Variable Phase Approach in Scattering and Control

• Variable Phase Approach is a suite of techniques that evolves a phase function to encode physical and statistical information, notably in quantum scattering and signal processing.
• The method enhances machine learning for periodic time-series by phase-folding data, ensuring robust classification and efficient data regularization.
• It underpins inverse problem solving, adaptive control in biomechanics, and numerical methods for variable-exponent PDEs, offering precise and scalable solutions.

A variable phase approach refers to a class of mathematical and computational techniques in which an evolving phase function, typically dependent on a spatial, temporal, or parametric variable, systematically encodes physical or statistical information about a system. The methodology has broad applications, from quantum and classical scattering theory (where it originated) to modern machine learning for periodic time-series classification, control of biomechatronic systems, electromagnetic wave localization, and variable-exponent partial differential equations. Essential to all these contexts is the use of phase—either as a physical observable, an algorithmic coordinate, or an analytic transformation—to reduce complexity, gain robustness, or extract interpretable physical quantities.

1. Foundations: Classical Variable Phase Equation and Generalization

The canonical variable phase approach was formulated in the context of quantum potential scattering, as a reformulation of the boundary-value problem for the radial Schrödinger equation. The central object is the running (distance-dependent) phase shift $\delta_\ell(r)$ for partial wave $\ell$, which satisfies a nonlinear first-order ODE:

$$\frac{d\delta_\ell(r)}{dr} = -\frac{V(r)}{k} \left[\cos \delta_\ell(r)\, j_\ell(kr) - \sin\delta_\ell(r)\, n_\ell(kr)\right]^2$$

with initial condition $\delta_\ell(0) = 0$. Here, $V(r)$ is the interaction potential, $k$ the wave number, and $j_\ell$, $n_\ell$ are Riccati–Bessel and Riccati–Neumann functions. The ODE accumulates the total physical phase shift, $\delta_\ell(\infty)$, directly as $r$ increases, providing a transparent mapping between local potential structure and observable scattering properties (Khachi, 14 Dec 2025, Khachi et al., 28 Mar 2024, Khachi, 2023).

This approach has been generalized to inverse problems (recovering $V(r)$ from measured phase shifts), coupled-channel systems, electromagnetic S-matrix calculations, and variable exponent, multi-phase PDEs in variational calculus (Forrow et al., 2012, Crespo-Blanco et al., 2022, Avci, 28 Jan 2025).

2. Variable Phase in Machine Learning: Periodic Time Series

Phase-based methods have been adapted to the automated classification of periodic astronomical light curves, where each time-series observation is folded into a phase coordinate:

$$\phi_i = \left(\frac{t_i - t_0}{P}\right) \bmod 1, \quad \phi_i \in [0,1)$$

with known period $P$ and a chosen zero point $t_0$. Points are then resampled onto a uniform phase grid, generally of fixed dimension (e.g., $M = 512$):

$\Phi_k = \frac{k}{M-1}, \;\; k=0,\dots,M-1$

$x_k = \frac{\widehat m(\Phi_k) - \mu}{\sigma}$

where $\widehat m(\Phi_k)$ is the linearly interpolated magnitude, and normalization enforces zero mean and unit variance. This fixed-input vector enables high-performing convolutional neural networks (with architectures such as sequential Conv1D + ReLU + BatchNorm + MaxPooling, repeated in blocks) to learn morphologically invariant, class-distinguishing features without handcrafted preprocessing. The method achieves an average accuracy of 90.1% and an $\mathrm{F1}$ score of 0.86 over six variable star types in diverse and irregular real survey data (Akhmetali et al., 16 Aug 2025).

Phase-folding directly aligns underlying physical cycles while regularizing sampling cadences and handling gaps, leading to a model that is robust, reproducible, and scalable for large datasets.

3. Variable Phase in Inverse Quantum Scattering and Nuclear Structure

Variable phase approaches are foundational in building inverse potentials from empirical phase shift data. The key workflow involves:

• Integrating the phase ODE with respect to $r$ for trial potentials (commonly parameterized as sums of Gaussians, Morse-type, or exponential wells).
• Comparing computed $\delta_\ell(\infty; E)$ to experimental $\delta^\text{exp}_\ell(E)$ at multiple energies with metrics such as mean absolute percentage error (MAPE).
• Optimizing potential parameters via stochastic or evolutionary algorithms (e.g., Variational Monte Carlo (Khachi et al., 28 Mar 2024), Genetic Algorithms (Awasthi et al., 15 May 2025)) to minimize the fit error.

High fidelity is demonstrated across multiple nuclear systems (e.g., $\alpha$–$\alpha$, $n$–$\alpha$, $\alpha$–$^{12}$C), with optimized potentials reproducing experimental phase shifts, resonance energies, and widths within a few percent (Khachi, 14 Dec 2025, Khachi et al., 28 Mar 2024, Awasthi et al., 15 May 2025). The approach is extensible to charged-particle systems (with appropriate Coulomb phase corrections), multichannel scattering, and data-driven inverse potential construction.

4. Extensions to Variable Exponent and Multi-phase PDEs

Generalizing variable phase principles to nonlinear PDEs with variable exponents leads to advanced analytical and variational structures. In the study of quasilinear elliptic equations with double or multi-phase Musielak–Orlicz-type operators (e.g.,

$$-\mathrm{div}\left(|\nabla u|^{p(x)-2}\nabla u + \mu(x) |\nabla u|^{q(x)-2}\nabla u\right) = f(x,u)$$

), the Nehari manifold and fibering map formalism act as variable-phase analogues. Solution branches are characterized via phase-sensitive functionals:

• Fibering maps $\phi_u(t) = I(tu)$, with phase-coherent critical points found by analyzing the sign of the second derivative along these orbits.
• Splitting of the solution set based on sign structure, yielding positive, negative, and nodal (sign-changing) solutions with precise domain information, following strict superlinearity and growth conditions on the nonlinearity $f$ (Crespo-Blanco et al., 2022, Avci, 28 Jan 2025).

These methodologies facilitate the proof of existence, multiplicity, and qualitative features of solutions to highly nonlinear and spatially heterogeneous equations.

5. Variable Phase in Control and Signal Processing

In applied biomechanics, variable phase serves as a real-time coordinate for rhythmic and non-rhythmic control of powered prosthetic devices. The approach involves:

• Computing a piecewise-holonomic phase variable, $\phi(t)$, as a monotonic, normalized function of biomechanical joint angles (e.g., thigh angle), segmented dynamically via a finite-state machine according to gait phase and sensory feedback.
• The phase variable parameterizes virtual constraints (desired joint trajectories), which are tracked by impedance controllers.
• This unified coordinate enables seamless transitions between walking speeds, start/stop maneuvers, backward locomotion, and volitional tasks (e.g., obstacle clearance, kicking), decreasing compensatory movement patterns and energy cost for the user (Rezazadeh et al., 2018).

In electromagnetic geophysics, variable phase propagation velocity is critical for correcting dispersive and multipath effects in long-range lightning location networks. By replacing the constant speed of light with empirically mapped, path-specific phase velocities, location accuracy is improved by approximately 1 km, accounting for regional ionospheric and ground conductivity variations (Liu et al., 2016).

6. Numerical Methods: Adaptive Phase Representation and Levin Techniques

For differential equations with rapidly varying solutions, variable phase methods underlie high-efficiency numerical algorithms. The Leveraged Phase Function approach (cf. Levin's method) recasts ODEs into Riccati-type ODEs for a slowly varying phase $\psi(t)$, allowing spectral or collocation representation that is independent of the coefficient magnitude:

• The approach systematically linearizes and solves for the desired phase branch using Newton iterations and Chebyshev polynomial expansions on adaptive subintervals.
• Both global and local Levin-method variants enable error control and robustness to turning points, maintaining computational cost independent of oscillation frequency or stiffness (Aubry et al., 2023).

This strategy is foundational for scalable simulation in high-frequency regimes or for strongly inhomogeneous media.

7. Impact, Limitations, and Future Prospects

Variable phase approaches provide unifying algorithmic and physical frameworks for:

• Extracting interpretable and quantifiable parameters (phase shifts, resonance positions, solution branches) tied directly to local properties.
• Enabling efficient and robust numerical computation in domains that would otherwise suffer from stiffness, high dimensionality, or sampling irregularity.
• Streamlining model pipelines in data-driven fields by aligning periodicities, regularizing representations, and eliminating the need for hand-engineered features (Akhmetali et al., 16 Aug 2025).

Limitations of the approach include the requirement of a known local or effective potential in quantum applications, need for period knowledge or phase reference in time-series contexts, and analytic complications when generalizing to nonlocal or multi-dimensional systems.

Emerging directions involve the fusion of variable phase principles with semi-supervised learning, multi-channel sequence analysis, adaptive control in robotics, and the scalable inference of complex, region-dependent physical models from sparse or irregular observational data.

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References

• For machine learning classification with phase folding: (Akhmetali et al., 16 Aug 2025)
• For quantum inverse scattering and phase ODEs: (Khachi, 14 Dec 2025, Khachi et al., 28 Mar 2024, Awasthi et al., 15 May 2025, Khachi, 2023, Forrow et al., 2012)
• For control of prostheses with phase variables: (Rezazadeh et al., 2018)
• For variable phase in electromagnetic/geophysical networks: (Liu et al., 2016)
• For variable exponent PDEs and the Nehari manifold: (Crespo-Blanco et al., 2022, Avci, 28 Jan 2025)
• For adaptive phase functions and numerical methods: (Aubry et al., 2023)