Amplitude Surrogates for Complex Systems

• Amplitude surrogates are computational constructs that approximate complex waveforms by projecting high-dimensional functions onto a reduced orthonormal basis.
• They leverage empirical interpolation and Fourier phase manipulation to faithfully preserve amplitude distributions and power spectra in time series and physics applications.
• Modern implementations use machine-learned regression and uncertainty quantification to accelerate analyses in gravitational-wave, collider, and nonlinear dynamics research.

An amplitude surrogate is a computational or statistical construct that approximates and interpolates the amplitude profile of a complex system—most frequently, waveforms arising in gravitational-wave physics or high-energy physics amplitudes—while faithfully preserving essential empirical or theoretical structures. In time-series analysis, the term also designates surrogate data generated to reproduce the empirical amplitude distribution and second-order statistics, serving as a robust null hypothesis for nonlinearity tests. Amplitude surrogates are foundational in gravitational-wave modeling, hypothesis testing for nonlinear dynamics, and machine-learned parametrizations of quantum field-theory amplitudes. Their mathematical and algorithmic foundations unify projection-based dimensionality reduction (reduced bases, empirical interpolation, singular value decomposition) with advanced regression and uncertainty quantification techniques.

1. Mathematical Foundations and Definitions

The amplitude surrogate formalism is predicated on the notion that a complex, high-dimensional amplitude function $A(t;\boldsymbol{\lambda})$—where $t$ is a temporal or phase variable and $\boldsymbol{\lambda}$ denotes physical parameters—can be efficiently approximated by projection onto a reduced orthonormal basis:

$$A_N(t;\boldsymbol{\lambda}) \approx \sum_{i=1}^{N} c_i(\boldsymbol{\lambda}) B_i(t),$$

where $\{B_i(t)\}$ are orthonormal basis functions (e.g., constructed via greedy algorithms or SVD) and the coefficients $c_i(\boldsymbol{\lambda})$ encode parameter dependence (Lackey et al., 2016, Gadre et al., 2022).

In the context of surrogate data methods for time series, an amplitude surrogate $x_s[t]$ is a randomized realization that exactly preserves:

• The amplitude distribution (empirical histogram) $p(x)$,
• The power spectrum $S(f)$ (or, equivalently, autocorrelation), with frequent enforcement of additional constraints, such as local mean and variance for nonstationary data (Raeth et al., 2008, Guarin et al., 2010).

In machine-learned settings, amplitude surrogates $f_\theta(x) \approx A_{\rm true}(x)$ are neural regression models trained on Monte Carlo or numerical data, optimized to minimize pointwise or distributional discrepancies while controlling uncertainty over parameter space (Bahl et al., 16 Dec 2024, Bahl et al., 29 Aug 2025).

2. Construction Methodologies

2.1 Reduced-Basis and Empirical Interpolation

Gravitational-wave surrogates employ a multistage pipeline:

• Reduced-basis selection: Construction of a compact set of $N$ basis functions $\{B_i(t)\}$ from training data via greedy L² minimization or proper orthogonal decomposition, yielding orthonormal time/function bases tailored to $A(t;\boldsymbol{\lambda})$ (Lackey et al., 2016, Gadre et al., 2022).
• Empirical interpolation method (EIM): Identification of empirical nodes $\{\tau_j\}$ such that

$$A_N(t;\boldsymbol{\lambda}) = \sum_{j=1}^{N} B_j(t)A(\tau_j;\boldsymbol{\lambda}),$$

allowing separation of parameter and temporal dependence.

• Parameter interpolation: Coefficients $A(\tau_j;\boldsymbol{\lambda})$ are interpolated across parameter space with tensor-product Chebyshev polynomials or, for more complex models, with Gaussian process regression or (multi-variate) polynomials (Lackey et al., 2016, Pacilio et al., 9 Aug 2024, Nee et al., 30 Sep 2025).

2.2 Surrogate Data Techniques for Nonlinearity Testing

Amplitude surrogate data generation for nonlinearity and stationarity hypothesis testing follows algorithmic procedures, e.g., AAFT/IAAFT or AATFT:

• Amplitude remapping: Rank-preserved mapping onto a target (often Gaussian) distribution.
• Fourier phase manipulation: Randomization (AAFT) or selective preservation (AATFT) of Fourier phases, maintaining spectral properties.
• Iterative adjustment: Recursion to minimize deviation of power spectrum and amplitude distribution (IAAFT, iAATFT), or more advanced iterative schemes to control Fourier phase correlations (IPAFT) (Raeth et al., 2008, Guarin et al., 2010).
• AATFT/iAATFT: Enforcement of spectrum, amplitude distribution, and local means/variances simultaneously for nonstationary time series (Guarin et al., 2010).

2.3 Machine-Learned Surrogates and Uncertainty Quantification

Modern amplitude surrogates for collider processes are built as high-capacity neural regression models:

• Architectures: MLP, GroupKAN, Lorentz-equivariant networks, typically with symmetry and physical-invariant preprocessing (Bahl et al., 16 Dec 2024).
• Losses: Heteroscedastic likelihood for systematic uncertainties, Bayesian or ensemble-based loss for statistical uncertainties, and evidential regression for conjugate-prior uncertainties (Bahl et al., 16 Dec 2024, Bahl et al., 29 Aug 2025).
• Uncertainty calibration: Pull distributions, global correction of systematic uncertainty in ensembles, evidential regression’s Student-t conjugate frameworks (Bahl et al., 29 Aug 2025).

3. Applications Across Scientific Domains

Domain	Amplitude Surrogate Role	Key Metric(s)
Gravitational-wave astronomy	Fast, accurate waveform generation for inference and data analysis	Time/Frequency mismatch, L²
Nonlinear time-series analysis	Generation of surrogate data for hypothesis testing of dynamical nonlinearity	Mutual information, NLPE
Collider physics (LHC amplitudes)	Acceleration and uncertainty quantification in matrix element calculations	Relative error, pull statistics

In gravitational-wave physics, surrogates replace numerical EOB, Teukolsky, or NR waveforms with order-$10^4$ speedup for parameter estimation, facilitating Bayesian inference for binary neutron stars, eccentric black holes, and postmerger ringdown modeling (Lackey et al., 2016, Nee et al., 30 Sep 2025, Pacilio et al., 9 Aug 2024, Rifat et al., 2019, Gadre et al., 2022).

In time-series analysis, amplitude surrogates underpin nonlinearity tests by providing null models with controlled preservation of amplitude, spectrum, and sometimes time-local structure, thereby clarifying the distinction between static nonlinearities and true dynamical ones (Raeth et al., 2008, Guarin et al., 2010).

In high-energy physics, amplitude surrogates trained with calibrated uncertainties allow for fast, robust estimation of scattering amplitudes across phase space, supporting systematic error quantification and stability to gaps or noise in training data (Bahl et al., 16 Dec 2024, Bahl et al., 29 Aug 2025).

4. Error Metrics, Calibration, and Validation

Error control is paramount:

• Waveform surrogates: Training and validation errors are quantified by maximum fractional amplitude error, L² time-domain errors, and mismatch (overlap) in the frequency domain weighted by detector noise curves (Lackey et al., 2016, Nee et al., 30 Sep 2025, Gadre et al., 2022). Achieved errors are $\sim0.04\%$ for neutron star inspirals (excluding highest-curvature merger regime), $<10^{-3}$–$10^{-2}$ for eccentric inspiral or BBH surrogates.
• Machine-learned amplitude surrogates: Pointwise relative errors reach $10^{-5}$–$10^{-6}$ with state-of-the-art symmetry-aware architectures (Bahl et al., 16 Dec 2024, Bahl et al., 29 Aug 2025). Systematic and statistical uncertainties are validated via pull distributions, ensuring uncertainties are neither under- nor overestimated.
• Surrogate data: Quantitative residuals such as phase–correlation covariance, spectral variability, and mutual information measures diagnose residual structure or nonlinearity beyond the null hypothesis (Raeth et al., 2008, Guarin et al., 2010).

5. Algorithmic Innovations and Adaptations

Recent developments extend amplitude surrogate methodologies to:

• Nonstationary/stochastic time series: The AATFT/iAATFT algorithms unify amplitude, power spectrum, and time-local statistics, enabling hypothesis testing for complex, nonstationary, and non-Gaussian systems (Guarin et al., 2010).
• Robust preservation of phase properties: Iterative phase-controlled methods (IPAFT-A/B) ensure fully random or ARMA-consistent amplitude surrogates, correcting limitations of AAFT/IAAFT in enforcing higher-order independence (Raeth et al., 2008).
• Hybrid models: Amplitude surrogates for gravitational waves now stitch together multiple basis expansions (radial-phase inspiral and time-domain merger), factor out periodic structure (e.g., through mean anomaly in eccentric systems), and deploy Gaussian-process regression to interpolate across high-dimensional physical parameter spaces (Nee et al., 30 Sep 2025, Islam et al., 16 Apr 2025).
• Calibration for NR agreement: Amplitude surrogates for black hole binaries are further calibrated in time and amplitude with explicit q-dependent factors to ensure quantitative alignment with full NR simulations in the moderate-mass-ratio regime (Islam et al., 2022, Rifat et al., 2019).

6. Limitations, Extensions, and Future Directions

Amplitude surrogate models are constrained by the dimensionality of parameter space (spin, eccentricity, precession), the availability/quality of training data (especially for numerical-relativity input), and, in the surrogate data context, by the assumption that non-dynamical nonlinearities are static and memoryless.

Extensions are anticipated in several directions:

• Gravitational-wave surrogates: Systematic inclusion of aligned and precessing-spin effects, higher-order modes, and joint inspiral–postmerger modeling as training datasets grow (Lackey et al., 2016, Nee et al., 30 Sep 2025, Gadre et al., 2022, Pacilio et al., 9 Aug 2024).
• Amplitudes in high-energy theory: Universal, uncertainty-aware surrogates for multiparticle and higher-loop amplitudes, leveraging equivariant neural architectures and advanced probabilistic frameworks (Bahl et al., 16 Dec 2024, Bahl et al., 29 Aug 2025).
• Surrogate data theory: Algorithmic generalization to preserve further empirical properties (e.g., non-Gaussian higher-order spectra), and robust validation against dynamic nonlinearities in nonstationary and multivariate settings (Guarin et al., 2010).

Amplitude surrogates will continue to play a pivotal role in facilitating efficient scientific inference and principled hypothesis testing across physics and nonlinear signal analysis.