Source-linked AI summary
PINNeik: Eikonal solution using physics-informed neural networks
Umair bin Waheed, Ehsan Haghighat, Tariq Alkhalifah, Chao Song, Qi Hao
TL;DR
Existing eikonal solvers can require substantial changes for added physics and remain costly for repeated traveltime computations. This paper trains a physics-informed neural network using the eikonal equation as a loss, achieving accurate traveltimes while using transfer learning and surrogate modeling for faster updates. The method also supports anisotropy and topography through the governing loss formulation.
Problem
Conventional eikonal solvers may require significant modifications for additional physics and repeated computations can be costly for updated velocity models or source locations.
Method
A feed-forward neural network is trained by minimizing the residual of the governing eikonal equation at selected training points.
Results
The solver produces sufficiently accurate traveltimes for most seismic applications and can use transfer learning and source-location surrogate models to accelerate repeated evaluations.
Takeaways & Limitations
The PINN framework offers a flexible forward-modeling approach that can incorporate anisotropy, topography, updated velocity models, and new source locations.
Takeaways & Limitations
Further study is needed to meet practical robustness and efficiency requirements, and the framework was demonstrated on 2D models.
Abstract
from arXiv · showhide
The eikonal equation is utilized across a wide spectrum of science and engineering disciplines. In seismology, it regulates seismic wave traveltimes needed for applications like source localization, imaging, and inversion. Several numerical algorithms have been developed over the years to solve the eikonal equation. However, these methods require considerable modifications to incorporate additional physics, such as anisotropy, and may even breakdown for certain complex forms of the eikonal equation, requiring approximation methods. Moreover, they suffer from computational bottleneck when repeated computations are needed for perturbations in the velocity model and/or the source location, particularly in large 3D models. Here, we propose an algorithm to solve the eikonal equation based on the emerging paradigm of physics-informed neural networks (PINNs). By minimizing a loss function formed by imposing the eikonal equation, we train a neural network to output traveltimes that are consistent with the underlying partial differential equation. We observe sufficiently high traveltime accuracy for most applications of interest. We also demonstrate how the proposed algorithm harnesses machine learning techniques like transfer learning and surrogate modeling to speed up traveltime computations for updated velocity models and source locations. Furthermore, we use a locally adaptive activation function and adaptive weighting of the terms in the loss function to improve convergence rate and solution accuracy. We also show the flexibility of the method in incorporating medium anisotropy and free-surface topography compared to conventional methods that require significant algorithmic modifications. These properties of the proposed PINN eikonal solver are highly desirable in obtaining a flexible and efficient forward modeling engine for seismological applications.
1. Introduction
The eikonal equation is a nonlinear PDE used across wave propagation, geometric optics, and many scientific and engineering applications. Conventional solvers such as fast marching and fast sweeping motivate a PINN-based approach that learns solutions from the governing PDE and supports repeated computations and additional physics.
- Background: The eikonal equation is a first-order nonlinear PDE arising in wave propagation and geometric optics.It can be derived from the first term of the WKB expansion or from Huygens’ principle.
- Applications: Applications include image processing, robotics, computer graphics, and seismological tasks such as tomography, source localization, and migration.These uses include distance-field computation, path planning, geo-time fields, and seismic imaging workflows.
- Conventional solvers: Fast marching is a single-pass solver, whereas fast sweeping uses Gauss–Seidel iterations with alternating sweep directions.Fast sweeping converges in a finite number of iterations irrespective of grid size.
- Proposed approach: The paper proposes training a deep neural network with a loss function formed from the underlying eikonal PDE to produce its solution.This physics-informed approach restricts admissible solutions through the governing physical equation rather than relying only on labeled data.
- Proposed approach: The proposed framework investigates transfer learning and surrogate modeling to accelerate repeated traveltime computations for changed velocity models or source locations.It also targets incorporation of additional physics and surface topography without the special treatments required by conventional methods.
2. Theory
This section formulates the eikonal equation and factorizes traveltime to remove the point-source singularity, then introduces neural-network approximation, automatic differentiation, and physics-informed training. The proposed workflow minimizes the factored eikonal residual at collocation points while enforcing positivity and the source condition.
- Eikonal equation: The eikonal equation is a first-order nonlinear PDE whose traveltime gradient is inversely proportional to wave velocity.For isotropic media, velocity does not depend on wave-propagation direction.
- Eikonal equation: Factoring traveltime as T(x) = T_0(x)τ(x) makes the unknown factor smooth near the point source and mitigates its singularity.T_0 is known analytically, while τ is the unknown function.
- Deep feed-forward neural networks: A feed-forward neural network represents the mapping from spatial input x to the unknown traveltime factor τ(x) through sequential layer evaluations.Its layers use weighted inputs, biases, and element-wise activation functions; universal approximation motivates this representation.
- Solving the eikonal equation: Finding weights and biases that solve a given PDE remains challenging despite neural networks’ theoretical ability to represent complex functions.The limitation concerns parameter determination rather than representational capacity.
- Automatic differentiation: Automatic differentiation computes the network-output derivatives required by PINNs using forward and reverse passes without numerical approximation error.The section contrasts automatic differentiation with hand-coded, symbolic, and numerical differentiation.
3. Numerical Tests
Numerical tests evaluate the PINN eikonal solver across synthetic, heterogeneous, anisotropic, and irregular-topography models. The tests examine accuracy, convergence improvements, transfer learning, and surrogate modeling for repeated traveltime computations.
- Test cases: The study tests the solver on heterogeneous, anisotropic, irregular-topography, and Marmousi velocity models.These cases assess both accuracy and flexibility across increasingly complex settings.
- Convergence strategies: Locally adaptive activation functions and adaptive loss-term weights markedly improve convergence over the standard PINN approach.The study uses these techniques in all reported examples to accelerate convergence.
- Accuracy: Using 25% of grid points, the PINN solution is significantly more accurate than first-order fast sweeping for the benchmark model.Fast-sweeping errors are large in the diagonal direction, whereas PINN errors are more randomly distributed.
- Repeated computations: Transfer learning produces faster convergence for a relocated source and updated velocity model without compromising solution accuracy.The pretrained network converges much faster than random initialization, while the resulting errors remain competitive with first-order fast sweeping.
- Repeated computations: A surrogate model accurately computes traveltimes for an unseen source location without additional training.The test source is deliberately placed farthest from the training sources to probe the surrogate model’s accuracy limits.
- Flexibility: PINNs handle elliptical anisotropy and irregular topography without the special treatment required by conventional solvers.Only grid points below the free surface are used for training and evaluation in the topographic case.
- Limitations: The Marmousi model exhibits considerably slower convergence because its sharp velocity variations require learning many local high-frequency features.The authors state that further advances are needed to make PINNs computationally feasible for such highly heterogeneous models.
4. Discussion
The paper presents a physics-informed neural-network solver for the eikonal equation, using PDE-residual training to obtain accurate traveltimes and machine-learning strategies for repeated computations. It also emphasizes flexibility for anisotropy and topography while noting unresolved efficiency and robustness challenges.
- Method: The proposed solver trains a feed-forward neural network by minimizing the residual of the governing eikonal equation at selected training points.Automatic differentiation is used to evaluate the PDE-based loss.
- Repeated computations: Transfer learning accelerates convergence for new source locations or updated velocity models by initializing the network with previously trained weights.A surrogate model can additionally use source locations as inputs to produce traveltime solutions for new sources.
- Repeated computations: 92.7 s of surrogate-model training and 0.11 s per new source evaluation make PINN solutions about 13.5 times faster than fast sweeping after 67 source locations.The fast sweeping solver requires about 1.5 s for one solution in this comparison.
- Accuracy and derivatives: The surrogate solution is considerably more accurate, while traveltime derivatives are obtained as a by-product of PINN training rather than computed separately.These derivatives support quantities such as amplitudes and take-off angles.
- Flexibility: The mesh-free framework simplifies incorporating anisotropic eikonal equations, surface topography, and source locations not restricted to a regular grid.Conventional algorithms require more adaptation for different computational platforms or architectures and special treatment for some extensions.
- Challenges: Convergence can be slow for sharply varying velocity models, and computational advantage depends on architecture, optimization, sampling, model complexity, and source count.The authors call for further study to establish robustness, efficiency, and comparative computational cost in practice.
5. Conclusions
The proposed deep-learning eikonal solver achieves high accuracy on benchmark models and supports efficient reuse across velocity models and source locations. The authors nevertheless present robustness and practical efficiency as areas requiring further study.
- Accuracy: The approach achieves better accuracy than the first-order fast sweeping solution on benchmark synthetic models.Training is needed for only a fraction of grid points, depending on velocity-model heterogeneity, to reliably reconstruct the solution.
- Reuse and acceleration: Transfer learning speeds convergence for new velocity models or source locations, while surrogate modeling can instantly provide traveltimes for new sources after training on several source solutions.These capabilities are not afforded by conventional numerical algorithms in the same way.
- Scope and limitations: The framework was demonstrated on 2D models, and further study is needed to reach the robustness and efficiency required in practice.The authors state that extension to 3D velocity models is straightforward.
CRediT authorship contribution statement
The contribution statement assigns distinct roles across conceptualization, methodology, validation, visualization, software, supervision, and writing.
- Umair bin Waheed led conceptualization, methodology, validation, visualization, software, and original-draft writing.
- Ehsan Haghighat contributed to conceptualization, methodology, software, and original-draft writing.
- Tariq Alkhalifah provided supervision, conceptualization, validation, and writing review and editing.
- Chao Song contributed methodology, visualization, and writing review and editing, while Qi Hao contributed validation and writing review and editing.