Learned Multiphysics Inversion with Differentiable Programming & Machine Learning
An open-source path from wave physics to CO₂ digital twins
2026-06-07
End-to-end multiphysics — every arrow differentiable
\[\mathbf{z} \xrightarrow{\;\mathcal{G}\;} \mathbf{K} \xrightarrow{\;\mathcal{S}\;} \mathbf{c} \xrightarrow{\;\mathcal{R}\;} \mathbf{m} \xrightarrow{\;\mathcal{F}(\cdot)\mathbf{q}\;} \mathbf{d}\]
Generative NF · multiphase flow · rock physics · wave physics — one AD graph, one gradient (Louboutin, Yin, et al., 2023; Yin et al., 2023).
ML-prior FWI — composing AD across physics & networks
\[\min_{\mathbf{z}}\ \tfrac{1}{2}\|\mathcal{F}(\mathcal{G}_{\theta^\ast}(\mathbf{z}))\mathbf{q} - \mathbf{d}\|_2^2 + \tfrac{\lambda}{2}\|\mathbf{z}\|_2^2\]
NF prior \(\mathcal{G}_{\theta^\ast}\) trained on Compass slices; gradient flows through wave physics and through the invertible net.
Truth 
Vanilla 
+ NF prior 
End-to-end permeability inversion from time-lapse seismic
One AD graph: wave physics → rock physics → multiphase flow (or FNO surrogate) → NF latent prior. No glue code, no manual adjoints (Grady II et al., 2023; Louboutin, Yin, et al., 2023).
Closing the [FR-1] gap with SAGE
SAGE — Subsurface AI-driven Geostatistical Extraction of proxy posteriors (Erdinc et al., 2026) — distills National Data Repository information into a trainable prior, plugged into WISE [variational FWI; Yin et al. (2024b); Yin et al. (2024a)] via WISE.jl.
[FR-1] data — UK National Data Repository curation
OpenSeisML (Bhar et al., 2026): automated pipeline curating imaged seismic + well logs from the UK NDR into a reproducible, ML-ready dataset for generative priors.
- time → depth via checkshots
- post-stack volumes + petrophysical logs
- North Sea geology, multiple surveys
- reproducible, open, at scale
Proxy prior — sampling plausible velocity realizations
Two-network distillation (Erdinc et al., 2026):
- Generative net on 3D imaged seismic → image realizations.
- Conditional 2D net on image ↔︎ well pairs → velocity / property realizations.
Result: a learned proxy prior \(p(\mathbf{m})\) used to train WISE / end-to-end inversion.
Currated training pairs from UK-NDR
Currated training samples (Bhar et al., 2026; Erdinc et al., 2026).
Guided inference with SAGE
Velocity models (third and fourth columns) sampled from the proxy prior conditioned on real unseen seismic (second colum) and guided by unseen wells (first column) produced by generative model trained on NDR data (Erdinc et al., 2026).
Samples from the proxy prior will be used to train Yin et al. (2024a) via
WISE.jl.
Closing the [FR-1] gap with SAGE
SAGE — Subsurface AI-driven Geostatistical Extraction of proxy posteriors (Erdinc et al., 2026) — distills National Data Repository information into a trainable prior, plugged into WISE [variational FWI; Yin et al. (2024b); Yin et al. (2024a)] via WISE.jl.
Colophon
These slides were generated with the assistance of Claude Code (Anthropic).
Source, bibliography, and reproducible Quarto build at github.com/slimgroup.
© 2026 the authors. Licensed under Creative Commons Attribution 4.0 International (CC BY 4.0).