Wave Propagation Operators

• Wave Propagation Operators are mappings that transform initial or source data into evolving wave fields while rigorously capturing dispersion and scattering properties.
• They employ spectral frameworks, high-order rational approximations, and operator splitting to achieve stability and high accuracy in variable media.
• WPOs underpin practical applications in seismic tomography, optical propagation, and acoustic modeling by integrating analytic, numerical, and neural operator techniques.

A wave propagation operator (WPO) is a (possibly time- or space-parametrized) operator that maps initial or source data for a given wave equation to the evolving field at later times or downstream positions, specifically encapsulating both the dispersion and scattering properties of the operator governing the system. WPOs provide a rigorous, unified formalism to analyze, compute, and generalize wave dynamics across variable media, types of governing operators, and numerical or learning-based frameworks. Their structural variants include analytic spectral representations, high-order rational approximations, operator splitting and factorization, integral operator forms, as well as recent neural operator surrogates.

1. Abstract Operator Formulation and Spectral Framework

The archetypal WPO arises from the abstract Cauchy problem for a self-adjoint, nonnegative operator $A$ with discrete spectrum on a Hilbert space $H$, evolving under a (possibly time-dependent) wave equation: $$\partial_t^2 u(t,x) + a(t) A u(t,x) = 0,\qquad u(0,x)=u_0(x),\ \partial_t u(0,x)=u_1(x),$$ where $a(t)$ is the (possibly irregular or distributional) coefficient serving as the propagation speed. The corresponding solution/propagation operator,

$$S(t,s)\! : (u_0,u_1) \mapsto (u(t), \partial_t u(t)),$$

is represented spectrally by

$$u(t,x) = \sum_{k=1}^\infty \phi_k(x)\left[ \cos(\sqrt{\lambda_k}\, \theta(t,s))\langle u_0,\phi_k\rangle + \frac{\sin(\sqrt{\lambda_k}\,\theta(t,s))}{\sqrt{\lambda_k}}\,\langle u_1, \phi_k\rangle \right],$$

where $\{\lambda_k,\phi_k\}$ are the spectral data of $A$ and $\theta(t,s)=\int_s^t \sqrt{a(\tau)}\,d\tau$ encodes the cumulative effect of the speed coefficient. The propagation operator enjoys a group property and well-posedness in Hilbert or Gevrey scales, with generalizations to "very weak solutions" in Colombeau-type algebras for distributional $a(t)$ (Ruzhansky et al., 2017).

2. Rational, High-Order and Exponential Propagators

Wave propagation over large time or space steps requires stable, high-accuracy approximations to the operator exponential $H$0 or analogous forms for one-way, envelope, or pseudo-differential models. Rational approximation methods, such as direct rational construction of time-evolution operators,

$H$1

where $H$2 is skew-Hermitian, enable large steps, unconditional stability, and high-order accuracy. Efficient computation is achieved via precomputed sparse or spectral direct solves for each pole (Haut et al., 2014). In vector Helmholtz settings with strong inhomogeneity and cross-polarization coupling, the exponential of pseudo-differential operators is rationally approximated using AAA-Lawson minimax interpolation over the relevant spectral domain, supporting accurate forward-arc propagation, evanescent content, and high-angle scattering (Keefe et al., 2024). This rational expansion is implemented via a moderate number of large, sparse linear solves per step.

3. Weak-Form, Finite and Spectral Discretization

Numerical WPOs often operate within spatial discretization schemes. An optimally accurate WPO stencil can be constructed in the weak form by Taylor-expanding field and material coefficients at grid nodes, integrating against basis functions (such as “hat” B-splines), and systematically assembling local Vandermonde-like systems. This yields compact algebraic stencils that recover or outperform classic methods (Galerkin, finite-difference, mass-lumped) in both homogeneous and heterogeneous media, maintaining high-order accuracy (O(Δx⁴) in favorable regimes) and demonstrating superior error properties (Fuji et al., 5 May 2025).

Spectrally, in periodic or high-contrast media, the WPO corresponds to the propagator associated with the unitary group $H$3, where $H$4 is a periodic or contrast-parameterized elliptic operator. This supports quantitative homogenization, analytic band structure perturbations, and non-local limit effective operators, each rigorously controlling wave dynamics and their spectral consequences (Cherednichenko et al., 2 Apr 2025, Godin et al., 14 Apr 2025).

4. Operator Factorizations and Paraxial/One-Way Evolution

WPOs provide the foundation for one-way and envelope modeling, frequently via operator factorization: $H$5 underscoring Helmholtz-to-paraxial reduction and the emergence of pseudo-differential square-root operators in the context of forward propagation (Keefe et al., 2024). The rigorous definition of operator square roots for parabolic/one-way models uses Dunford–Taylor calculus, sectoriality theory, and generation of strongly continuous semigroups, underpinning the stability and correctness of such marching schemes (Ehrhardt et al., 30 Apr 2025).

5. Neural and Data-Driven Wave Propagation Operators

Recent advances position WPOs as differentiable learned surrogates. Neural operator architectures such as Fourier Neural Operators (FNOs) (Yang et al., 2021), Windowed Fourier Propagators (WFPs) (Cai et al., 15 Mar 2026), and DeepONets (Borrel-Jensen et al., 2023) encode the propagation operator as a parameterized composition of local or global spectral-feature mixing, often exploiting a frequency-locality principle. For example, WFPs learn block-sparse, local dictionaries mapping active spectral windows, attaining O(N r^d) complexity, strict superposition preservation, and strong explainability in terms of underlying scattering physics.

Such neural WPOs generalize over material or source variation and deliver order-of-magnitude speedups in simulation or inversion, as in seismic tomography and 3D room acoustics; their primary limitations are the need for large, diverse training data and challenges in strongly discontinuous or unresolved regimes.

6. Stochastic, Random, and Lossy Extensions

The WPO framework extends to stochastic and dissipative systems. For waves in random media, the propagator is formulated as a Fourier integral operator (FIO), with randomness entering via additive perturbations to its phase and amplitude, enabling direct statistical inference for parameter estimation and hypothesis testing (damage detection) (Oberguggenberger et al., 2020). In lossy media, the operator model separates propagation and dissipation, with attenuation compensation strategies designed to invert dominant exponential decay effects, restoring lossless-like data for reduced-order model construction (Shao et al., 31 Oct 2025). The precise split between closed-form propagation and non-closed-form dissipation is established at the operator level, with rigorous quantification of attendant errors.

7. Applications and Exemplars across Physical Regimes

WPOs are foundational across numerous domains:

• Harmonic oscillators, Landau Hamiltonians, subelliptic and sub-Laplacian operators on manifolds, and elliptic boundary-value problems, via spectral expansions and energy methods (Ruzhansky et al., 2017);
• Paraxial and nonparaxial light fields, including Bessel–Gauss and Helmholtz–Gauss beam families, via spatial evolution operators and algebraic closed-form propagation (Silva et al., 18 Feb 2026);
• Periodic lattices or crystals with lower-dimensional defects, modeled through Bloch–Floquet reduction and analytic defect-operator constructions, yielding full propagating, guided, and localized spectra (Kutsenko, 2013);
• High-contrast composites, via analytic perturbation and nonlocal limit operators, capturing altered group velocities, emergence of band gaps, and novel spectral phenomena (Godin et al., 14 Apr 2025).

WPO constructions are thus both a common thread and a unifying abstraction enabling advanced modeling, sharp spectral analysis, quantitative homogenization, optimal numerical integration, and modern operator-learning approaches. Their architecture and analytic properties continue to draw on, and motivate, intertwined advances in spectral theory, pseudo-differential and microlocal analysis, numerical PDE, and neural operator learning.