Neural Post-Einsteinian Framework

• The Neural Post-Einsteinian framework is a deep learning extension of the ppE method, enabling continuous, theory-agnostic testing of general relativity with gravitational-wave data.
• It employs variational autoencoders and neural tensor fields to model waveform deformations and 4D spacetime metrics, enhancing accuracy and computational efficiency.
• The approach supports hierarchical Bayesian inference and explores emergent dualities between quantum dynamics and gravity, advancing gravitational-wave astrophysics and numerical relativity.

The Neural Post-Einsteinian (npE) Framework refers to a broad class of methodologies that extend or generalize the "parameterized post-Einsteinian" (ppE) approach using neural network architectures or higher-order geometric/algorithmic enhancements. These frameworks are developed for efficient, theory-agnostic, and robust testing of general relativity (GR) with gravitational-wave (GW) data and for more expressive, compressed modeling of Einstein’s field equations and spacetime geometry. The npE framework encompasses deep-learning surrogates for waveform modeling, continuous latent representations for generic deviations from GR, mesh-free neural approaches to numerical relativity, and even fundamental, dual quantum-gravitational formalisms. The following sections summarize the principal methodologies, mathematical structures, and implications of these frameworks.

1. Foundations: From Parameterized post-Einsteinian to Neural Approaches

The original ppE framework was designed to address "fundamental theoretical bias" in gravitational-wave astrophysics—the systematic error introduced by assuming GR is correct in all dynamical regimes, despite weak-field confirmation only in regimes such as the Solar System or binary pulsars (0909.3328). The ppE approach generalizes GR waveform templates by introducing amplitude and phase corrections parameterized as: $$\tilde{h}(f) = \tilde{h}_{\mathrm{GR}}(f) \, [1 + \alpha u^a] \exp\{i[\Psi_{\mathrm{GR}}(f) + \beta u^b]\},$$ where $u = \pi \mathcal{M}f$ is a dimensionless frequency, $\alpha,\beta$ are the non-GR parameters, and $a,b$ are exponents determined by the theory.

While effective in interpolating between GR and specific alternatives (e.g., Brans-Dicke, massive graviton, Chern-Simons scenarios), this approach is primarily tailored to PN-expandable modifications in the inspiral regime and relies on discrete testing over theoretical templates. The neural post-Einsteinian (npE) approach overcomes these limitations by embedding ppE or more general deviations into neural network latent spaces, enabling continuous, theory-agnostic exploration and substantial computational advantages (Xie et al., 27 Mar 2024, Xie et al., 2 Oct 2025).

2. Neural Representations: Architecture and Methodology

a. Latent Space Construction via Deep Learning

The core of the npE framework is a variational autoencoder (VAE) trained on a dataset of theoretical waveform deviations—each with a discrete ppE index (effectively a PN order) and amplitude (Xie et al., 27 Mar 2024, Xie et al., 2 Oct 2025). The VAE encodes each waveform modification as a continuous set of latent parameters ($\mathbf{z}$), naturally organizing them into a two-dimensional space:

• Radial coordinate ($\|\mathbf{z}\|$): Encodes deviation magnitude.
• Angular coordinate (theory angle $\varphi$ or a bilateral parameter): Encodes the modification type (e.g., effective PN order, even admitting non-PN behaviors such as sharp, step-like deviations from hidden sector physics).

The decoder network reconstructs an analytic "shape function" representing the phase (or amplitude) deformation as a pseudo-PN expansion: $$S(\bar{f};\mathbf{n}) = \sum_{n=1}^{N_p} U_n(\mathbf{n})\, \bar{f}^{V_n(\mathbf{n})},$$ where $\mathbf{n} = \mathbf{z} / \|\mathbf{z}\|$, $\bar{f}$ is the rescaled frequency, and $U_n, V_n$ are the learned amplitude and exponent functions for each latent direction.

An auxiliary scaling network $\mathcal{T}(\mathbf{\Xi}, \mathbf{n})$ modulates the overall size of the deviation depending on physical source parameters (chirp mass, mass ratio, spins). The resulting modification to the GW waveform is

$$\tilde{h}_{\mathrm{mod}}(f) = \tilde{h}_{\mathrm{GR}}(f) \cdot \exp\left[i\, \|\mathbf{z}\|\,\mathcal{T}(\mathbf{\Xi}, \mathbf{n})\, S(\bar{f};\mathbf{n})\right],$$

enabling efficient parameter estimation over a continuous, theory-agnostic space of deviations (Xie et al., 27 Mar 2024).

b. Implicit Neural Tensor Field Models

Complementing the waveform-centric models, "Einstein Fields" encode the spacetime metric tensor itself as a neural field in $\mathbb{R}^4 \to \mathrm{Sym}^2(T^*\mathcal{M})$, outputting the full 4D metric at arbitrary spacetime points (Cranganore et al., 15 Jul 2025). Training incorporates not only the metric values but their derivatives (via Sobolev supervision), such that Christoffel symbols, curvature tensors, and geodesic equations can be obtained by automatic differentiation. This approach compactly stores and differentiates solutions to Einstein’s field equations, eliminates the need for meshes, and supports high-order accuracy.

3. Mathematical Formalism and Physical Content

a. Generalized Deformation Model

In the npE pipeline, deviations from GR are not coded at fixed PN powers but as arbitrary analytic (or even discontinuous) functions learned from theoretical or synthetic training data. For example, corrections due to higher-order curvature terms, dark-sector couplings (e.g., dark-photon interactions with step-function activations), or strong-field, high-PN phenomena can all be jointly modeled and tested. The latent space thus continuously interpolates between known theories and allows for robust exploration even in the absence of analytic templates.

b. Population-Level and Hierarchical Inference

The npE framework naturally generalizes to population analysis (Xie et al., 2 Oct 2025). Deviations across multiple GW events are modeled as being drawn from an underlying hyperdistribution, with e.g., the bilateral deviation parameter $\zeta_b$ assumed Gaussian and the theory angle $\varphi$ universal. This structure supports hierarchical Bayesian inference: $$\mathcal{L}_{\mathrm{hierarchical}}(\{\mathrm{data}\} \mid \mu, \sigma, \varphi) = \prod_{i} \int d\zeta_b^i\, d\varphi^i\, \mathcal{L}_i(\zeta_b^i, \varphi^i) p_{\mathrm{pop}}(\zeta_b^i, \varphi^i|\mu, \sigma, \varphi)$$ where $(\mu, \sigma, \varphi)$ are population hyperparameters. This approach enables combined catalog-level bounds on both GR-consistent and exotic (non-PN) classes of deviations.

c. Neural Dynamics and Dual Descriptions

Certain formulations treat the neural network itself as the physical substrate, with trainable and hidden variables corresponding to quantum and gravitational degrees of freedom, respectively (Vanchurin, 2020, Vanchurin, 2021):

• Stochastic dynamics of trainable variables approximate Madelung or Schrödinger equations (in the limit of large numbers, learning near equilibrium).
• Fast-evolving neuron states give rise to emergent Lorentzian geometry, with collective variables satisfying geodesic and Einstein-Hilbert equations.
• Entropy production, Onsager symmetries, and large deviation theory generate an effective (bulk) gravitational action with emergent curvature terms, and a holographic duality between boundary quantum/learning dynamics and the bulk spacetime geometry is suggested.

4. Applications and Impact in Gravitational Wave Astrophysics

a. Gravitational-Wave Data Analysis and Testing GR

The npE framework supports comprehensive tests of GR with both individual events and entire GW catalogs. Waveform deviations not constrained to PN power-law corrections can be efficiently marginalized over, removing theoretical bias and enabling simultaneous constraint of conventional and exotic modifications. For GW signals in LIGO/Virgo/KAGRA runs (e.g., GWTC-3) (Xie et al., 2 Oct 2025), no significant deviation from GR has been observed; population analysis, combining multiple binary black hole events, provides competitive constraints on both PN and non-PN deviations across the latent space.

b. Quantum Gravity, Duality, and Emergent Spacetime

Neural network interpretations offer a compelling route toward quantum gravity and emergent geometry. The macroscopic duality between quantum evolution (trainable variables) and Einsteinian dynamics (hidden neuron states) provides a novel, non-perturbative approach for unifying quantum mechanics and gravity in a statistical learning context (Vanchurin, 2020, Vanchurin, 2021). This synthesis suggests the possibility that spacetime and its dynamics are emergent from underlying learning systems.

c. Numerical Relativity and Spacetime Modeling

Neural tensor field approaches accelerate high-accuracy modeling of complex spacetimes, including black hole mergers, by encoding 4D spacetimes in compact neural representations (Cranganore et al., 15 Jul 2025). The mesh-free, differentiable nature of the approach offers benefits in storage efficiency, derivative accuracy, and unification of metric modeling with geometric analysis (e.g., geodesic propagation, calculation of curvature invariants), with the potential for future integration into hybrid numerical–neural simulation pipelines.

5. Limitations and Further Extensions

While the npE framework provides efficient coverage of a broad theory space, several challenges remain:

• Accurate representation outside the training set depends on the diversity and physics completeness of the waveform/metric samples used in training (Xie et al., 27 Mar 2024).
• Extensions to higher-dimensional latent spaces may be required for full inclusion of merger–ringdown phases, precession, or amplitude deformations.
• In applied GW data analysis, systematics such as noise artifacts, calibration errors, and selection effects must be incorporated for robust population inference (Xie et al., 2 Oct 2025).
• Interpretability of the latent coordinates with respect to underlying physical quantities can be limited; empirical mapping to theoretical models must be carefully validated.

6. Future Directions

Proposed future developments for the neural post-Einsteinian program include:

• Scaling to hybrid neural–numerical relativity solvers for extreme events, such as neutron-star mergers and higher-precision waveform surrogates (Cranganore et al., 15 Jul 2025).
• Enriching training datasets with non-PN, sharp, or environmental effects (e.g., dark matter, dark-photon, or high-curvature-inspired models) (Xie et al., 27 Mar 2024).
• Expanding latent parameterizations to include precession, amplitude, and higher harmonic content (Loutrel et al., 2022).
• Embedding the framework into real-time, low-latency data pipelines and hierarchical Bayesian inference for next-generation GW observatories.
• Deeper exploration of emergent dualities between quantum theory, gravity, and learning-theoretic neural architectures (Vanchurin, 2020, Vanchurin, 2021).

In summary, the neural post-Einsteinian framework unifies deep-learning, geometric, and statistical methodologies to deliver efficient, flexible, and maximally agnostic probes of fundamental physics with gravitational waves, as well as compact, mesh-agnostic models of Einsteinian and post-Einsteinian spacetimes. These developments extend the reach of gravitational theory inference, numerical relativity, and theoretical physics by leveraging advances in neural networks and data-driven modeling.