Scientific Machine Learning for Elastic and Acoustic Wave Propagation: Neural Operator and Physics-Guided Neural Network
Abstract
:1. Introduction
2. Data-Intensive SciML Models: Architecture and Algorithms
2.1. Concept of Operator Learning
2.2. DeepONet
Algorithm 1 DeepONet |
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11: else |
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13: end if |
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15: end for |
16: end for |
2.3. Physical Understanding of DeepONet
2.4. Fourier Neural Operator (FNO)
Algorithm 2 FNO |
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8: end for |
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12: end for |
13: end for |
2.5. Physical Understanding of FNOs
2.6. Application Cases of FNO
3. NO Applications in Wave Propagation
3.1. Wave Propagation with DeepONet
3.2. Wave Propagation with FNOs
4. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Aspects | DeepONet | FNO |
---|---|---|
Operator type | Approximates via finite basis expansion | Approximates via Fourier-domain convolution |
Kernel function | Implicit via trunk net basis and branch net coefficients | Explicit via learned Fourier multipliers |
Integral approximation | Discrete latent expansion into number of modes and their contributions | Fourier transform, multiply in spectral space |
Global modeling | Through learned basis functions in trunk net | Through Fourier modes capturing global functional behavior |
Miscellaneous understanding | Very general; works for arbitrary operators | Most efficient when operator is translation-invariant (convolution-like PDEs) |
Parametric PDEs | Yes Input: Parametric function (e.g., wave speed map , initial profile) | Yes Input: Parametric function (e.g., wave speed map , initial profile) |
Nonparametric PDEs | Yes Input: Prior field value (e.g., ) | Yes Input: Prior field value (e.g., ) |
Inverse Problem | No: tough to converge | Yes |
Authors | Year | Key Objectives | Model Architecture | Type of Wave | Dimension | Type of Medium |
---|---|---|---|---|---|---|
Aldirany et al. [91] | 2024 | Transient wave propagation modeling | DeepONet and GreenONet | Acoustic wave | 2D | Homogeneous |
Zhu et al. [94] | 2024 | Full waveform inversion with noise-robust generalization | Fourier DeepONet | Acoustic wave | 2D | Heterogeneous |
Guo et al. [96] | 2024 | Improve generalization across source locations and frequencies | Inversion DeepONet | Acoustic wave | 2D | Heterogeneous |
Li et al. [97] | 2024 | Accelerated global seismic forward modeling and inversion | DeepONet, Physics-Informed DeepONet | Acoustic wave and Elastic wave | 3D | Heterogeneous |
Wagner et al. [98] | 2023 | Fast surrogate modeling of transmission loss in sonic crystals | DeepONet | Acoustic wave | 2D | Homogeneous |
Authors | Year | Key Objectives | Model Architecture | Type of Wave | Dimension | Type of Medium |
---|---|---|---|---|---|---|
Yang et al. [104] | 2021 | Fast inference of 2D seismic wavefields across varying source and velocity | Vanilla FNO | Acoustic wave | 2D | Heterogeneous |
Song and Yang [105] | 2022 | Predicting high-frequency wavefields from low-frequency inputs | Vanilla FNO | Acoustic wave | 2D | Heterogeneous |
Zhang et al. [106] | 2022 | Time extrapolation of wavefields for seismic analysis | Vanilla FNO | Elastic wave | 2D | Heterogeneous |
Li et al. [107] | 2023 | Forward modeling across diverse velocity models for full waveform inversion | Parallel FNO (PFNO) | Acoustic wave | 2D | Heterogeneous |
Lehmann et al. [108] | 2023 | Simulating 3D elastic ground motion for earthquake hazard assessment | U-Shaped FNO (UNO) | Elastic wave | 3D | Heterogeneous |
Kong et al. [109] | 2023 | Real-time simulation of 3D ground motion for subsurface imaging and seismic inversion | UNO and Vanilla FNO | Elastic wave | 3D | Homogeneous |
Middleton et al. [110] | 2023 | Learning long-term acoustic wave propagation from short input in a free-field simulation | Tensorized FNO (TFNO) | Acoustic wave | 2D | Homogeneous |
Rosofsky et al. [111] | 2023 | Surrogate modeling of wave equation | Physics-Informed FNO (PIFNO) | Elastic wave | 1D, 2D | Homogeneous |
Konuk and Shragge [112] | 2023 | Generalizing frequency domain AWE solutions for anisotropic media across frequencies | PIFNO | Acoustic wave | 2D | Anisotropic VTI |
Guan et al. [113] | 2023 | Fast modeling of broadband photoacoustic wave propagation for image reconstruction applications | Vanilla FNO | Acoustic waves | 2D | Homogeneous |
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Mehtaj, N.; Banerjee, S. Scientific Machine Learning for Elastic and Acoustic Wave Propagation: Neural Operator and Physics-Guided Neural Network. Sensors 2025, 25, 3588. https://doi.org/10.3390/s25123588
Mehtaj N, Banerjee S. Scientific Machine Learning for Elastic and Acoustic Wave Propagation: Neural Operator and Physics-Guided Neural Network. Sensors. 2025; 25(12):3588. https://doi.org/10.3390/s25123588
Chicago/Turabian StyleMehtaj, Nafisa, and Sourav Banerjee. 2025. "Scientific Machine Learning for Elastic and Acoustic Wave Propagation: Neural Operator and Physics-Guided Neural Network" Sensors 25, no. 12: 3588. https://doi.org/10.3390/s25123588
APA StyleMehtaj, N., & Banerjee, S. (2025). Scientific Machine Learning for Elastic and Acoustic Wave Propagation: Neural Operator and Physics-Guided Neural Network. Sensors, 25(12), 3588. https://doi.org/10.3390/s25123588