Mesh-based simulations are central to modeling complex physical systems in many disciplines across science and engineering. Mesh representations support powerful numerical integration methods and their resolution can be adapted to strike favorable trade-offs between accuracy and efficiency. However, high-dimensional scientific simulations are very expensive to run, and solvers and parameters must often be tuned individually to each system studied. Here we introduce MeshGraphNets, a framework for learning mesh-based simulations using graph neural networks. Our model can be trained to pass messages on a mesh graph and to adapt the mesh discretization during forward simulation. Our results show it can accurately predict the dynamics of a wide range of physical systems, including aerodynamics, structural mechanics, and cloth. The model's adaptivity supports learning resolution-independent dynamics and can scale to more complex state spaces at test time. Our method is also highly efficient, running 1-2 orders of magnitude faster than the simulation on which it is trained. Our approach broadens the range of problems on which neural network simulators can operate and promises to improve the efficiency of complex, scientific modeling tasks.

Learning Mesh-Based Simulation with Graph Networks

A goal-oriented adaptive finite-element approach for plane wave 3-D electromagnetic modelling

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SUMMARY We present a nodal finite-element method that can be used to compute in parallel highly accurate solutions for 3-D controlled-source electromagnetic forward-modelling problems in anisotropic media. Secondary coupled-potential formulation of Maxwell’s equations allows to avoid the singularities introduced by the sources, while completely unstructured tetrahedral meshes and mesh refinement support an accurate representation of geological and bathymetric complexity and improve the solution accuracy. Different complex iterative solvers and an efficient pre-conditioner based on the sparse approximate inverse are used for solving the resulting large sparse linear system of equations. Results are compared with the ones of other researchers to check the accuracy of the method. We demonstrate the performance of the code in large problems with tens and even hundreds of millions of degrees of freedom. Scalability tests on massively parallel computers show that our code is highly scalable.

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A parallel finite-element method for three-dimensional controlled-source electromagnetic forward modelling

Just a few meters below the Earth's surface lie features of great importance, from geological faults which can produce devastating earthquakes, to lost archaeological treasures. This refreshing, up-to-date book explores the foundations of interpretation theory and the latest developments in near-surface techniques, used to complement traditional geophysical methods for deep-exploration targets. Clear but rigorous, the book explains theory and practice in simple physical terms, supported by intermediate-level mathematics. Techniques covered include magnetics, resistivity, seismic reflection and refraction, surface waves, induced polarization, self-potential, electromagnetic induction, ground-penetrating radar, magnetic resonance, interferometry, seismoelectric and more. Sections on data analysis and inverse theory are provided and chapters are illustrated by case studies, giving students and professionals the tools to plan, conduct and analyze a near-surface geophysical survey. This is an important textbook for advanced-undergraduate and graduate students in geophysics and a valuable reference for practising geophysicists, geologists, hydrologists, archaeologists, and civil and geotechnical engineers.

Near-Surface Applied Geophysics

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Three-dimensional parallel distributed inversion of CSEM data using a direct forward solver

SUMMARY

We present a new 3-D vector finite element code and demonstrate its strength by modelling a realistic marine CSEM scenario. Unstructured tetrahedral meshes easily allow for the inclusion of arbitrary seafloor bathymetry so that natural environments are mapped into the model in a close-to-reality way. A primary/secondary field approach, an adaptive mesh refinement strategy as well as a higher order polynomial finite element approximation improve the solution accuracy. A convergence study strongly indicates that the use of higher order finite elements is beneficial even if the solution is not globally smooth. The marine CSEM scenario also shows that seafloor topography gives an important response which needs to be reproduced by numerical modelling to avoid the misinterpretation of measurements.

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Three-dimensional adaptive higher order finite element simulation for geo-electromagnetics—a marine CSEM example

SUMMARY We introduce the concept of multi-objective optimization to cast the regularized inverse direct current resistivity problem into a general formulation. This formulation is suitable for the efficient application of a genetic algorithm, which is known as a global and non-linear optimization tool. The genetic inverse algorithm generates a set of solutions reflecting the trade-off between data misfit and some measure of model features. Examination of such an ensemble is highly preferable to classical approaches where just one ‘optimal’ solution is examined since a better overview over the range of possible inverse models is gained. However, the computational cost to obtain this ensemble is enormous. We demonstrate that at the current state of computer performance inversion of 2-D direct current resistivity data using genetic algorithms is possible if state-of-the-art computational techniques such as parallelization and efficient 2-D forward operators are applied.

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Two‐dimensional inversion of direct current resistivity data using a parallel, multi‐objective genetic algorithm

Improved upscaling of steel-cased wells through inversion

When simulating controlled-source electromagnetic data at frequencies significantly lower than 1 Hz, standard formulations based, e.g., on the vector Helmholtz equation for the electric field, may fail to produce faithful results, because the term in the Helmholtz equation involving conductivity decays linearly with frequency. To stabilize low-frequency simulations, we introduce an auxiliary potential, constrained to be zero by boundary conditions, and explicitly enforce a divergence condition. We present an implementation of our stabilization technique within a finite-difference frequency-domain algorithm. The utility of the stabilized approach is demonstrated by showing synthetic data for frequencies as low as 0.001 Hz for a 3D model roughly mimicking the geometry of the CO2 sequestration pilot site in Ketzin, Germany. In these synthetic data, we observe that instability occurring primarily in the air for non-stabilized simulations disappears for stabilized computations. As expected, the auxiliary potential assumes near-zero values, but its amplitude increases with decreasing frequency due to numerical limitations. The amplitude characteristics of the auxiliary potential, and good accuracy of our synthetic data in comparison with 1D simulation results, suggest that the stabilization is usable down to frequencies at which electromagnetic fields can effectively be considered static.

Christoph Schwarzbach

Papers

Three-dimensional adaptive higher order finite element simulation for geo-electromagnetics—a marine CSEM example

Two‐dimensional inversion of direct current resistivity data using a parallel, multi‐objective genetic algorithm

Improved upscaling of steel-cased wells through inversion

Controlled-source Electromagnetic Modelling Studies – Utility of Auxiliary Potentials for Low-frequency Stabilization

Monitoring hydraulic-fracture volume using surface-to-borehole EM and conductive proppant