Overview Three Constructs EFE Scale Regimes Quick Setup Live Dashboard GitLab

Covariant Neural Operator

GRAVI-NEURAL introduces the first physics-informed AI framework for learning solutions to the Einstein Field Equations in strongly perturbed spacetimes — the Covariant Neural Operator (CNO). Built on three mathematically rigorous constructs spanning Gravitational Neural Operator, Space-Time Covariant Network, and Micro-Gravity Anomaly Network.

GitHub Repository Live Dashboard DOI: 10.5281/zenodo.19871822

0.31%

Mean EFE Residual

3-regime cross-validation

2.1×10⁻³

GW Mismatch

Below detection threshold

10⁷×

Speedup vs NR

47 ms per waveform

1.8 cm

Geodesic Residual

24-hour GPS prediction

The Three GRAVI-NEURAL Constructs

GNO

Gravitational Neural Operator · Fourier Neural Operator

g_μν(x) = η_μν + h_μν^AI(x; θ) — Maps stress-energy T_μν to metric perturbation with 12 FNO layers

S-TCN

Space-Time Covariant Network · Diffeomorphism Invariance

Tensor equivariant neural network for GL(4,ℝ) covariance — <0.1% deviation across 50 coordinate charts

M-GAN

Micro-Gravity Anomaly Network · CVAE Gravity Inversion

Conditional VAE for subsurface mass-density perturbation recovery from satellite gravity gradiometry

Covariant Neural Operator (CNO) Formula

g_μν(x) = η_μν + h_μν^AI(x; θ)

            G_μν ≡ R_μν − (1/2)g_μνR = 8π T_μν

            ∇^μ G_μν = 0 (Bianchi identity — hard constraint)

Python Interface

            
from gravineural import CovariantNeuralOperator

cno = CovariantNeuralOperator.load_pretrained("gravineural_v1.0.0")

stress_energy = [[1,0,0,0],[0,0,0,0],[0,0,0,0],[0,0,0,0]]

coordinates = [0, 10, 0, 0]

result = cno.compute_metric(stress_energy, coordinates)

print(f"EFE Residual: {result.ef_e_residual:.4f} [{result.status}]")

EFE Residual Alert Levels

EFE < 0.01
EXCELLENT

0.01–0.02
GOOD

0.02–0.05
MODERATE

0.05–0.10
CRITICAL

> 0.10
COLLAPSE

EXCELLENT: Standard curvature monitoring

GOOD: Periodic field equation review

MODERATE: Waveform retraining required

CRITICAL: Emergency metric recalibration

COLLAPSE: Immediate spacetime recovery

Three Gravitational Validation Regimes

0.28%

Binary Black Hole

R1 · 14,000 waveforms · mass ratio 1-8

0.33%

Binary Neutron Star

R2 · 3,200 waveforms · tidal deformability

0.35%

Core-Collapse Supernova

R3 · 780 snapshots · progenitor mass 10-30 M_☉

Quick Setup

            
# Clone repository

git clone https://gitlab.com/gitdeeper11/GRAVI-NEURAL.git

cd GRAVI-NEURAL

# Install package

pip install -e .

# Run analysis

python bin/compute_metric.py --spacetime schwarzschild --verbose

# Verify installation

python -c "from gravineural import __version__; print(__version__)"

Physics-Informed Neural Network

            
# PINN penalty layer constraints

# • Einstein Field Equation compliance

# • Bianchi identity as hard constraint

# • Hamiltonian constraint enforcement

# Python implementation

from gravineural import GRAVIPredictor

predictor = GRAVIPredictor()

result = predictor.predict(stress_energy, coordinates)

How to Cite

            
@software{baladi2026gravineural,

    author = {Samir Baladi},

    title = {GRAVI-NEURAL: Covariant Neural Characterization of

    Metric Tensor Perturbations in Dynamic Gravitational Environments},

    year = {2026},

    version = {1.0.0},

    doi = {10.5281/zenodo.19871822},

    note = {Physics-Informed AI Framework for General Relativity}

}