Physics-Informed XNet for Seismic Ray Tracing

This repository contains the implementation and reproducibility materials for the paper:

Physics-Informed XNet: A Cauchy Activation-Based Neural Architecture for Seismic Ray Tracing under Multiple Initial Conditions

The authors are Rosélia N. da Silva, Lucas T. Marques, Luiz Affonso Guedes and Tiago Barros, from the Department of Computer Engineering and Automation at the Federal University of Rio Grande do Norte (UFRN), Brazil.

The proposed Physics-Informed XNet (PIXNet) is a neural surrogate for seismic ray tracing in heterogeneous velocity models. It predicts the ray state $(x, z, p_x, p_z)$ from the initial position, take-off angle and travel time, while a physics-informed loss constrains the predictions using the Hamiltonian ray equations.

The article PDF is intentionally not included in this repository.

Method overview

PIXNet combines:

• XNet with four hidden layers of 64 neurons and trainable Cauchy activation functions;
• a data loss based on mean squared error;
• a physics loss based on the Hamiltonian ray-tracing equations and automatic differentiation;
• density-based spatial weights that increase the contribution of sparsely sampled regions;
• second-order Runge–Kutta (RK2) ray tracing to generate reference data.

The experiments use the Marmousi velocity model. A two-dimensional B-spline filter smooths the velocity field before ray tracing. The paper evaluates subsampling factors 20, 25, 30, 35 and 40, and uses factor 30 for the main model because it balances predictive accuracy and geological detail. Data are split by complete ray trajectory into training, validation and test partitions to avoid leakage between points from the same ray.

The evaluation reports $R^2$, MAPE, MAE, MSE and RMSE for the four predicted state variables, together with qualitative trajectory comparisons and inference-time measurements. In the reported benchmark, RK2 requires approximately 16.1 s for 323 rays, while the trained PIXNet requires approximately 14.3 ms on the stated workstation.

Repository structure

data/
├── marmousi_vp.bin                 # Raw Marmousi velocity model
└── splits/                         # Generated ray-level data partitions

output/
├── checkpoints/                    # Saved PIXNet and MLP checkpoints
├── figures/                        # Static figures generated by notebooks
├── interactive/                    # Interactive Plotly HTML outputs
└── tables/                         # CSV/LaTeX metrics and tables

src/
├── marmousi_xnet.ipynb             # Main PIXNet experiment
├── marmousi_xnet_mlp_comparison.ipynb
├── marmousi_xnet_test_smooth_factor.ipynb
├── rt_python/                      # RK2 ray tracing and B-spline routines
└── utils/                          # XNet, training, preprocessing and metrics

Large generated datasets, checkpoints and interactive plots are excluded from version control. Static figures and selected tables can be committed for publication or archived as supplementary material.

Environment

The experiments use Python 3.10 and PyTorch. Create the environment with:

conda env create -f conda.yaml
conda activate xnet-ray-tracing

The Marmousi velocity model must be available at data/marmousi_vp.bin. If it is not included in a distribution of this repository, download it from The Marmousi Experience and place it at that path.

Reproducing the experiments

Launch Jupyter from either the repository root or src/:

jupyter lab

Run the notebooks in this order:

1. marmousi_xnet.ipynb trains and evaluates the main PIXNet model.
2. marmousi_xnet_mlp_comparison.ipynb compares PIXNet with a matched MLP baseline.
3. marmousi_xnet_test_smooth_factor.ipynb evaluates the smoothing-factor experiment.

The notebooks resolve the project root automatically, so imports and outputs do not depend on the directory from which Jupyter was launched. They save generated artifacts to data/splits/ and output/.

Implementation

• src/utils/xnet_model.py implements XNet and the parameterized Cauchy activation.
• src/utils/architecture_xnet.py implements PIXNet training, physics-informed loss integration, automatic differentiation, checkpointing and prediction.
• src/utils/architecture_mlp.py implements the MLP comparison baseline.
• src/rt_python/ray_tracing.py generates reference trajectories from the Marmousi model.
• src/rt_python/adjointlib.py contains the B-spline interpolation and ray-tracing numerical routines.