# Microseismic source imaging using physics-informed neural networks with hard constraints

arXiv: https://arxiv.org/abs/2304.04315 · HTML: https://mlsanity.com/paper/2304.04315/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 52 references that are in the map.
- First posted on arXiv 9 Apr 2023.

## Builds on (most cited first, 5 of 5)

- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Implicit Full Waveform Inversion with Deep Neural Representation](https://mlsanity.com/paper/2209.03525/) (2022)
- [PINNup: Robust neural network wavefield solutions using frequency upscaling and neuron splitting](https://mlsanity.com/paper/2109.14536/) (2021)
- [MLReal: Bridging the gap between training on synthetic data and real data applications in machine learning](https://mlsanity.com/paper/2109.05294/) (2021)

## Built on by (most cited first, 5 of 5)

- [LatentPINNs: Generative physics-informed neural networks via a latent representation learning](https://mlsanity.com/paper/2305.07671/) (2023)
- [Joint Microseismic Event Detection and Location with a Detection Transformer](https://mlsanity.com/paper/2307.09207/) (2023)
- [Physics-Informed Neural Networks for Joint Source and Parameter Estimation in Advection-Diffusion Equations](https://mlsanity.com/paper/2512.07755/) (2025)
- [Double Coupling Architecture and Training Method for Optimization Problems of Differential Algebraic Equations with Parameters](https://mlsanity.com/paper/2603.22724/) (2026)
- [Generative wave propagator](https://mlsanity.com/paper/2607.04440/) (2026)

## Abstract (arXiv)

Microseismic source imaging plays a significant role in passive seismic monitoring. However, such a process is prone to failure due to aliasing when dealing with sparsely measured data. Thus, we propose a direct microseismic imaging framework based on physics-informed neural networks (PINNs), which can generate focused source images, even with very sparse recordings. We use the PINNs to represent a multi-frequency wavefield and then apply inverse Fourier transform to extract the source image. To be more specific, we modify the representation of the frequency-domain wavefield to inherently satisfy the boundary conditions (the measured data on the surface) by means of a hard constraint, which helps to avoid the difficulty in balancing the data and PDE losses in PINNs. Furthermore, we propose the causality loss implementation with respect to depth to enhance the convergence of PINNs. The numerical experiments on the Overthrust model show that the method can admit reliable and accurate source imaging for single- or multiple- sources and even in passive monitoring settings. Compared with the time-reversal method, the results of the proposed method are consistent with numerical methods but less noisy. Then, we further apply our method to hydraulic fracturing monitoring field data, and demonstrate that our method can correctly image the source with fewer artifacts.