# Seismic wave propagation and inversion with Neural Operators

arXiv: https://arxiv.org/abs/2108.05421 · HTML: https://mlsanity.com/paper/2108.05421/

## Key facts

- Cited by 51 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 26 references that are in the map.
- First posted on arXiv 11 Aug 2021.

## Builds on (most cited first, 7 of 7)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [EikoNet: Solving the Eikonal equation with Deep Neural Networks](https://mlsanity.com/paper/2004.00361/) (2020)
- [HypoSVI: Hypocenter inversion with Stein variational inference and Physics Informed Neural Networks](https://mlsanity.com/paper/2101.03271/) (2021)

## Built on by (most cited first, 30 of 51)

- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Towards Multi-spatiotemporal-scale Generalized PDE Modeling](https://mlsanity.com/paper/2209.15616/) (2022)
- [Fast Sampling of Diffusion Models via Operator Learning](https://mlsanity.com/paper/2211.13449/) (2022)
- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [Clifford Neural Layers for PDE Modeling](https://mlsanity.com/paper/2209.04934/) (2022)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [Generative Adversarial Neural Operators](https://mlsanity.com/paper/2205.03017/) (2022)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [Learning to Solve PDE-constrained Inverse Problems with Graph Networks](https://mlsanity.com/paper/2206.00711/) (2022)
- [Guided Diffusion Sampling on Function Spaces with Applications to PDEs](https://mlsanity.com/paper/2505.17004/) (2025)
- [Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equation](https://mlsanity.com/paper/2301.11509/) (2023)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (2022)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Solving Seismic Wave Equations on Variable Velocity Models with Fourier Neural Operator](https://mlsanity.com/paper/2209.12340/) (2022)
- [SeismicNet: Physics-informed neural networks for seismic wave modeling in semi-infinite domain](https://mlsanity.com/paper/2210.14044/) (2022)
- [Stochastic Process Learning via Operator Flow Matching](https://mlsanity.com/paper/2501.04126/) (2025)
- [Broadband Ground Motion Synthesis via Generative Adversarial Neural Operators: Development and Validation](https://mlsanity.com/paper/2309.03447/) (2023)
- [Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning](https://mlsanity.com/paper/2506.10973/) (2025)
- [Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagation](https://mlsanity.com/paper/2304.10242/) (2023)
- [Understanding the Expressivity and Trainability of Fourier Neural Operator: A Mean-Field Perspective](https://mlsanity.com/paper/2310.06379/) (2023)
- [Universal Functional Regression with Neural Operator Flows](https://mlsanity.com/paper/2404.02986/) (2024)
- [PaCMO: Partner Dependent Human Motion Generation in Dyadic Human Activity using Neural Operators](https://mlsanity.com/paper/2211.16210/) (2022)
- [Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML](https://mlsanity.com/paper/2301.13331/) (2023)
- [BiLO: Bilevel Local Operator Learning for PDE Inverse Problems](https://mlsanity.com/paper/2404.17789/) (2024)
- [Component Fourier Neural Operator for Singularly Perturbed Differential Equations](https://mlsanity.com/paper/2409.04779/) (2024)
- [Generative Neural Operators of Log-Complexity Can Simultaneously Solve Infinitely Many Convex Programs](https://mlsanity.com/paper/2508.14995/) (2025)

## Abstract (arXiv)

Seismic wave propagation forms the basis for most aspects of seismological research, yet solving the wave equation is a major computational burden that inhibits the progress of research. This is exacerbated by the fact that new simulations must be performed when the velocity structure or source location is perturbed. Here, we explore a prototype framework for learning general solutions using a recently developed machine learning paradigm called Neural Operator. A trained Neural Operator can compute a solution in negligible time for any velocity structure or source location. We develop a scheme to train Neural Operators on an ensemble of simulations performed with random velocity models and source locations. As Neural Operators are grid-free, it is possible to evaluate solutions on higher resolution velocity models than trained on, providing additional computational efficiency. We illustrate the method with the 2D acoustic wave equation and demonstrate the method's applicability to seismic tomography, using reverse mode automatic differentiation to compute gradients of the wavefield with respect to the velocity structure. The developed procedure is nearly an order of magnitude faster than using conventional numerical methods for full waveform inversion.