Publications

2026

1. arXiv

   Enforcing Reciprocity in Operator Learning for Seismic Wave Propagation

   Caifeng Zou , Yaozhong Shi , Zachary E. Ross, and 2 more authors

   2026

   Abs arXiv Bib

   Accurate and efficient wavefield modeling underpins seismic structure and source studies. Traditional methods comply with physical laws but are computationally intensive. Data-driven methods, while opening new avenues for advancement, have yet to incorporate strict physical consistency. The principle of reciprocity is one of the most fundamental physical laws in wave propagation. We introduce the Reciprocity-Enforced Neural Operator (RENO), a transformerbased architecture for modeling seismic wave propagation that hard-codes the reciprocity principle. The model leverages the cross-attention mechanism and commutative operations to guarantee invariance under swapping source and receiver positions. Beyond improved physical consistency, the proposed architecture supports simultaneous realizations for multiple sources without crosstalk issues. This yields an order-of-magnitude inference speedup at a similar memory footprint over an reciprocity-unenforced neural operator on a realistic configuration. We demonstrate the functionality using the reciprocity relation for particle velocity fields under single forces. This architecture is also applicable to pressure fields under dilatational sources and travel-time fields governed by the eikonal equation, paving the way for encoding more complex reciprocity relations.

   @misc{zou2026enforcingreciprocityoperatorlearning,
     title = {Enforcing Reciprocity in Operator Learning for Seismic Wave Propagation},
     author = {Zou, Caifeng and Shi, Yaozhong and Ross, Zachary E. and Clayton, Robert W. and Azizzadenesheli, Kamyar},
     year = {2026},
     archiveprefix = {arXiv},
     primaryclass = {physics.geo-ph},
     url = {https://arxiv.org/abs/2602.11631}
   }

2025

1. ICLR

   InverseBench: Benchmarking Plug-and-Play Diffusion Models for Scientific Inverse Problems

   Hongkai Zheng , Wenda Chu , Bingliang Zhang, and 9 more authors

   In The Thirteenth International Conference on Learning Representations, 2025

   Abs Bib

   Plug-and-play diffusion prior methods have emerged as a promising research direction for solving inverse problems. However, current studies primarily focus on natural image restoration, leaving the performance of these algorithms in scientific inverse problems largely unexplored. To address this gap, we introduce \textscInverseBench, a unified framework that evaluates diffusion models across five distinct scientific inverse problems. These problems present unique structural challenges that differ from existing benchmarks, arising from critical scientific applications such as black hole imaging, seismology, optical tomography, medical imaging, and fluid dynamics. With \textscInverseBench, we benchmark 15 inverse problem algorithms that use plug-and-play diffusion prior methods against strong, domain-specific baselines, offering valuable new insights into the strengths and weaknesses of existing algorithms. We open-source the datasets, pre-trained models, and the codebase to facilitate future research and development.

   @inproceedings{zheng2025inversebench,
     title = {InverseBench: Benchmarking Plug-and-Play Diffusion Models for Scientific Inverse Problems},
     author = {Zheng, Hongkai and Chu, Wenda and Zhang, Bingliang and Wu, Zihui and Wang, Austin and Feng, Berthy and Zou, Caifeng and Sun, Yu and Kovachki, Nikola Borislavov and Ross, Zachary E and Bouman, Katherine and Yue, Yisong},
     booktitle = {The Thirteenth International Conference on Learning Representations},
     year = {2025},
     url = {https://openreview.net/forum?id=U3PBITXNG6}
   }

2. SRL

   Reducing Frequency Bias of Fourier Neural Operators in 3D Seismic Wavefield Simulations Through Multistage Training

   Qingkai Kong , Caifeng Zou , Youngsoo Choi, and 5 more authors

   Seismological Research Letters, Jul 2025

   Abs DOI Bib

   The recent development of neural operator (NeurOp) learning for solutions to the elastic wave equation shows promising results and provides the basis for fast large‐scale simulations for different seismological applications. In this article, we use the Fourier neural operator (FNO) model to directly solve the 3D Helmholtz wave equation for fast seismic ground‐motion simulations on different frequencies and show the frequency bias of the FNO model, that is, it learns the lower frequencies better comparing to the higher frequencies. To reduce the frequency bias, we adopt the multistage FNO training, that is, after training a stage 1 FNO model for estimating the ground motion, we use a second FNO model as the stage 2 to learn from the residual, which greatly reduced the errors on the higher frequencies. By adopting this multistage training, the FNO models show reduced biases on higher frequencies, which enhanced the overall results of the ground‐motion simulations. Thus the multistage training FNO improves the accuracy and realism of the ground‐motion simulations.

   @article{10.1785/0220250085,
     author = {Kong, Qingkai and Zou, Caifeng and Choi, Youngsoo and Matzel, Eric M. and Azizzadenesheli, Kamyar and Ross, Zachary E. and Rodgers, Arthur J. and Clayton, Robert W.},
     title = {Reducing Frequency Bias of Fourier Neural Operators in 3D Seismic Wavefield Simulations Through Multistage Training},
     journal = {Seismological Research Letters},
     volume = {97},
     number = {1},
     pages = {272-282},
     year = {2025},
     month = jul,
     issn = {0895-0695},
     doi = {10.1785/0220250085},
     url = {https://doi.org/10.1785/0220250085},
   }

3. JGR Solid Earth

   Ambient Noise Full Waveform Inversion With Neural Operators

   Caifeng Zou , Zachary E. Ross , Robert W. Clayton, and 2 more authors

   Journal of Geophysical Research: Solid Earth, 2025

   e2025JB031624 2025JB031624

   Abs DOI Bib

   Numerical simulations of seismic wave propagation are crucial for investigating velocity structures and improving seismic hazard assessment. However, standard methods such as finite difference or finite element are computationally expensive. Recent studies have shown that a new class of machine learning models, called neural operators, can solve the elastodynamic wave equation orders of magnitude faster than conventional methods. Full waveform inversion is a prime beneficiary of the accelerated simulations. Neural operators, as end-to-end differentiable operators, combined with automatic differentiation, provide an alternative approach to the adjoint-state method. State-of-the-art optimization techniques built into PyTorch provide neural operators with greater flexibility to improve the optimization dynamics of full waveform inversion, thereby mitigating cycle-skipping problems. In this study, we demonstrate the first application of neural operators for full waveform inversion on a real seismic data set, which consists of several nodal transects collected across the San Gabriel, Chino, and San Bernardino basins in the Los Angeles metropolitan area.

   @article{https://doi.org/10.1029/2025JB031624,
     author = {Zou, Caifeng and Ross, Zachary E. and Clayton, Robert W. and Lin, Fan-Chi and Azizzadenesheli, Kamyar},
     title = {Ambient Noise Full Waveform Inversion With Neural Operators},
     journal = {Journal of Geophysical Research: Solid Earth},
     volume = {130},
     number = {11},
     pages = {e2025JB031624},
     keywords = {full waveform inversion, neural operators, seismic interferometry, sedimentary basins, linear nodal arrays, automatic differentiation},
     doi = {https://doi.org/10.1029/2025JB031624},
     url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2025JB031624},
     note = {e2025JB031624 2025JB031624},
     year = {2025}
   }

2024

1. GJI

   Deep neural Helmholtz operators for 3-D elastic wave propagation and inversion

   Caifeng Zou , Kamyar Azizzadenesheli , Zachary E Ross, and 1 more author

   Geophysical Journal International, Sep 2024

   Abs DOI Bib

   Numerical simulations of seismic wave propagation in heterogeneous 3-D media are central to investigating subsurface structures and understanding earthquake processes, yet are computationally expensive for large problems. This is particularly problematic for full-waveform inversion (FWI), which typically involves numerous runs of the forward process. In machine learning there has been considerable recent work in the area of operator learning, with a new class of models called neural operators allowing for data-driven solutions to partial differential equations. Recent work in seismology has shown that when neural operators are adequately trained, they can significantly shorten the compute time for wave propagation. However, the memory required for the 3-D time domain equations may be prohibitive. In this study, we show that these limitations can be overcome by solving the wave equations in the frequency domain, also known as the Helmholtz equations, since the solutions for a set of frequencies can be determined in parallel. The 3-D Helmholtz neural operator is 40 times more memory-efficient than an equivalent time-domain version. We use a Helmholtz neural operator for 2-D and 3-D elastic wave modelling, achieving two orders of magnitude acceleration compared to a baseline spectral element method. The neural operator accurately generalizes to variable velocity structures and can be evaluated on denser input meshes than used in the training simulations. We also show that when solving for wavefields strictly at the free surface, the accuracy can be significantly improved via a graph neural operator layer. In leveraging automatic differentiation, the proposed method can serve as an alternative to the adjoint-state approach for 3-D FWI, reducing the computation time by a factor of 350.

   @article{10.1093/gji/ggae342,
     author = {Zou, Caifeng and Azizzadenesheli, Kamyar and Ross, Zachary E and Clayton, Robert W},
     title = {Deep neural Helmholtz operators for 3-D elastic wave propagation and inversion},
     journal = {Geophysical Journal International},
     volume = {239},
     number = {3},
     pages = {1469-1484},
     year = {2024},
     month = sep,
     issn = {1365-246X},
     doi = {10.1093/gji/ggae342},
     url = {https://doi.org/10.1093/gji/ggae342},
   }

2. SRL

   Imaging the Northern Los Angeles Basins with Autocorrelations

   Caifeng Zou and Robert W. Clayton

   Seismological Research Letters, Nov 2024

   Abs DOI Bib

   We show reflectivity cross sections for the San Gabriel, Chino, and San Bernardino basins north of Los Angeles (LA), California, determined from autocorrelations of ambient noise and teleseismic earthquake waves. These basins are thought to channel the seismic energy from earthquakes on the San Andreas fault to LA, and a more accurate model of their depth is important for hazard mitigation. We use the causal side of the autocorrelation function (ACF) to determine the zero‐offset reflection response. To minimize the smoothing effect of the source time function, we remove the common mode from the autocorrelation to reveal the zero‐offset reflection response. We apply this to 10 temporary nodal lines consisting of a total of 758 geophones with an intraline spacing of 250–300 m. We also show that the ACF from teleseismic events can provide illumination on the subsurface that is consistent with ambient noise. Both autocorrelation results compare favorably to receiver functions.

   @article{10.1785/0220240140,
     author = {Zou, Caifeng and Clayton, Robert W.},
     title = {Imaging the Northern Los Angeles Basins with Autocorrelations},
     journal = {Seismological Research Letters},
     volume = {96},
     number = {3},
     pages = {1791-1801},
     year = {2024},
     month = nov,
     issn = {0895-0695},
     doi = {10.1785/0220240140},
     url = {https://doi.org/10.1785/0220240140},
   }

2023

1. Geophysics

   A comparison of machine learning methods to predict porosity in carbonate reservoirs from seismic-derived elastic properties

   Caifeng Zou , Luanxiao Zhao , Fei Hong, and 3 more authors

   Geophysics, Mar 2023

   Abs DOI Bib

   Porosity prediction from seismic data in carbonate reservoirs is challenging because the common presence of heterogeneities in carbonates makes it difficult to establish a clear physical relationship between reservoir properties and elastic responses. Regarding the strong nonlinearities underlying the relationship, machine learning is considered to be a good alternative to traditional methods. We compare several representative supervised machine learning algorithms (light gradient boosting machine [LightGBM], extreme gradient boosting, categorical boosting, random forest, multilayer perceptron, and convolutional neural network) in terms of predictive accuracy and runtime for crosswell blind tests and seismic prediction in a heterogeneous carbonate reservoir, offshore Brazil. The machine learning models are trained with the porosity and elastic parameters (P-impedance and VP/VS ratio) from the smoothed and standardized logging data. Then, we apply the trained model on the inverted elastic properties to predict the porosity profile from seismic data. In the crosswell blind tests for the studied reservoir, LightGBM clearly stands out from the compared machine learning methods with the highest predictive accuracy and the shortest runtime, showing potential for fast and reliable porosity prediction from seismic data. In addition, we analyze the geologic factors (clay content, oil saturation, and relative depth) that possibly affect the predictive accuracy. We find that, with the constraints from clay content and fluids saturation, the performance of porosity prediction from elastic parameters can be further improved to a certain degree.

   @article{10.1190/geo2021-0342.1,
     author = {Zou, Caifeng and Zhao, Luanxiao and Hong, Fei and Wang, Yirong and Chen, Yuanyuan and Geng, Jianhua},
     title = {A comparison of machine learning methods to predict porosity in carbonate reservoirs from seismic-derived elastic properties},
     journal = {Geophysics},
     volume = {88},
     number = {2},
     pages = {B101-B120},
     year = {2023},
     month = mar,
     issn = {0016-8033},
     doi = {10.1190/geo2021-0342.1},
     url = {https://doi.org/10.1190/geo2021-0342.1},
   }

2021

1. JGR Solid Earth

   Porosity Prediction With Uncertainty Quantification From Multiple Seismic Attributes Using Random Forest

   Caifeng Zou , Luanxiao Zhao , Minghui Xu, and 2 more authors

   Journal of Geophysical Research: Solid Earth, 2021

   e2021JB021826 2021JB021826

   Abs DOI Bib

   Inferring porosity of subsurface from seismic data is of profound significance to many fields of Earth science and engineering applications, including but not limited to: hydrocarbon reservoir characterization, underground water flow modeling, geological CO2 storage, and geothermal energy exploitation. Traditional model-driven approaches confront the problems of strong nonlinearity and geological heterogeneity, while machine learning is good at nonlinear mapping, providing higher efficiency and accuracy as well. We propose a Random Forest (RF) based method using multiple seismic attributes to predict the underground porosity distribution with uncertainty quantification. The standard deviation of base models’ predictions is used to quantify the regression uncertainty of RF. The uncertainty can robustly indicate the prediction quality in numerous experiments, where low uncertainty corresponds to relatively precise prediction and high uncertainty gives a possibility of larger errors. Furthermore, we utilize the quantified uncertainty to improve the RF regression accuracy by correcting the originally predicted porosity according to the statistical relationship between the absolute error and the standard deviation. The application of the proposed method on seismic data shows its potential to characterize spatially varying reservoir parameters, and the quantified uncertainty profile offers insights into risk evaluation for hydrocarbon exploration and development.

   @article{https://doi.org/10.1029/2021JB021826,
     author = {Zou, Caifeng and Zhao, Luanxiao and Xu, Minghui and Chen, Yuanyuan and Geng, Jianhua},
     title = {Porosity Prediction With Uncertainty Quantification From Multiple Seismic Attributes Using Random Forest},
     journal = {Journal of Geophysical Research: Solid Earth},
     volume = {126},
     number = {7},
     pages = {e2021JB021826},
     keywords = {Machine Learning, porosity prediction, seismic attributes, uncertainty},
     doi = {https://doi.org/10.1029/2021JB021826},
     url = {https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2021JB021826},
     note = {e2021JB021826 2021JB021826},
     year = {2021}
   }

2. Geophysics

   Fluid and lithofacies prediction based on integration of well-log data and seismic inversion: A machine-learning approach

   Luanxiao Zhao , Caifeng Zou , Yuanyuan Chen, and 4 more authors

   Geophysics, Jul 2021

   Abs DOI Bib

   Seismic prediction of fluid and lithofacies distribution is of great interest to reservoir characterization, geologic model building, and flow unit delineation. Inferring fluids and lithofacies from seismic data under the framework of machine learning is commonly subject to issues of limited features, imbalanced data sets, and spatial constraints. As a consequence, an extreme gradient boosting-based workflow, which takes feature engineering, data balancing, and spatial constraints into account, is proposed to predict the fluid and lithofacies distribution by integrating well-log and seismic data. The constructed feature set based on simple mathematical operations and domain knowledge outperforms the benchmark group consisting of conventional elastic attributes of P-impedance and VP/VS ratio. A radial basis function characterizing the weights of training samples according to the distances from the available wells to the target region is developed to impose spatial constraints on the model training process, significantly improving the prediction accuracy and reliability of gas sandstone. The strategy combining the synthetic minority oversampling technique and spatial constraints further increases the F1 score of gas sandstone and also benefits the overall prediction performance of all of the facies. The application of the combined strategy on prestack seismic inversion results generates a more geologically reasonable spatial distribution of fluids, thus verifying the robustness and effectiveness of our workflow.

   @article{10.1190/geo2020-0521.1,
     author = {Zhao, Luanxiao and Zou, Caifeng and Chen, Yuanyuan and Shen, Wenlong and Wang, Yirong and Chen, Huaizhen and Geng, Jianhua},
     title = {Fluid and lithofacies prediction based on integration of well-log data and seismic inversion: A machine-learning approach},
     journal = {Geophysics},
     volume = {86},
     number = {4},
     pages = {M151-M165},
     year = {2021},
     month = jul,
     issn = {0016-8033},
     doi = {10.1190/geo2020-0521.1},
     url = {https://doi.org/10.1190/geo2020-0521.1},
   }