NO. 45 · Computational & Data

Learned Solvers and Synthetic Wavefields

Networks that make wavefields. Forward modeling as data generation, eikonal nets, DeepONets and Fourier neural operators, and generative velocity priors: simulation as a product, with uncertainty attached.

You can train a network to generate wavefields and travel times, say exactly when a learned solver beats a per-instance one, use generative priors to manufacture plausible velocity models, and put honest uncertainty on everything the machinery produces.

16 competencies · 6 interactive widget challenges · 8 to 13 hours of guided study

For researchers building surrogates and synthetic-data engines for seismology

Draws on: Physics-Informed Neural Networks for Exploration Seismology

Forward modeling as data generation

PINN versus analytic and FDTDwidget challenge

A learned wavefield is only worth having if you know how it compares to the solvers geophysics already trusts; here is the honest head-to-head.

Reads: 1D layered medium: PINN vs analytic 2D smooth velocity: PINN vs FDTD

The cost-accuracy front and failure modes

Every surrogate lives somewhere on the cost-accuracy front, and multi-scale failure is where forward PINNs quietly fall off it.

Reads: The cost-vs-accuracy front Multi-scale failure modes in pure forward PINN

When a learned forward model earns its keep

Meshless domains, many queries, differentiable outputs: the short honest list of places a learned solver actually wins.

Reads: When forward PINN earns its keep

Travel times

The eikonal equation and EikoNetwidget challenge

Travel-time fields are the cheapest synthetic data in seismology, and the eikonal equation is the PDE networks solve most gracefully.

Reads: The eikonal equation and why PINNs love it EikoNet: from clicked source to travel-time field

Factored eikonal and tomography

Removing the source singularity makes the network's job fair, and tomography turns the travel-time trick into a velocity model.

Reads: Factored eikonal: removing the source singularity Travel-time tomography

Microseismic location and joint inversion

Locating events and inverting dispersion curves are the two field problems where learned travel times already do real work.

Reads: Microseismic event location Dispersion-curve and joint inversion

Operator learning

From solvers to operators, and DeepONet

A per-instance PINN solves one earth; an operator learns the map from any earth to its wavefield, and DeepONet is the founding architecture.

Reads: From per-instance solvers to operator learners DeepONet: branch and trunk networks

Fourier neural operatorswidget challenge

FNOs do their learning in the frequency domain, which is where wave physics kept its structure all along.

Reads: Fourier Neural Operators (FNO)

Learned propagators and parametric explorers

Once the operator is trained, forward modeling becomes interactive: thousand-fold speedups that turn synthetics from a batch job into a slider.

Reads: Learned propagators for fast forward modelling Parametric PDE explorers in real time

When operator learning wins

The training set is the price and the query count is the payoff; the crossover arithmetic tells you which projects should buy an operator.

Reads: When operator learning beats per-instance training

Generative and Bayesian

PINN roles and learned priors

Regulariser, initialiser, or solver: the same network plays three roles, and learned priors are how training data whispers geology into an inversion.

Reads: PINN as regulariser, initialiser, or solver Learned regularisers and ML priors

Generative priors for velocity modelswidget challenge

Generative models manufacture plausible earths on demand: the purest form of synthetic data this library teaches, and the prior modern inversion increasingly leans on.

Reads: Generative priors for velocity models

Bayesian PINNs and uncertaintywidget challenge

Ensembles and Bayesian networks turn one confident answer into a distribution of defensible ones; synthetic data with error bars is the deliverable.

Reads: Bayesian PINNs and ensemble PINNs Uncertainty-aware velocity inversion

Capstones

Capstone: Marcellus microseismicwidget challenge

A frac stage monitored in real time by learned travel times: the eikonal work of this path deployed on a live field problem.

Reads: Microseismic monitoring of a Marcellus frac stage

Capstone: USArray tomography

Continental-scale surface-wave tomography with learned machinery: the same tools, three thousand kilometers wide.

Reads: USArray surface-wave tomography of the western US

Capstone: eikonal basin tomography

A full basin solved with the factored eikonal network: the closing argument for travel times as a learned product.

Reads: Eikonal basin tomography