NNNN: Neural Networks for Newtonian Noise Mitigation at the Einstein Telescope

Technical Summary: Neural Networks for Newtonian Noise Mitigation at the Einstein Telescope

Problem Statement
Newtonian noise (NN), arising from the gravitational coupling of seismic density fluctuations and ground displacements to the test masses of underground gravitational wave detectors, is projected to be the dominant noise source for the Einstein Telescope (ET) at frequencies below 15 Hz. To mitigate this, it is proposed to predict the NN at the test mass using measurements from an array of surrounding seismometers. While Wiener filters (WF) have been the standard benchmark for this prediction task, they rely on stationarity assumptions that may not hold for transient or non-stationary seismic events. This paper investigates whether deep learning architectures, specifically Convolutional Neural Networks (CNNs) and Graph Neural Networks (GNNs), can outperform Wiener filters in predicting and mitigating NN, particularly in non-stationary scenarios.

Methodology
The authors developed a simulation framework to generate synthetic datasets for training and evaluation.

• Simulation Physics: The simulation models density fluctuations ($\delta\rho$) in a homogeneous medium as monochromatic plane waves and Gaussian wave packets. These fluctuations generate ground displacement fields ($\vec{\xi}$) and the resulting Newtonian force on a mirror. The simulation accounts for both P-waves (longitudinal, causing bulk density fluctuations) and S-waves (shear, causing interface displacements).
• Data Generation: The dataset includes time series of 3D displacements from $N_S$ seismometers and the target NN force at a single mirror. Three distinct data scenarios were created:
  1. Stationary Case: Sums of 10 randomly sampled plane waves.
  2. Dominant Plane Waves: Single plane wave events.
  3. Dominant Wave Packets: Single Gaussian wave packet events (transients).
• Benchmark: A Wiener filter was implemented in the frequency domain, trained on the cross-power spectral density (CPSD) of the seismometer data and the target signal.
• Neural Architectures:
  • CNNs: Utilized spatiotemporal convolutions with separate spatial and temporal kernels. Designed for regular seismometer grids (e.g., cubic or cuboid arrangements).
  • GNNs: Utilized Graph Attention Layers (GATs) to handle irregular seismometer grids. The network ingests an adjacency matrix representing the inverse distances between sensors, allowing for optimized, non-uniform sensor placements.
• Training: Models were trained to minimize mean absolute error between the predicted and true NN force. Seismometer noise (white noise with SNR=15) was added to the input data.

Key Results
The performance was evaluated using the residual ratio ($\sqrt{r_{ind}}$) and the Amplitude Spectral Density (ASD) reduction factors.

1. Stationary Wave Fields:

   • For approximately stationary fields (overlaid plane waves), CNNs and WFs demonstrated comparable performance.
   • Both methods achieved noise suppression factors between 10 and 20.
   • CNNs performed slightly better than WFs (residuals were 1.1 to 1.4 times lower), but the difference was marginal.

2. Non-Stationary/Dominant Events (Plane Waves & Wave Packets):

   • In scenarios dominated by single events (non-stationary), neural networks significantly outperformed the Wiener filter.
   • Plane Waves: CNNs reduced residuals by factors of up to 40 compared to WFs. While WFs achieved suppression factors of only ~2, CNNs reached factors of 10–40.
   • Gaussian Wave Packets: The performance gap widened. CNNs achieved suppression factors of 10–30 (up to a factor of 30 at low frequencies), whereas WFs barely exceeded a factor of 1.6.
   • The ratio of WF to CNN residuals ranged from 15 to 80 depending on the event strength and frequency, with CNNs maintaining residuals in the single-digit to low double-digit percent range, while WF residuals often exceeded 100% (amplification) for low-RMS events.

3. Array Configuration and Architecture:

   • Array Size: Increasing the array from 8 to 32 seismometers improved performance for both methods. However, the WF benefited more from the increased data density than the CNN, suggesting CNNs can extract information efficiently even from smaller arrays.
   • Regular vs. Irregular Grids: GNNs were tested on irregular grids optimized for NN mitigation. While GNNs showed slightly lower performance than CNNs on regular grids (likely due to fewer trainable parameters), they still significantly outperformed WFs on irregular grids.
   • Optimization: Using an optimized irregular grid with GNNs improved performance over a standard regular grid, achieving suppression factors of 9–16 compared to the WF's ~1.5.

Significance and Claims
The paper claims that neural networks are a viable and often superior alternative to Wiener filters for Newtonian noise mitigation, particularly when the seismic wave field is non-stationary.

• Performance: The authors assert that for transient or dominating noise sources, neural networks can suppress NN amplitudes by factors of 10–30, outperforming Wiener filters by factors of 15–80.
• Flexibility: The study highlights the utility of GNNs in handling optimized, irregular seismometer arrays, which are theoretically more effective for NN mitigation than regular grids but difficult to process with standard CNNs.
• Practical Implications: The ability of neural networks to perform well on smaller arrays suggests a potential reduction in the number of required boreholes, thereby lowering the cost of detector construction.
• Limitations and Future Work: The authors modestly note that current GNNs have fewer trainable parameters than their CNN counterparts, which may account for the performance gap. Future work is proposed to include more realistic geological models (layered media, surface waves, mode conversions) and to compare these results against adaptive Wiener filters. The study is based on synthetic data from a simplified homogeneous medium; validation with real subsurface data is identified as a necessary next step.