# EikoNet: Solving the Eikonal equation with Deep Neural Networks

arXiv: https://arxiv.org/abs/2004.00361 · HTML: https://mlsanity.com/paper/2004.00361/

## Key facts

- Cited by 50 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Mar 2020.

## Built on by (most cited first, 30 of 50)

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- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
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- [Seismic wave propagation and inversion with Neural Operators](https://mlsanity.com/paper/2108.05421/) (2021)
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- [A Framework for Machine Learning of Model Error in Dynamical Systems](https://mlsanity.com/paper/2107.06658/) (2021)
- [Eikonal Fields for Refractive Novel-View Synthesis](https://mlsanity.com/paper/2202.00948/) (2022)
- [Delta-PINNs: physics-informed neural networks on complex geometries](https://mlsanity.com/paper/2209.03984/) (2022)
- [NTFields: Neural Time Fields for Physics-Informed Robot Motion Planning](https://mlsanity.com/paper/2210.00120/) (2022)
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- [A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements](https://mlsanity.com/paper/2204.00205/) (2022)
- [SeismicNet: Physics-informed neural networks for seismic wave modeling in semi-infinite domain](https://mlsanity.com/paper/2210.14044/) (2022)
- [Physics-informed Value Learner for Offline Goal-Conditioned Reinforcement Learning](https://mlsanity.com/paper/2509.06782/) (2025)
- [Machine learning and domain decomposition methods -- a survey](https://mlsanity.com/paper/2312.14050/) (2023)
- [Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturing](https://mlsanity.com/paper/2405.09572/) (2024)
- [State-of-the-Art Review of Design of Experiments for Physics-Informed Deep Learning](https://mlsanity.com/paper/2202.06416/) (2022)
- [Non-equispaced Fourier Neural Solvers for PDEs](https://mlsanity.com/paper/2212.04689/) (2022)
- [Progressive Learning for Physics-informed Neural Motion Planning](https://mlsanity.com/paper/2306.00616/) (2023)
- [Explorable INR: An Implicit Neural Representation for Ensemble Simulation Enabling Efficient Spatial and Parameter Exploration](https://mlsanity.com/paper/2504.00904/) (2025)
- [The False Promise of Zero-Shot Super-Resolution in Machine-Learned Operators](https://mlsanity.com/paper/2510.06646/) (2025)
- [Physics Informed RNN-DCT Networks for Time-Dependent Partial Differential Equations](https://mlsanity.com/paper/2202.12358/) (2022)
- [Physics-informed Neural Motion Planning on Constraint Manifolds](https://mlsanity.com/paper/2403.05765/) (2024)
- [HypoSVI: Hypocenter inversion with Stein variational inference and Physics Informed Neural Networks](https://mlsanity.com/paper/2101.03271/) (2021)
- [Separable PINN: Mitigating the Curse of Dimensionality in Physics-Informed Neural Networks](https://mlsanity.com/paper/2211.08761/) (2022)
- [Deep Decomposition for Stochastic Normal-Abnormal Transport](https://mlsanity.com/paper/2111.14777/) (2021)
- [Learning Relaxation for Multigrid](https://mlsanity.com/paper/2207.11255/) (2022)
- [Physics-informed neural network for seismic wave inversion in layered semi-infinite domain](https://mlsanity.com/paper/2305.05150/) (2023)

## Abstract (arXiv)

The recent deep learning revolution has created an enormous opportunity for accelerating compute capabilities in the context of physics-based simulations. Here, we propose EikoNet, a deep learning approach to solving the Eikonal equation, which characterizes the first-arrival-time field in heterogeneous 3D velocity structures. Our grid-free approach allows for rapid determination of the travel time between any two points within a continuous 3D domain. These travel time solutions are allowed to violate the differential equation - which casts the problem as one of optimization - with the goal of finding network parameters that minimize the degree to which the equation is violated. In doing so, the method exploits the differentiability of neural networks to calculate the spatial gradients analytically, meaning the network can be trained on its own without ever needing solutions from a finite difference algorithm. EikoNet is rigorously tested on several velocity models and sampling methods to demonstrate robustness and versatility. Training and inference are highly parallelized, making the approach well-suited for GPUs. EikoNet has low memory overhead, and further avoids the need for travel-time lookup tables. The developed approach has important applications to earthquake hypocenter inversion, ray multi-pathing, and tomographic modeling, as well as to other fields beyond seismology where ray tracing is essential.