Discovering Explicit Magnetic Core Loss Equations via Learnable Symbolic Sparse Identification

Technical Summary: Discovering Explicit Magnetic Core Loss Equations via Learnable Symbolic Sparse Identification

Problem Statement
The continuous push toward higher power densities in power electronics has elevated magnetic component operating frequencies into the megahertz domain, making magnetic core losses a dominant factor in efficiency and thermal limits. Accurately modeling volumetric core loss ($P_v$) across broad spectra of frequency ($f$) and flux density ($B$) remains a fundamental challenge due to highly nonlinear physical mechanisms. Traditional empirical approaches, such as the Steinmetz Equation (SE) and its variants, offer simplicity but suffer from significant prediction errors due to their fixed structures and inability to capture fractional power laws. Conversely, modern machine learning methods (e.g., Random Forests, Neural Networks) achieve high precision but function as "mathematical black boxes," lacking physical interpretability and the analytical insights required for efficient magnetic design. There is a critical need for a method that bridges this gap, providing explicit, physically interpretable equations with the accuracy of data-driven models.

Methodology: Learnable Symbolic Sparse Identification (LSSI)
The paper proposes the Learnable Symbolic Sparse Identification (LSSI) framework, which reformulates magnetic core loss modeling as a symbolic regression problem derived directly from experimental data. The methodology consists of three core components:

1. Physics-Inspired Candidate Library:
   Instead of relying on a fixed dictionary of integer exponents, LSSI constructs a library of $M=10$ candidate functions based on electromagnetic physics. These include:

   • Hysteresis Loss: Modeled with a dynamically modified exponent to account for nonlinear structural shifts near saturation.
   • Eddy Current and Anomalous Loss: Derived from Maxwell equations and domain wall dynamics, typically exhibiting fractional exponents.
   • Saturation Terms: Unconstrained power-law and exponential terms to map highly nonlinear regimes.
   • Auxiliary Terms: Polynomial cross-coupling and bias terms to isolate measurement offsets and thermal drift.

2. Learnable Parameters:
   A key innovation of LSSI is treating the internal parameters of the candidate functions (specifically the exponents $\alpha$ and $\beta$ in terms like $f^\alpha B^\beta$) as learnable parameters ($\Lambda$) rather than fixed constants.

   • Geometric Interpretation: While conventional sparse identification pins candidates to a discrete integer lattice in the $(\alpha, \beta)$ space, forcing fractional physical laws to be approximated by superpositions of integer terms, LSSI releases these exponents to migrate continuously within an admissible domain. This allows a single active term to precisely match the true fractional exponent (e.g., $\alpha=1.20, \beta=2.68$) rather than requiring multiple terms to approximate it.

3. Joint Optimization and Sparsity:
   The framework employs a hybrid Log-Mean Absolute Percentage Error (Log-MAPE) loss function to ensure uniform percentage accuracy across the wide dynamic range of core losses.

   • Optimization: The sparse scaling coefficients ($\Xi$) and the learnable parameters ($\Lambda$) are jointly optimized using the AdamW algorithm.
   • Sparsity Enforcement: A decoupled weight decay is applied exclusively to the coefficients $\Xi$ to promote sparsity (driving most coefficients to zero), while a threshold pruning mechanism removes uninformative terms. This ensures the final model is compact and physically interpretable.

Key Contributions

1. Reformulation of Core Loss Modeling: The paper reframes magnetic core loss modeling as an ML-based function identification problem that retains the analytical spirit of the classic Steinmetz Equation.
2. Sparse Identification Framework: It introduces a framework that automatically discovers dominant candidate functions, achieving both sparsity and physical interpretability.
3. Learnable Parameters for Fractional Laws: By introducing learnable parameters into the candidate functions, the method aligns with the underlying physics of fractional power laws, significantly enhancing accuracy without sacrificing simplicity.

Experimental Results
The LSSI framework was validated using high-fidelity core loss datasets from Fair-Rite 95 ferrite cores, collected via automated parallel-resonant methods. The dataset included over 1,000 data points across various frequencies and flux densities at 25°C.

• Accuracy: LSSI achieved a state-of-the-art $R^2$ of 0.9999 and a Mean Absolute Percentage Error (MAPE) of 1.04% on the test set.
• Compactness: The discovered model condensed the underlying physics into an explicit equation containing only 4 active terms.
• Parameter Efficiency: The method utilized only 15 parameters, a drastic reduction compared to Neural Network methods (4,417 parameters) and Random Forests (199,555 parameters). As detailed in Table 2, these 15 parameters comprise a mix of scaling coefficients (e.g., $\xi^*_1, \xi^*_2$), fractional exponents (e.g., $\alpha^*_h, \beta^*_h$), and auxiliary physical parameters (e.g., cutoff frequencies $f^*_{c1}, f^*_{c2}$ and roll-off powers $p^*_{c1}, p^*_{c2}$).
• Comparison: LSSI outperformed the traditional Steinmetz Equation, Random Forest, Feedforward Neural Networks, and a standard Sparse Identification (SSI) framework (which lacks learnable parameters) in both accuracy and model compactness.

Significance and Claims
The paper claims that LSSI provides a physically transparent and highly accurate solution suitable for complex modern magnetic characterization and design. By successfully balancing the expressiveness of machine learning with the interpretability of physical laws, LSSI offers a design-friendly alternative to black-box models. The authors emphasize that the framework's ability to discover explicit equations with fractional exponents directly from data allows for superior accuracy while maintaining the speed and intuition required for practical engineering applications. The work suggests that future extensions could address temperature-dependent and non-sinusoidal excitation conditions.