# Multiple-Input Fourier Neural Operator (MIFNO) for source-dependent 3D elastodynamics

arXiv: https://arxiv.org/abs/2404.10115 · HTML: https://mlsanity.com/paper/2404.10115/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 47 references that are in the map.
- First posted on arXiv 15 Apr 2024.

## Builds on (most cited first, 11 of 11)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Frequency Bias in Neural Networks for Input of Non-Uniform Density](https://mlsanity.com/paper/2003.04560/) (2020)
- [Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning](https://mlsanity.com/paper/2305.19913/) (2023)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [Graph Neural Networks for Multivariate Time Series Regression with Application to Seismic Data](https://mlsanity.com/paper/2201.00818/) (2022)
- [Fourier Neural Operator Surrogate Model to Predict 3D Seismic Waves Propagation](https://mlsanity.com/paper/2304.10242/) (2023)

## Built on by (most cited first, 8 of 8)

- [Integrating Fourier Neural Operators with Diffusion Models to improve Spectral Representation of Synthetic Earthquake Ground Motion Response](https://mlsanity.com/paper/2504.00757/) (2025)
- [Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling](https://mlsanity.com/paper/2510.00233/) (2025)
- [Enforcing Reciprocity in Operator Learning for Seismic Wave Propagation](https://mlsanity.com/paper/2602.11631/) (2026)
- [Towards real-time assessment of infrasound event detection capability using deep learning-based transmission loss estimation](https://mlsanity.com/paper/2506.06358/) (2025)
- [Constitutive parameterized deep energy method for solid mechanics problems with random material parameters](https://mlsanity.com/paper/2603.26030/) (2026)
- [Correcting Neural Operator Spectral Bias via Diffusion Posterior Sampling with Sparse Observations](https://mlsanity.com/paper/2606.03936/) (2026)
- [An Agentic Interface for End-to-End Probabilistic Seismic Hazard and Risk Analysis](https://mlsanity.com/paper/2607.16249/) (2026)
- [A Research Prototype for Closed-Loop Generative Design of Customized Foot Orthoses via Semantic-Physics Alignment](https://mlsanity.com/paper/2607.16631/) (2026)

## Abstract (arXiv)

Numerical simulations are essential tools to evaluate the solution of the wave equation in complex settings, such as three-dimensional (3D) domains with heterogeneous properties. However, their application is limited by high computational costs and existing surrogate models lack the flexibility of numerical solvers. This work introduces the Multiple-Input Fourier Neural Operator (MIFNO) to deal with structured 3D fields representing material properties as well as vectors describing the source characteristics. The MIFNO is applied to the problem of elastic wave propagation in the Earth's crust. It is trained on the HEMEW^S-3D database containing 30000 earthquake simulations in different heterogeneous domains with random source positions and orientations. Outputs are time- and space-dependent surface wavefields. The MIFNO predictions are assessed as good to excellent based on Goodness-Of-Fit (GOF) criteria. Wave arrival times and wave fronts' propagation are very accurate since 80% of the predictions have an excellent phase GOF. The fluctuations amplitudes are good for 87% of the predictions. The envelope score is hindered by the small-scale fluctuations that are challenging to capture due to the complex physical phenomena associated with high-frequency features. Nevertheless, the MIFNO can generalize to sources located outside the training domain and it shows good generalization ability to a real complex overthrust geology. When focusing on a region of interest, transfer learning improves the accuracy with limited additional costs, since GOF scores improved by more than 1 GOF unit with only 500 additional specific samples. The MIFNO is the first surrogate model offering the flexibility of an earthquake simulator with varying sources and material properties. Its good accuracy and massive speed-up offer new perspectives to replace numerical simulations in many-query problems.