Enhanced WT-RDF+ Framework

• The enhanced WT-RDF+ framework integrates physics-based wavelet analysis with machine learning to remediate amplitude inaccuracies in RDF predictions.
• It reduces second-peak amplitude errors dramatically in Ge–Se and Ag–Ge–Se glassy systems, ensuring improved peak localization and coordination number estimates.
• This approach accelerates materials design by enabling precise structural characterization essential for phase-change device and nonvolatile memory applications.

The enhanced WT-RDF⁺ framework integrates physics-based wavelet-transform analysis of radial distribution functions (RDFs) with machine learning–assisted parameter tuning to achieve robust structural characterization of amorphous materials. Its application addresses critical amplitude inaccuracies in WT-RDF predictions, especially for Ge–Se and Ag–Ge–Se glassy systems, and yields substantial improvements in precise peak estimation and coordination number calculation, outperforming benchmark ML regressors and facilitating accelerated materials design in phase-change device applications (Senjaya et al., 19 Dec 2025).

1. Theoretical Foundations of WT-RDF

The radial distribution function (RDF), denoted by $g(r)$, encapsulates the probability density of finding an atom a distance $r$ from a reference atom in a homogeneous amorphous phase:

$g(r) = \frac{dN(r)}{dr} = 4\pi\,r^{2}\,\rho(r)$

where $N(r)$ is the cumulative atom count within radius $r$ and $\rho(r)$ is the local number density. Peaks in $g(r)$ correspond to preferred coordination shell distances.

WT-RDF applies a continuous wavelet transform (CWT) to the reduced structure factor $S_R(q) \equiv q(S(q)-1)$ to improve real-space resolution:

$G(r)=\Lambda\int_{q_\min}^{q_\max} S_R(q)\, D^{*}\!\Bigl(\frac{q\,r-b}{a}\Bigr)\, dq$

Key parameters are $a$ (dilation/scale), $b$ (translation/shift), $\Lambda$ (scaling factor), and $D(x)$ (wavelet kernel). The kernel,

$$D(x) = \sum_{s=0}^{\infty} \beta_{s,p}\, x^{2s+p} + K_f\, \Phi(x) + \tilde{C}$$

with $\Phi(x) = x^4\, e^{-x^2}$, is rigorously constructed to satisfy zero-mean and orthogonality wavelet conditions:

$$\int x\, |D(x)|^2 dx = 0, \qquad \int D^*_i(x)\, D_j(x)\,dx = \delta_{ij}$$

This formulation enables direct mapping of the structure factor to real-space PDF profiles, capturing both short- and medium-range order.

2. Quantitative Limitations of WT-RDF

While WT-RDF reliably recovers the positions of principal RDF peaks (2.4–10.2% error), it exhibits systematic and substantial amplitude inaccuracies. Specifically, for binary $\mathrm{Ge}_{0.25}\mathrm{Se}_{0.75}$:

• First-peak amplitude error (FPE): ≈0.0057 (low, systematic bias)
• Second-peak amplitude error (SPE): ≈0.3207 (significant, underestimation)
• Resultant errors in experimentally relevant coordination numbers (CN), computed as

$$\mathrm{CN} = \int_{\mathrm{1st\,min}}^{\mathrm{1st\,max}} 4\pi\, r^2\, g(r)\, dr$$

can reach 10–20%.

For ternary Ag-doped compositions $$\mathrm{Ag}_x(\mathrm{Ge}_{0.25}\mathrm{Se}_{0.75})_{100-x}$$ ($x = 5,10,15,20,25$), the original WT-RDF retains peak localization but the amplitudes remain misaligned, rendering it less reliable for mixed coordination environment analyses.

3. Machine Learning Parameter Optimization

To remediate amplitude deviations, key WT-RDF parameters $\theta = \{ a, b, K_f, \tilde{C}, \Lambda \}$ are re-conceptualized as learnable variables and jointly optimized using PyTorch-based gradient descent.

Benchmark regressors for comparison include:

• Radial Basis Function (RBF) network: hidden units $M$, Gaussian width $\beta$; $\phi_j(r)=\exp[-\beta\|r-\mu_j\|^2]$, output $\hat y = \sum w_j \phi_j + b$.
• Long Short-Term Memory (LSTM) network: sequence input $r_t$, hidden state $h_t$, update $(h_t, c_t) = \mathrm{LSTM}(r_t, h_{t-1}, c_{t-1})$, output $\hat y_t = W h_t + b$.

Training utilizes a 59-point RDF dataset, normalized by Z-score, with train splits of 100%, 75%, 50%, 25%. Optimization uses Adam (lr = 0.01 → 0.001, 500 epochs).

Loss functions:

• Global mean-absolute-error (MAE):

$$\mathrm{MAE} = \frac{1}{N} \sum_{n=1}^{N} |y_n - \hat y_n|$$

• Selective peak loss ($\mathcal{L}_{SL}$), emphasizing accurate reproduction of the two primary RDF peaks:

$$\mathcal{L}_{SL} = \sum_{p=1}^2 \sum_{n=1}^N m_{n,p} |y_{n,p} - \hat y_{n,p}|$$

with mask $m_{n,p} \in \{0,1\}$ for peak bins.

WT-RDF⁺ combines both losses, targeting global curve accuracy while specifically improving peak amplitude fidelity.

4. WT-RDF⁺ Framework Construction

Four principal modifications define the enhanced WT-RDF⁺ pipeline:

1. Parameterization: All $\theta$ values are formulated as PyTorch learnable tensors within the wavelet integral.
2. Selective loss incorporation: $\mathcal{L}_{SL}$ augments the objective, penalizing errors at primary peaks.
3. Parameter bounding: Parameters are constrained per iteration to physically plausible ranges:

$$\theta^{(i)} = \min(\max(\theta^{(i-1)}, n_2^{(i)}), n_1^{(i)})$$

e.g., $a \in [0.600, 0.610]$, $K_f \in [0.010, 0.300]$.

1. Two-stage training: Preliminary phase at lr = 0.01 (100 epochs), followed by refinement at lr = 0.001.

Performance improvements on binary $\mathrm{Ge}_{0.25}\mathrm{Se}_{0.75}$:

Method	MAE	FPE	SPE
WT-RDF	0.9463	0.0057	0.3207
WT-RDF⁺	0.6598	0.0065	0.0199

For sparse training (25% data input):

Method	MAE	FPE	SPE
RBF	0.8522	>1.0	1.2345
LSTM	0.8365	>5.3	0.6486
WT-RDF⁺	0.7296	0.0620	0.0444

WT-RDF⁺ demonstrates superior amplitude prediction and reduced error under limited training data; for ternary Ag-doped variants, a 20–50% reduction in SPE and stable FPE is observed.

5. Structural Characterization and Materials Applications

WT-RDF⁺ yields both accurate peak positions and amplitudes, critical for reliable computation of local and medium-range atomic ordering. This precision facilitates robust determination of coordination numbers and mixed environments in amorphous Ge–Se and Ag–Ge–Se films.

For phase-change thin films, enhanced RDF estimation enables improved predictions of glass–crystal transition temperatures, thermal conductivities, and mechanical characteristics, informing device engineering requirements. The resilience of WT-RDF⁺ to data sparsity (accurate predictions from just 25% of X-ray dataset) streamlines experimental workflows and promotes efficient materials screening.

Device design for nonvolatile memories and neuromorphic components directly benefits via expedited materials-by-design cycles and reduced empirical data demands. This suggests the framework can substantially lower both costs and time-to-market for next-generation electronic elements.

6. Interpretability, Generality, and Future Implications

WT-RDF⁺ maintains the interpretability and transferability of a physics-based model, while leveraging data-driven selective optimization to resolve key quantitative deficiencies in original WT-RDF. A plausible implication is the migration of similar wavelet-transform–machine learning hybrid approaches to other classes of disordered and noncrystalline systems. The methodological separation between global and selective losses may inspire broader adoption in fields where curve shape and local feature fidelity are jointly essential.

The architecture also supports future extensions to adaptivity in parameter bounds and loss weighting, targeting more complex amorphous systems with multicomponent disorder.

7. Common Misconceptions and Comparative Analysis

A frequent misconception is that amplitude accuracy in RDF modeling can be trivially achieved by neural network regressors. WT-RDF⁺ evidences that generic ML models (RBF, LSTM) underperform compared to physics-guided WT-RDF⁺ both in overall MAE and peak-selective metrics, especially under data scarcity. This supports the continuing value of physics-based representations augmented by targeted machine learning optimization (Senjaya et al., 19 Dec 2025). There is no indication in the reported data of loss of interpretability or generality, contrary to concerns often voiced about ML integration with physically inspired models.

In summary, enhanced WT-RDF⁺ systematically addresses longstanding amplitude precision limitations in wavelet-transform RDFs through selective machine learning–based parameter tuning, producing a robust tool for amorphous structural analysis and next-generation materials design.