22 SciML Applications in Geoscience
This chapter surveys the rapidly growing literature on scientific machine learning (SciML) applied to geoscience problems. Unlike the previous chapters — which introduced methods with worked code examples — here we focus on what has been done and where the field is heading, organized by geoscience sub-discipline.
The methods covered in this survey correspond directly to the tools introduced earlier in this book: PINNs, physics-based ML inversion, DeepONet, PI-DeepONet, and FNO.
22.1 Seismology and seismic imaging
Seismology is one of the most active areas for SciML adoption, driven by the computational cost of wave-equation solves and the abundance of observational data.
22.1.1 PINNs for wave propagation
- Waheed et al. (2021) developed PINNeik, a PINN that solves the eikonal equation for seismic travel-time computation. The mesh-free formulation handles complex velocity models naturally and avoids the grid artifacts of fast-marching methods.
- Smith et al. (2021) trained EikoNet, a deep neural network constrained by the eikonal equation, to predict seismic travel times in 3D heterogeneous velocity models. Once trained, it provides travel times at arbitrary source–receiver pairs without re-solving.
- Song et al. (2021) extended PINNs to solve the frequency-domain acoustic wave equation for anisotropic (VTI) media, demonstrating that PINNs can handle the additional complexity of directional velocity dependence.
- Rasht-Behesht et al. (2022) applied PINNs to full waveform modeling and inversion, showing that PDE-constrained neural networks can simultaneously recover velocity models and wavefields from sparse seismic observations.
- Song & Alkhalifah (2023) used Fourier-feature-enhanced PINNs to simulate multifrequency seismic wavefields, overcoming the spectral bias that limits standard PINNs for high-frequency wave solutions.
22.1.2 Neural operators for seismic modeling
- Yin et al. (2023) developed learned surrogates based on neural operators for multiphysics-based seismic inverse problems, replacing expensive wave-equation solves with fast neural operator evaluations during gradient-based inversion.
- FNO-based surrogates have been trained to map velocity models to seismograms, enabling rapid scenario testing for seismic hazard assessment where thousands of forward simulations are needed.
22.1.3 Seismic inversion
- Yang & Ma (2019) applied deep learning to full-waveform inversion (FWI), using neural networks to directly map seismic data to velocity models while embedding physical constraints to improve the inversion quality.
- Implicit neural representations (INRs) have been used to parameterize velocity models during FWI, providing implicit regularization through the network architecture and eliminating the need for hand-tuned regularization parameters.
22.2 Hydrogeology and subsurface flow
Groundwater modeling and subsurface flow simulation are natural targets for SciML because the governing equations (Darcy’s law, Richards’ equation) are well established but computationally expensive for heterogeneous media.
22.2.1 PINNs for groundwater modeling
- He et al. (2020) applied PINNs to multiphysics data assimilation in subsurface transport, demonstrating that physics-informed training can recover contaminant concentration fields from sparse monitoring well data while honoring the advection-diffusion equation.
- Zhu et al. (2019) developed physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification in subsurface transport, showing that PINNs can work without labeled data when the governing PDE is known.
22.2.2 Neural operator surrogates for reservoir simulation
- Wen et al. (2022) introduced U-FNO, an enhanced Fourier neural operator with U-Net-style skip connections, applied to multiphase flow in CO₂ storage simulations. The surrogate learned the mapping from injection scenarios to pressure and saturation fields, enabling real-time reservoir management decisions.
- Neural operators have been trained as surrogates for history matching in reservoir engineering, where the operator maps permeability fields to production data, enabling Bayesian inversion with thousands of forward evaluations that would be infeasible with traditional simulators.
- Wang et al. (2023) demonstrated reliable extrapolation of neural operators informed by physics for CO₂ sequestration modeling, showing that physics constraints improve generalization beyond the training distribution.
22.3 Weather and climate science
Global weather prediction has emerged as a flagship application for neural operators, with several models now rivaling or exceeding traditional numerical weather prediction (NWP) systems.
22.3.1 Neural operator weather models
- Pathak et al. (2022) developed FourCastNet, using adaptive Fourier neural operators (AFNO) for global weather prediction. The model produces 10-day forecasts in seconds rather than the hours required by traditional NWP systems, at comparable accuracy for many variables.
- Bi et al. (2023) introduced Pangu-Weather, a 3D neural architecture that achieved competitive medium-range weather forecasting. The model was trained on 39 years of ERA5 reanalysis data and demonstrated skillful forecasts up to 7 days.
- Lam et al. (2023) developed GraphCast, a graph-neural-network-based weather model that outperformed ECMWF’s HRES operational system on most metrics, marking a milestone in data-driven weather prediction.
22.3.2 Physics-informed climate modeling
- Kashinath et al. (2021) presented case studies of physics-informed machine learning for weather and climate modeling, demonstrating how embedding physical constraints (conservation laws, symmetries) improves forecast accuracy and ensures physical consistency of predictions.
- Physics-informed approaches have been applied to climate downscaling — using neural operators to map coarse-resolution climate model output to fine-resolution fields while respecting physical constraints like energy conservation.
22.4 Geothermal energy
Geothermal reservoir characterization and management require solving coupled thermo-hydro-mechanical (THM) equations that are computationally demanding.
- Sun et al. (2023) applied PINNs to geothermal reservoir simulation, using physics-informed training to model subsurface temperature and pressure distributions from sparse borehole measurements. The PINN naturally combines the heat equation with observational data, producing physically consistent predictions between measurement points.
- Neural operator surrogates have been explored for geothermal uncertainty quantification, where the mapping from subsurface property fields to temperature/pressure responses must be evaluated thousands of times during Bayesian inversion.
22.5 Solid Earth geophysics
22.5.1 Gravity and magnetic inversion
- INR-based parameterizations have been applied to 3D gravity inversion, where the density distribution is represented as a neural field and trained by minimizing the misfit between predicted and observed gravity anomalies. The network architecture provides implicit regularization, eliminating the need for Tikhonov-style penalty terms.
- Physics-informed approaches to potential field inversion exploit the Laplace/Poisson equation as an additional constraint, improving the depth resolution of recovered density models.
22.5.2 Electromagnetic methods
- PINNs have been applied to magnetotelluric (MT) forward modeling, solving Maxwell’s equations in the frequency domain for layered and 2D conductivity structures.
- Neural operator surrogates for electromagnetic induction problems enable rapid evaluation of forward responses during probabilistic inversion of controlled-source electromagnetic (CSEM) data.
22.6 Geodynamics and mantle convection
- PINNs have been applied to solve the Stokes equations for mantle convection, where the high computational cost of traditional finite element solvers limits the number of simulations that can be run for parameter studies.
- Neural operators have been explored as surrogates for mantle convection simulations, learning the mapping from rheological parameters and boundary conditions to flow fields and temperature distributions.
22.7 Earthquake science
- Mousavi et al. (2020) developed Earthquake Transformer (EQTransformer), an attention-based deep learning model for simultaneous earthquake detection and phase picking. While not strictly a SciML method (it is data-driven), it demonstrates the power of neural architectures for seismological signal processing.
- PINNs have been applied to earthquake source characterization, where the wave equation is used as a physics constraint to invert for source parameters (location, mechanism, rupture history) from seismogram recordings.
- Moseley et al. (2020) used deep learning for fast simulation of seismic waves in complex media, training neural networks as fast surrogates for finite-difference wave propagation codes.
22.8 Cross-cutting themes
Several themes emerge across these applications:
22.8.1 1. The speed–accuracy trade-off
Neural surrogates (FNO, DeepONet) sacrifice some accuracy for dramatic speedups — typically \(10^3\)–\(10^6\times\) faster than traditional solvers. This trade-off is acceptable when the surrogate is used within an outer loop (inversion, UQ) where thousands of forward evaluations are needed.
22.8.2 2. Physics as regularization
In data-scarce geoscience settings, physics-informed losses (PINNs, PI-DeepONet) act as powerful regularizers, preventing the network from producing physically impossible predictions. This is especially valuable in subsurface characterization where direct measurements are sparse and expensive.
22.8.3 3. Mesh-free flexibility
PINNs and INRs naturally handle the irregular geometries common in geoscience — topography, coastlines, fault surfaces, borehole trajectories — without requiring mesh generation, which is often the most time-consuming part of traditional numerical modeling.
22.8.4 4. Differentiable pipelines
Because the entire SciML pipeline is differentiable (from model parameters through the PDE solve to predictions), gradient-based optimization for inversion and optimal experimental design becomes straightforward. This is a fundamental advantage over traditional solvers that require adjoint formulations for gradient computation.
22.8.5 5. Foundation models for Earth science
An emerging trend is the development of foundation models — large neural operators pre-trained on massive simulation or reanalysis datasets — that can be fine-tuned for specific tasks. Weather prediction (FourCastNet, Pangu-Weather, GraphCast) is leading this trend, with analogous efforts underway for ocean modeling, seismology, and climate science.
22.9 Choosing the right SciML approach
| Problem type | Recommended approach | Key advantage |
|---|---|---|
| Solve one PDE instance | PINN | No training data, mesh-free |
| Recover model parameters from data | Physics-ML inversion (INR) | Implicit regularization |
| Repeated forward solves (regular grid) | FNO | Fast inference, resolution-invariant |
| Repeated forward solves (irregular data) | DeepONet | Flexible geometry |
| Operator learning without solver data | PI-DeepONet | Physics replaces training data |
| Real-time forecasting at scale | FNO / AFNO | Sub-second global predictions |
The SciML landscape is changing fast. New architectures, training strategies, and geoscience applications appear regularly. The references in this chapter provide entry points into the literature as of 2025; readers are encouraged to track developments through venues such as Journal of Computational Physics, Nature Machine Intelligence, Geophysical Journal International, and the ICLR/NeurIPS workshops on AI for science.