-LSTM-Bayes core loss prediction model, the physical model of the ISE is considered as an input feature supplement to optimize the prediction model. Figure 14 presents the results of the Bi-LSTM-Bayes-ISE core loss prediction model, the statistics of the error values, and the visualization results of the kernel density.

The accuracy of the prediction model optimized by the ISE physical model has been significantly improved, and the error values have been significantly reduced (Figure 14).

Table 9. Core loss prediction performance based on Bi-LSTM-Bayes-ISE.

RMSE	MSE	MAE	MAPE	SMAPE	R2
43,615.13	1.90 × 109	26,296.84	148.58	59.30	0.9622

presents the performance analysis results based on the Bi-LSTM-Bayes-ISE core loss prediction model. Compared with Table 8, the values of RMSE, MSE, MAE, and SMAPE were further reduced, and the R² value was also increased (reaching 0.9622). It can be seen that after considering the ISE physical model, the prediction accuracy of the core loss model has been further improved.

The predicted values of the core loss prediction models of Bi-LSTM, Bi-LSTM-Bayes, and Bi-LSTM-Bayes-ISE were compared with the true values, and the results are shown in Figure 15. Before the Bayesian hyperparameter optimization, the predicted core loss based on Bi-LSTM deep learning still deviated greatly from the true value. After the hyperparameter optimization, the calculated value of the core loss was significantly closer to the true value, which proved the correctness of the hyperparameter optimization. After introducing the ISE modified equation, the accuracy of the prediction model was further improved, making the best predicted results.

3.3. Core Loss Prediction Optimization Based on NSGA-II-CSA

Multiple objectives to be solved need to be transformed into solvable extremum objective functions. Firstly, the objective functions regarding P and E are defined as follows:

$$\mathrm{Min} P = g\left(T,f,W,B_m,M\right)$$

(27)

$$\mathrm{Max} E=f\times B_m$$

(28)

where P is the core loss; g(·) is a function of P; E is the transmitted magnetic energy, and T, f, M, W, and B_m are, respectively, the temperature, frequency, material type, excitation waveform type, and peak magnetic flux density. Meanwhile, these decision variables need to meet the following conditions:

$$\begin{array}{l}\mathrm{S}.\mathrm{t} \\ T=\left\{25,50,70,90\right\}\\ f\in \left[49990,501180\right]\\ W=\left\{sine,triangle,trapezoid\right\}=\left\{1,2,3\right\}\\ M=\left\{Material1,Material2,Material3,Material4\right\}=\left\{1,2,3,4\right\}\\ B_m\in \left[0.0096,0.3133\right]\end{array}$$

(29)

The NSGA-II algorithm randomly generates a series of initial populations of a certain scale based on the search space and constraints of the problem. This population contains the starting point of algorithm optimization, and each individual represents a potential solution. Furthermore, non-dominant sorting is carried out based on the dominance relationship to divide multiple Pareto front. Then, a series of crowding degree calculations (such as the sum of total distances, see Equation (30)), selection of crowding comparison operators, crossover and mutation (such as individual c mutation, see Equation (31)), population mutation, and update are carried out until the maximum population convergence or the maximum algebra G is reached.

$$\begin{array}{l}{d}_i={\displaystyle \sum _{k=1}^m{d}_i^k=}{\displaystyle \sum _{k=1}^m\frac{f_k^{i+1}-f_k^{i-1}}{f_k^{\max }-f_k^{\min }}}\\ d_1^k=d_N^k=\infty \end{array}$$

(30)

$$\begin{array}{l}{\delta }_m=\left\{\begin{array}{cc}{\left(2u\right)}^{1/\left(n_m+1\right)}-1,& \mathrm{if} u<0.5 \hfill \\ 1-\left(2{\left(1-u\right)}^{1/\left(n_m+1\right)}\right),& \mathrm{if} u\ge 0.5\hfill \end{array}\right.\\ c_i=c_i+{\delta }_m\cdot \left(c_i^{\max }-c_i^{\min }\right)\end{array}$$

(31)

where d_i is the total congestion distance, u is a uniformly distributed random number, and n_m is the mutation release index.

3.3.1. Pareto Front

Based on the NSGA-II algorithm, the Bi-LSTM-Bayes-ISE core loss prediction model was adopted as the core loss function objective, and MOO training was carried out. The parameter selection of related algorithms will affect the number of selected Pareto frontiers and the degree of training. Referring to the current data volume and operating costs, the algorithm parameters are shown in

Table 10. Hyperparameter settings of the NSGA-II algorithm.

Hyperparameters	Value
Number of populations	200
Number of iterations	100
Crossed factors	0.8
Variation factors	0.25

.

Taking into account P and E comprehensively, the Pareto front diagram of the optimization result is shown in Figure 16. It shows 70 Pareto frontier values, represented by a “gradient red” to indicate the frequency distribution histogram of the related core losses, highlighting the negative impact of core losses on magnetic components. Correspondingly, the frequency distribution histogram of the relevant transmitted magnetic energy is represented by “gradient blue” to demonstrate the positive effect of the transmission magnetic energy. The magnetic flux loss distribution corresponding to the value of the optimal solution is uniform, and the transmission magnetic energy shows a normal distribution law.

3.3.2. Optimal Condition Solution Based on Eight Decision-Making Methods

Figure 17 shows the selection of the optimal solution for eight decision-making methods. Among them, WSM, IPM, EWM, TOPSIS method, and HOM have selected the same point under the default setting conditions, where both the core loss and the transmitted magnetic energy are at the lowest values. This is obviously not the most suitable optimal solution.

Taking into account the scattered point distribution in Figure 17 comprehensively, the optimal solution selection of UFM, RBSM, and IM would be more appropriate. The optimal solutions and influencing factors for all eight types of decision-making methods are selected (

Table 11. Optimal solutions based on different decision-making methods.

Methods	Temperature (°C)	Frequency (Hz)	Materials	Waveform	Peak Magnetic Flux Density (T)	Core Loss (W/m3)	Transmit Magnetic Energy (T·Hz)
WSM	70	85,244	2	Sinusoidal	0.03245	22.71	0.00036
IPM	70	85,244	2	Sinusoidal	0.03245	22.71	0.00036
EWM	70	85,244	2	Sinusoidal	0.03245	22.71	0.00036
TOPSIS	70	85,244	2	Sinusoidal	0.03245	22.71	0.00036
UFM	90	476,910	1	Sinusoidal	0.07748	551,221	36,949
RBSM	90	477,686	1	Sinusoidal	0.1083	1,201,030	51,729
IM	90	477,150	1	Sinusoidal	0.15031	2,384,220	71,721.1
HOM	70	85,244	2	Sinusoidal	0.03245	22.71	0.00036

and Figure 18), respectively. The optimal solutions selected by WSM, IPM, EWM, TOPSIS method, and HOM are only theoretically reasonable. Although the core loss (22.71 W/m³) is very low, the actual magnetic energy is only 0.00036 T·Hz. On the contrary, the optimal solutions of UFM, RBSM, and IM are more suitable for the actual requirements of magnetic components. These three optimal solutions are all sinusoidal waves, material 1, and 90 °C, with specific frequencies all around 470,000 Hz. The peak magnetic flux densities are different, and the corresponding core losses and transmitted magnetic energy basically maintain a positive correlation state.

3.3.3. NSGA-II-CSA

Based on the NSGA-II-CSA model, Figure 19 shows the Pareto front results calculated by the improved model. It can be seen from the figure that the relevant scattered points are concentrated. Especially when the core loss is between 10⁶ and 2 × 10⁶ W/m³ and the transmitted magnetic energy is between 4 × 10⁴ and 6 × 10⁴ T·Hz, the number of Pareto frontiers reaches 30.

To further compare the model effects before and after optimization, a Pareto front scatter comparison chart before and after improvement was drawn (as shown in Figure 20). From the comparison chart, it can be seen that the similarity of the relevant optimal condition solution sets is relatively large, and the overall deviation is small. However, we can find that the Pareto front after CSA optimization is more concentrated in the middle of the two objectives, and the number of solutions in the lower left and upper right corners of Figure 20 is sparser. It is impossible to determine the optimal solution merely from the solution set graph. At this time, eight optimization strategies are necessary. The optimal conditions of the NSGA-II-CSA model were solved by using the same eight decision-making methods, and the multi-decision optimal solution selection graph was drawn as shown in Figure 21.

Through these eight types of methods, decisions can be made to select the optimal conditions. To further understand the relevant optimal solution values, we have supplemented

Table 12. Optimal solutions of NSGA-II-CSA based on different decision-making methods.

Methods	Temperature (°C)	Frequency (Hz)	Materials	Waveform	Peak Magnetic Flux Density (T)	Core Loss (W/m3)	Transmit Magnetic Energy (T·Hz)
WSM	70	132,047	1	Sinusoidal	0.06287	731.508	8301.84
IPM	70	132,047	1	Sinusoidal	0.06287	731.508	8301.84
EWM	70	132,047	1	Sinusoidal	0.06287	731.508	8301.84
TOPSIS	70	132,047	1	Sinusoidal	0.06287	731.508	8301.84
UFM	90	489,674	1	Sinusoidal	0.0841	659,555	41,201.9
RBSM	90	491,283	1	Sinusoidal	0.1504	2,447,990	73,888
IM	90	486,189	1	Sinusoidal	0.1445	2,259,440	70,263.4
HOM	70	132,047	1	Sinusoidal	0.06287	731.508	8301.84

and Figure 22 to demonstrate the optimal conditions of the NSGA-II-CSA algorithm. According to the optimal decision conditions in Table 12, the magnetic components selected under 90 °C, material 1, and sine wave conditions can approach the optimum in terms of core loss and transmission magnetic energy.

For the specific selection of the optimal solution, refer to the bolded part in Table 12. The scientific analysis of optimal solution selection based on UFM, RBSM, and IM methods reveals critical trade-offs between core loss minimization and magnetic energy transmission maximization. UFM achieves the lowest core loss (659,555 W/m³) while simultaneously maximizing transmitted magnetic energy (41,201.9 T·Hz). This dual superiority suggests UFM effectively balances the conflicting objectives of loss reduction and energy efficiency.

UFM operates at lower frequency (489,674 Hz) and peak flux density (0.0841 T) compared to RBSM and IM. These conservative parameters likely reduce hysteresis/eddy current losses while maintaining sufficient energy transmission through optimized waveform utilization. RBSM and IM use higher flux densities (0.1445–0.1504 T) and frequencies (486,189–491,283 Hz), which increase energy transmission but at the cost of significantly higher core losses (3.7–3.8× UFM’s loss). UFM’s utility function framework appears to capture nonlinear relationships between operational parameters more effectively than ranking-based (RBSM) or interactive (IM) approaches. This enables simultaneous optimization of multiple objectives without requiring predefined weights or iterative adjustments. The Pareto front analysis demonstrates that UFM provides the optimal trade-off under the given operational constraints (Material 1, sinusoidal waveform, 90 °C). Its superior performance stems from parameter selection that avoids excessive flux/frequency levels while maintaining energy efficiency through mathematical formulation advantages. UFM represents the scientifically optimal solution as it uniquely achieves both lowest core loss and highest magnetic energy transmission among the three methods. This finding validates the utility function approach for multi-objective core loss optimization under industrial conditions.

4. Limitations and Future Work

This study presents a comprehensive methodology for magnetic core loss prediction and multi-objective optimization through interdisciplinary integration. Firstly, we established a physics-informed modeling framework by solving the sinusoidal Steinmetz equation (SE) using four fitting approaches (linear, nonlinear least squares, annealing, and genetic algorithms) and six temperature correction strategies. The nonlinear least squares method proved optimal for deriving ISE with a square-root temperature correction model (exponent = 0.5), effectively capturing thermal-loss coupling. Building on this, we developed a hybrid Bi-LSTM-Bayes-ISE prediction model combining deep learning with physical constraints, achieving exceptional accuracy (R² = 96.22%). Furthermore, we enhanced NSGA-II-CSA to improve initial population diversity, enabling systematic comparison of eight MCDM methods. UFM emerged as the optimal decision strategy, achieving the dual objectives of minimum core loss and maximum energy transmission. This integrated approach demonstrates how combining physics-informed machine learning with advanced multi-objective optimization provides a solution for addressing conflicting engineering demands in magnetic component design.

In our research, BiLSTM was selected as the base model for core prediction, fully considering that other simple machine learning models (Linear, SVR, Decision Tree, LSTM, GRU) were insufficient to achieve high prediction performance. This can provide a reference for other research on magnetic core prediction. The robustness of the model is of vital importance, especially when there is a large volume of data and various types of materials [30]. Although the existing models have achieved good results in predicting and optimizing core losses, it would be more beneficial to continuously explore other more suitable models. For example, Shen et al. [36] combined the enhanced method of GAN and NSGA-II to calculate its multi-objective optimization. Furthermore, how to select more interpretive machine learning methods has always been a direction that researchers are constantly exploring.

Judging from the prediction performance indicators of Bi-LSTM-Bayes-ISE, the overall value of MSE is relatively large. Although it has a better R² prediction performance when viewed on a larger data scale, it should be recognized that MSE is more sensitive to larger errors. In future research, more attention should be paid to controlling the overall quality of the data to avoid the overall impact of outliers on the prediction performance.

In future research, when determining the optimal operating condition with the least core loss and the maximum magnetic energy transmission, it is advisable to consider applying other multi-objective optimization algorithms for comparative studies to enhance the reliability of the results. We hope to introduce methods such as SHAP or PDPs in subsequent research to enhance interpretability. This is a feasible strategy for transitioning the “black box” model to the “white box” model. If more core loss data with stronger noise or other materials are collected in the later research, it is necessary to improve its robustness by means such as k-fold cross-validation.

5. Conclusions

Based on the magnetic loss data of magnetic components, this study conducts the influence of different factors on core loss, constructs a core loss model considering multiple factors, and reaches the following conclusions:

(1)

The fitting coefficients of the sinusoidal waveform SE equation are solved based on four fitting methods (the linear fitting method, nonlinear least squares method, annealing algorithm, and genetic algorithm). In terms of equation correction, six different temperature correction strategies are provided. The best temperature correction equation (ISE) is solved through the optimal fitting method (nonlinear least square method), and finally a square root (with an exponent of 0.5) correction model is adopted.

(2)

A core loss prediction model based on Bi-LSTM was constructed. On this basis, the parameter range and the optimal hyperparameters for Bayesian hyperparameter optimization were given. Furthermore, an improved model with physical equation (Bi-LSTM-Bayes-ISE) was discussed and proposed. The performance R² is 96.22%, with strong robustness and high prediction accuracy.

(3)

Based on NSGA-II to solve the problem of optimization conditions, the search for the optimal solution by eight decision-making methods has been expanded. On this basis, CSA was adopted to improve the initial population, effectively improving the initial solution distribution of NSGA-II. Taking into account various factors, under the conditions of a temperature of 90 °C, a frequency of 489,674 Hz, a sinusoidal wave, a peak magnetic flux density of 0.0841 T, and material 1 by UFM, the minimum core loss (659,555 W/m³) and the maximum transmission magnetic energy (41,201.9 T·Hz) can be achieved. UFM’s superior performance stems from parameter selection that avoids excessive flux/frequency levels while maintaining energy efficiency through mathematical formulation advantages.