366 research outputs found
Using deep generative neural networks to account for model errors in Markov chain Monte Carlo inversion
Most geophysical inverse problems are non-linear and rely upon numerical forward solvers involving discretization and simplified representations of the underlying physics. As a result, forward modelling errors are inevitable. In practice, such model errors tend to be either completely ignored, which leads to biased and over-confident inversion results, or only partly taken into account using restrictive Gaussian assumptions. Here, we rely on deep generative neural networks to learn problem-specific low-dimensional probabilistic representations of the discrepancy between high-fidelity and low-fidelity forward solvers. These representations are then used to probabilistically invert for the model error jointly with the target geophysical property field, using the computationally cheap, low-fidelity forward solver. To this end, we combine a Markov chain Monte Carlo (MCMC) inversion algorithm with a trained convolutional neural network of the spatial generative adversarial network (SGAN) type, whereby at each MCMC step, the simulated low-fidelity forward response is corrected using a proposed model-error realization. Considering the crosshole ground-penetrating radar traveltime tomography inverse problem, we train SGAN networks on traveltime discrepancy images between: (1) curved-ray (high fidelity) and straight-ray (low fidelity) forward solvers; and (2) finite-difference-time-domain (high fidelity) and straight-ray (low fidelity) forward solvers. We demonstrate that the SGAN is able to learn the spatial statistics of the model error and that suitable representations of both the subsurface model and model error can be recovered by MCMC. In comparison with inversion results obtained when model errors are either ignored or approximated by a Gaussian distribution, we find that our method has lower posterior parameter bias and better explains the observed traveltime data. Our method is most advantageous when high-fidelity forward solvers involve heavy computational costs and the Gaussian assumption of model errors is inappropriate. Unstable MCMC convergence due to non-linearities introduced by our method remain a challenge to be addressed in future work
Efficient Bayesian travel-time tomography with geologically-complex priors using sensitivity-informed polynomial chaos expansion and deep generative networks
Monte Carlo Markov Chain (MCMC) methods commonly confront two fundamental
challenges: the accurate characterization of the prior distribution and the
efficient evaluation of the likelihood. In the context of Bayesian studies on
tomography, principal component analysis (PCA) can in some cases facilitate the
straightforward definition of the prior distribution, while simultaneously
enabling the implementation of accurate surrogate models based on polynomial
chaos expansion (PCE) to replace computationally intensive full-physics forward
solvers. When faced with scenarios where PCA does not offer a direct means of
easily defining the prior distribution alternative methods like deep generative
models (e.g., variational autoencoders (VAEs)), can be employed as viable
options. However, accurately producing a surrogate capable of capturing the
intricate non-linear relationship between the latent parameters of a VAE and
the outputs of forward modeling presents a notable challenge. Indeed, while PCE
models provide high accuracy when the input-output relationship can be
effectively approximated by relatively low-degree multivariate polynomials,
this condition is typically unmet when utilizing latent variables derived from
deep generative models. In this contribution, we present a strategy that
combines the excellent reconstruction performances of VAE in terms of prio
representation with the accuracy of PCA-PCE surrogate modeling in the context
of Bayesian ground penetrating radar (GPR) travel-time tomography. Within the
MCMC process, the parametrization of the VAE is leveraged for prior exploration
and sample proposal. Concurrently, modeling is conducted using PCE, which
operates on either globally or locally defined principal components of the VAE
samples under examination.Comment: 25 pages, 15 figure
Neural Eikonal Solver: improving accuracy of physics-informed neural networks for solving eikonal equation in case of caustics
The concept of physics-informed neural networks has become a useful tool for
solving differential equations due to its flexibility. There are a few
approaches using this concept to solve the eikonal equation which describes the
first-arrival traveltimes of acoustic and elastic waves in smooth heterogeneous
velocity models. However, the challenge of the eikonal is exacerbated by the
velocity models producing caustics, resulting in instabilities and
deterioration of accuracy due to the non-smooth solution behaviour. In this
paper, we revisit the problem of solving the eikonal equation using neural
networks to tackle the caustic pathologies. We introduce the novel Neural
Eikonal Solver (NES) for solving the isotropic eikonal equation in two
formulations: the one-point problem is for a fixed source location; the
two-point problem is for an arbitrary source-receiver pair. We present several
techniques which provide stability in velocity models producing caustics:
improved factorization; non-symmetric loss function based on Hamiltonian;
gaussian activation; symmetrization. In our tests, NES showed the
relative-mean-absolute error of about 0.2-0.4% from the second-order factored
Fast Marching Method, and outperformed existing neural-network solvers giving
10-60 times lower errors and 2-30 times faster training. The inference time of
NES is comparable with the Fast Marching. The one-point NES provides the most
accurate solution, whereas the two-point NES provides slightly lower accuracy
but gives an extremely compact representation. It can be useful in various
seismic applications where massive computations are required (millions of
source-receiver pairs): ray modeling, traveltime tomography, hypocenter
localization, and Kirchhoff migration.Comment: The paper has 14 pages and 6 figures. Source code is available at
https://github.com/sgrubas/NE
Tracking Black Holes in Numerical Relativity
This work addresses and solves the problem of generically tracking black hole
event horizons in computational simulation of black hole interactions.
Solutions of the hyperbolic eikonal equation, solved on a curved spacetime
manifold containing black hole sources, are employed in development of a robust
tracking method capable of continuously monitoring arbitrary changes of
topology in the event horizon, as well as arbitrary numbers of gravitational
sources. The method makes use of continuous families of level set viscosity
solutions of the eikonal equation with identification of the black hole event
horizon obtained by the signature feature of discontinuity formation in the
eikonal's solution. The method is employed in the analysis of the event horizon
for the asymmetric merger in a binary black hole system. In this first such
three dimensional analysis, we establish both qualitative and quantitative
physics for the asymmetric collision; including: 1. Bounds on the topology of
the throat connecting the holes following merger, 2. Time of merger, and 3.
Continuous accounting for the surface of section areas of the black hole
sources.Comment: 14 pages, 16 figure
Digital twinning of cardiac electrophysiology models from the surface ECG: a geodesic backpropagation approach
The eikonal equation has become an indispensable tool for modeling cardiac
electrical activation accurately and efficiently. In principle, by matching
clinically recorded and eikonal-based electrocardiograms (ECGs), it is possible
to build patient-specific models of cardiac electrophysiology in a purely
non-invasive manner. Nonetheless, the fitting procedure remains a challenging
task. The present study introduces a novel method, Geodesic-BP, to solve the
inverse eikonal problem. Geodesic-BP is well-suited for GPU-accelerated machine
learning frameworks, allowing us to optimize the parameters of the eikonal
equation to reproduce a given ECG. We show that Geodesic-BP can reconstruct a
simulated cardiac activation with high accuracy in a synthetic test case, even
in the presence of modeling inaccuracies. Furthermore, we apply our algorithm
to a publicly available dataset of a rabbit model, with very positive results.
Given the future shift towards personalized medicine, Geodesic-BP has the
potential to help in future functionalizations of cardiac models meeting
clinical time constraints while maintaining the physiological accuracy of
state-of-the-art cardiac models.Comment: 9 pages, 5 figure
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