Wave-Based Representations: Theory & Applications

• Wave-based representations are mathematical frameworks that encode functions and signals using the principles of wave propagation, superposition, and interference.
• They integrate physics-informed models, such as the wave and Helmholtz equations, with data-driven methods to achieve robust modal analysis and signal recovery.
• Applications span structural health monitoring, quantum dynamics, and harmonic analysis, enabling interpretable, compact, and efficient representations across disciplines.

Wave-based representations are mathematical, computational, and physical frameworks that encode functions, signals, physical fields, probability distributions, or abstract numbers using the structure, symmetries, and analytical properties of waves. These representations exploit the fundamental mechanisms of superposition, interference, modal decomposition, and propagation inherent in the wave equation and its generalizations across physics, applied mathematics, and data-driven sciences. They include both explicit physical models (wave equation, Helmholtz equation, Green’s functions), structured matrix factorizations regularized by wave physics, neural and low-rank parametric encodings informed by wave dynamics, harmonic analytic constructions (wavelets, cluster expansions), and novel perspectives such as smooth analytic arithmetic using kernel waveforms.

1. Foundational Physical and Geometric Principles

Wave-based representations are grounded in both the analytical structure of the wave equation and abstract geometric frameworks.

A universal foundation is the multidimensional oscillator viewpoint, wherein any classical, nondissipative wave may be described as an abstract vector $|\psi\rangle$ in a complex Hilbert (or pseudo-Hilbert) space $\Psi$ equipped with a Hermitian (possibly indefinite) metric. The wave equations themselves can be derived from geometric properties of $\Psi$, and are shown to be equivalent to quantumlike Schrödinger equations: $i\,\partial_t\ket{\psi} = \hat H\,\ket{\psi},$ where $\hat H$ is a Hermitian operator associated with the energy functional of the system (Dodin, 2013). Wave action is described as a density operator $\hat{\rho} = |\psi\rangle\langle\psi|$, and kinetic equations (Wigner, Liouville, Whitham) appear as various projections of the von Neumann equation for $\hat{\rho}$.

Coordinate and momentum (phase space) notions appear as emergent structures, constructed through appropriate operator bases. The Wigner function and star-product (Moyal) brackets arise naturally, so that classical and quantum wave kinetic theory, as well as geometrical optics and ray tracing, are unified under a single invariant framework (Dodin, 2013, McCaul et al., 2023).

2. Physics-Informed and Data-Driven Factorizations

Recent developments have established rigorous techniques for learning representations of wave-dominated signals and fields which are explicitly constrained by the governing PDEs. Unlike generic low-rank or dictionary models, these methods integrate the discrete Helmholtz/wave equation as a soft or hard constraint in the factorization process.

Wave-Informed Matrix Factorization (WIMF):

Given spatially and temporally sampled data $Y \in \mathbb{R}^{L \times T}$, WIMF seeks a decomposition

$Y \approx \sum_{i=1}^N D_i X_i^T$

with spatial modes $D_i \in \mathbb{R}^L$, temporal coefficients $X_i \in \mathbb{R}^T$, and wavenumbers $k_i$, such that $D_i$ are regularized to satisfy the discrete Helmholtz eigenproblem

$L D_i + k_i^2 D_i \approx 0,$

where $L$ is the Laplacian matrix (Tetali et al., 2021, Tetali et al., 2023). The objective is

$$\min_{N, D, X, k}\; \frac12 \|Y-D X^T\|_F^2 + \frac\lambda2 \sum_{i=1}^N \left( \|D_i\|_2^2 + \|X_i\|_2^2 + \gamma \|L D_i + k_i^2 D_i\|_2^2 \right).$$

Non-convexity is overcome by leveraging the "structured matrix factorization" framework, yielding a meta-algorithm with a global optimality certificate based on the polar problem; the optimal modes are identified via a 1D line search in $k$ and SVD in the filtered domain.

Modal Analysis and Filtering Connections:

This structured regularization essentially extracts modes via optimal band-pass Butterworth filters, thus connecting wave-based representations to classic filter theory. In modal analysis and structural health monitoring, these approaches exhibit superior mode recovery, noise robustness, and interpretability compared to PCA, ICA, DMD, or heuristic dictionary learning (Tetali et al., 2023).

3. Joint Time–Frequency and Multi-Resolution Representations

Wavelet-based representations regularize or analyze wave phenomena in joint time–frequency (or space–momentum) domains, offering optimal localization for transient, non-stationary, or spatially inhomogeneous wavefields.

Examples include:

• Affine Poincaré wavelets: For the 2D wave equation, arbitrary solutions are represented as integrals over affine Poincaré group–transformed mother wavelets. Lorentz boosts, dilations, and translations simultaneously encode propagation angle, spectral scale, and localization. The full solution is built as an explicit superposition of localized, causally classified propagating and surface-evanescent "mother" packets (Perel et al., 2012).
• Continuous wavelet transforms in joint (x,k): The wavelet transform of a wavefunction $\psi(x)$,

$$\mathbb{W}(x, k) = \frac{1}{\sqrt{|k|}} \int \psi(x')\, \mathcal{G}^*((x'-x)/k)\, dx',$$

provides fine-grained structure for detecting self-interference, beam acceleration, and soliton formation in nonstationary wave packets and nonlinear regimes (Colas et al., 2019).

• Wavelet-enhanced representations in SHM: The CWT of ultrasonic structural signals yields dense scalograms, which, when used as inputs to deep autoencoders or unsupervised outlier detectors, provide robust damage localization and anomaly detection in real-world composite systems (Rautela et al., 2022, Fan et al., 15 Apr 2025).

4. Parametric, Tensor, and Neural Representations

A class of wave-based representations encodes functions, distributions, or operators as structured parametric expansions or neural architectures motivated by wave physics.

Atomic Cluster Expansion (ACE) for many-body wavefunctions:

ACE generalizes from invariant polynomials to fully antisymmetric functions, capturing all known linear (e.g., configuration-interaction, Slater determinant) and nonlinear (Jastrow, backflow, Vandermonde) ansätze for quantum wavefunctions. All such representations are compressible using cluster order, basis size, and symmetry constraints, providing systematic convergence while respecting physical invariances (Drautz et al., 2022).

Wave-function representations of probability densities:

Univariate densities $p(x)$ are encoded as

$$\psi(x) = \sqrt{p(x)} = \sum_{n=0}^{N} c_n \varphi_n(x),$$

where $\{\varphi_n\}$ is the Hermite basis. This expansion is formally identical to quantum wavefunction representations, with statistical constraints $\|\psi\|^2=1$ and $p(x)=|\psi(x)|^2$ (Thompson, 2017).

Low-Rank Neural Representations (LRNR):

For hyperbolic wave dynamics, LRNRs construct expressively efficient architectures—wherein neural network layers are parameterized by low-rank tensors, with temporal evolution encoded by a hypernetwork generating the low-dimensional parameter trajectory. The learned "hypermodes" are interpretable as characteristic wave features (translations, modulations), and enable both compression and scalable inference (Cho et al., 29 Oct 2025).

5. Analytic and Harmonic Encodings in Abstract Domains

Wave-based principles extend beyond classical PDEs to domains such as number theory and arithmetic via integrals of analytic kernels.

• Wave Arithmetic:

Each number $N$ is realized as the integral of a structured kernel,

$N = \int_D K_N(\mathbf{x})\, d\mathbf{x},$

with arithmetic operations realized as geometric concatenation (addition), tensor/slice integration (multiplication), and higher tensor products (exponentiation). Primality and divisibility correspond to geometric irreducibility or spectral resonance. Thus, arithmetic becomes an analysis of analytic resonance patterns rather than symbolic discrete tokens (Semenov, 21 Apr 2025).

6. Unified Operator and Field Representations

Full field solutions and operator actions for diverse wave phenomena can be encoded compactly via matrix–vector first-order systems and associated Green’s functions, propagator matrices, and focusing functions:

• Matrix–vector wave equation:

$$\partial_z\, \mathbf{q}(x) = \mathbf{L}(x)\, \mathbf{q}(x) + \mathbf{d}(x)$$

with operator $\mathbf{L}$ covering acoustic, elastodynamic, electromagnetic, and other wave types by appropriate block structure (Wapenaar, 2017). Symmetry properties of $\mathbf{L}$ (including adjoint relations) enforce universal reciprocity theorems.

• Green’s matrix and propagator representations:

Volume and boundary integral representations for wave fields, boundary-only single-sided representations (using propagator matrices $\mathbf{W}$), and data-driven Marchenko-type focusing functions all provide different but related "wave-based" forms for field recovery, imaging, and inverse problems (Wapenaar, 2021).

• Global reciprocity and time reversal:

Boundary integral expressions (correlational and convolutional types) relate fields throughout the domain, encoding forward modeling, imaging, redatuming, and Green’s function retrieval, including via single-boundary data.

7. Applications, Extensions, and Open Problems

Wave-based representations are deployed in:

• Structural health monitoring (modal decomposition, anomaly detection, inverse design) (Tetali et al., 2021, Tetali et al., 2023, Rautela et al., 2022, Rautela et al., 2022, Fan et al., 15 Apr 2025)
• Quantum/classical hybrid dynamics and density-matrix evolution (McCaul et al., 2023)
• Unsupervised and deep-learning driven inversion and data generation (Rautela et al., 2022, Cho et al., 29 Oct 2025)
• Number-theoretic and analytic frameworks (Wave Arithmetic) (Semenov, 21 Apr 2025)
• Unified and multi-physics field modeling (Wapenaar, 2017, Wapenaar, 2021)

Prominent open questions include the generalization of physics-informed factorization to non-linear PDEs, automated model/hyperparameter selection, extension to multidimensional and non-separable domains, physically interpretable latent representations in learned architectures, and the rigorous interplay between analytic-harmonic and deep/neural approaches.

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Wave-based representations unify diverse phenomena by encoding structure, symmetry, and propagation properties of waves in mathematical, computational, and physical systems. They serve as a backbone for robust, interpretable, and often optimally compact descriptions across domains from PDE-based physics to probabilistic modeling, signal analysis, and even arithmetic.