CO$_2$ sequestration hybrid solver using isogeometric alternating-directions and collocation-based robust variational physics informed neural networks (IGA-ADS-CRVPINN)

Askold Vilkha; Maciej Paszy\'nski; Marcin {\L}o\'s; Tomasz S{\l}u\.zalec

arxiv: 2604.20731 · v1 · submitted 2026-04-22 · 🧮 math.NA · cs.NA· cs.NE

CO₂ sequestration hybrid solver using isogeometric alternating-directions and collocation-based robust variational physics informed neural networks (IGA-ADS-CRVPINN)

Askold Vilkha , Tomasz S{\l}u\.zalec , Marcin {\L}o\'s , Maciej Paszy\'nski This is my paper

Pith reviewed 2026-05-09 23:31 UTC · model grok-4.3

classification 🧮 math.NA cs.NAcs.NE

keywords CO2 sequestrationhybrid solverisogeometric analysisphysics-informed neural networksDarcy's lawporous media flowalternating directionsvariational methods

0 comments

The pith

A hybrid solver pairs an isogeometric alternating-directions method for saturation with a pretrained neural network for pressure to simulate CO2 flow more than three times faster than a direct solver baseline.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a hybrid numerical approach for tracking the physical movement of CO2 through porous rock structures using only Darcy's law and no chemical reactions. An explicit isogeometric alternating-directions solver advances the saturation field at each time step while a collocation-based robust variational physics-informed neural network, pretrained once on the starting pressure, computes each new pressure field in just 100 Adam iterations. On a single compute node this combination finishes the full run in less than one-third the time required by the same geometric solver paired with a standard direct linear solver. Readers would care because the speed gain makes repeated, longer, or larger-scale forecasts of underground carbon storage practical on modest hardware.

Core claim

The central claim is that the IGA-ADS-CRVPINN hybrid solver produces accurate time-dependent saturation and pressure fields for the two-phase Darcy flow problem while delivering more than a threefold reduction in wall-clock time relative to the IGA-ADS-MUMPS baseline; the reduction is obtained by performing a single pretraining of the neural network on the initial pressure configuration and then limiting each subsequent pressure update to 100 Adam steps.

What carries the argument

The CRVPINN component, a collocation-based robust variational physics-informed neural network that approximates the pressure scalar field after one-time pretraining and requires only a fixed small number of optimization steps at every new time level.

If this is right

The complete simulation finishes in less than one-third the wall-clock time of the baseline direct-solver run on identical hardware.
Saturation is advanced explicitly by the alternating-directions isogeometric method without reported stability loss in the presented cases.
Pressure accuracy is preserved across many time steps by the lightweight neural updates rather than full retraining or direct solves.
The formulation deliberately excludes chemical reactions so that the same splitting can be reused for related storage problems such as hydrogen.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the fixed-iteration neural updates prove stable, the method could support ensembles of simulations over varied porous geometries to quantify storage reliability.
The same explicit-plus-neural split could accelerate other multiphase porous-media models in which pressure solves dominate runtime.
Faster forward runs would make repeated solution of inverse problems, such as inferring rock properties from observed CO2 plumes, computationally affordable.

Load-bearing premise

The neural-network pressure updates remain accurate and do not accumulate unacceptable error over the full duration of the simulation when limited to 100 Adam iterations after the initial pretraining.

What would settle it

A side-by-side run in which the hybrid solver's pressure and saturation fields after several hundred time steps are compared against a reference solution from the direct solver, checking whether mass-conservation error or plume-front location begins to diverge beyond a chosen tolerance.

Figures

Figures reproduced from arXiv: 2604.20731 by Askold Vilkha, Maciej Paszy\'nski, Marcin {\L}o\'s, Tomasz S{\l}u\.zalec.

**Figure 2.** Figure 2: Permeability distributions for three different reservoir configurations. The units are in [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Figure 3: CO2 saturation evolution for configuration K1 with uniform permeability at different time [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: CO2 saturation evolution for configuration K2 with uniform permeability at different time [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: CO2 saturation evolution for configuration K3 with uniform permeability at different time [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗

**Figure 6.** Figure 6: CO2 saturation evolution for configuration K1 with non-uniform permeability at different [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: CO2 saturation evolution for configuration K2 with non-uniform permeability at different [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗

**Figure 8.** Figure 8: CO2 saturation evolution for configuration K3 with non-uniform permeability at different [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗

**Figure 9.** Figure 9: CRVPINN pressure solution comparison with IGA-ADS solver for configuration K1 with [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗

**Figure 10.** Figure 10: CRVPINN pressure solution comparison with IGA-ADS solver for configuration K2 with [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗

**Figure 11.** Figure 11: CRVPINN pressure solution comparison with IGA-ADS solver for configuration K3 with [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗

**Figure 12.** Figure 12: CRVPINN pressure solution comparison with IGA-ADS solver for configuration K1 with [PITH_FULL_IMAGE:figures/full_fig_p023_12.png] view at source ↗

**Figure 13.** Figure 13: CRVPINN pressure solution comparison with IGA-ADS solver for configuration K2 with [PITH_FULL_IMAGE:figures/full_fig_p024_13.png] view at source ↗

**Figure 14.** Figure 14: CRVPINN pressure solution comparison with IGA-ADS solver for configuration K3 with [PITH_FULL_IMAGE:figures/full_fig_p025_14.png] view at source ↗

**Figure 16.** Figure 16: CRVPINN loss for uniform permeability field (a) Configuration K1 (b) Configuration K2 (c) Configuration K3 [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗

**Figure 18.** Figure 18: Execution times for IGA-ADS solver for saturation coupled with MUMPS solver executed [PITH_FULL_IMAGE:figures/full_fig_p032_18.png] view at source ↗

**Figure 19.** Figure 19: Execution times for IGA-ADS solver for saturation coupled with CVPINN solver executed [PITH_FULL_IMAGE:figures/full_fig_p033_19.png] view at source ↗

read the original abstract

This paper presents the hybrid solver for a $CO_2$ sequestration problem. The solver uses the IGA-ADS (IsoGeometric Analysis Alternating Directions solver) to compute the saturation scalar field update using the explicit method, and CRVPINN (Collocation-based Robust Variational Physics Informed Neural Networks solver) to compute the pressure scalar field. The study focuses on simulating the physical behavior of $CO_2$ in porous structures, excluding chemical reactions. The mathematical model is based on Darcy's Law. The CRVPINN is pretrained on the initial pressure configuration, and the time step pressure updates require only 100 iterations of the Adam method per time step. We compare our hybrid IGA-ADS solver, coupled with the CRVPINN method, with a baseline of the IGA-ADS solver coupled with the MUMPS direct solver. Our hybrid solver is over 3 times faster on a single computational node from the ARES cluster of ACK CYFRONET. Future work includes extensive testing, inverse problem solving, and potential application to $H_2$ storage problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The hybrid IGA-ADS plus CRVPINN solver reports a 3x wall-clock gain for Darcy CO2 flow but supplies no error histories or field comparisons to show the 100-iteration pressure updates stay accurate over time.

read the letter

The main point to take away is that this paper puts together an existing IGA-ADS explicit scheme for saturation with a collocation-based robust VPINN for pressure, pretrained once and then run with a fixed 100 Adam steps per time step. On their ARES cluster node the hybrid version finishes more than three times faster than the same IGA-ADS code calling MUMPS for pressure. That concrete pairing and the restricted iteration budget are the new implementation details; both component techniques already exist in the literature, but the limited per-step optimization and the reported timing on this specific problem are not previously documented in the cited works. The authors keep the model simple by dropping chemistry, which is reasonable for a first test of the hybrid idea. The writing is straightforward and the focus on a single-node speed-up is practical. The soft spot is exactly the one flagged in the stress-test note. The abstract and the available description give no L2 errors, pointwise comparisons, mass-conservation checks, or final-time field plots between the CRVPINN pressure and the direct solver. Without those numbers it is impossible to tell whether the faster run is still solving the same PDE or whether the pressure field drifts enough to corrupt the saturation update over many steps. The 100-iteration budget is aggressive; if error accumulates, the claimed advantage becomes irrelevant to the actual physics. The paper also omits any discussion of how the collocation points are chosen or adapted across time steps, which would help readers judge stability. This work is aimed at people already working on hybrid PINN-traditional solvers for porous-media flow or on isogeometric methods for Darcy problems. A reader who wants to try the same split could extract the architecture and the 100-iteration recipe, but would have to add their own validation. The manuscript is coherent on its own terms and shows honest engagement with the literature, so it deserves a serious referee rather than a desk reject. Referees can ask for the missing error data and decide whether the speedup holds up once accuracy is checked.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a hybrid solver for CO2 sequestration in porous media using Darcy's law, where IGA-ADS handles explicit saturation updates and CRVPINN computes the pressure field after initial pretraining with only 100 Adam iterations per time step. The central claim is that this IGA-ADS-CRVPINN hybrid is over 3 times faster than the baseline IGA-ADS coupled with the MUMPS direct solver on a single node of the ARES cluster.

Significance. If the CRVPINN pressure solutions remain accurate and stable under the limited per-step optimization budget, the hybrid approach could offer meaningful efficiency gains for time-dependent porous media simulations, enabling larger-scale or parametric studies in geoscience applications. The combination of isogeometric alternating-directions methods with collocation-based variational PINNs is a novel hybrid strategy that merits further exploration if properly validated.

major comments (2)

[Abstract] Abstract: The claim that the hybrid solver 'is over 3 times faster' is presented without any error metrics, L2 norms, pointwise field comparisons, or convergence histories between the CRVPINN pressure and the MUMPS reference solution. This omission makes it impossible to verify that the speedup preserves numerical accuracy over the simulation horizon.
[Numerical results] Numerical results: No quantitative evidence (e.g., error histories, conservation checks, or final-time field comparisons) is supplied to show that the pressure field produced by CRVPINN after pretraining plus 100 Adam iterations per step remains sufficiently close to the direct solver solution, preventing error accumulation that would invalidate the saturation updates.

minor comments (1)

[Abstract] Abstract: Adding a brief statement of the total simulation duration or number of time steps would help readers assess the practical significance of the reported wall-clock improvement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

Thank you for the opportunity to respond to the referee's report. We value the feedback and will revise the manuscript to address the concerns regarding the presentation of accuracy metrics alongside the performance claims.

read point-by-point responses

Referee: [Abstract] Abstract: The claim that the hybrid solver 'is over 3 times faster' is presented without any error metrics, L2 norms, pointwise field comparisons, or convergence histories between the CRVPINN pressure and the MUMPS reference solution. This omission makes it impossible to verify that the speedup preserves numerical accuracy over the simulation horizon.

Authors: We agree that the abstract should be supported by evidence of maintained accuracy. In the revised version, we will augment the abstract and the results section with L2 error norms, pointwise comparisons, and convergence histories demonstrating that the CRVPINN pressure solutions, after pretraining and 100 Adam iterations, closely match the MUMPS solutions throughout the time steps. This will substantiate that the reported speedup preserves the required numerical fidelity. revision: yes
Referee: [Numerical results] Numerical results: No quantitative evidence (e.g., error histories, conservation checks, or final-time field comparisons) is supplied to show that the pressure field produced by CRVPINN after pretraining plus 100 Adam iterations per step remains sufficiently close to the direct solver solution, preventing error accumulation that would invalidate the saturation updates.

Authors: The referee correctly identifies a gap in the current manuscript. We will include in the numerical results section detailed error histories (L2 and pointwise), checks for conservation properties where applicable, and comparisons at final time to confirm that the pressure fields from CRVPINN do not deviate significantly from the reference, ensuring no invalidating error accumulation in the coupled saturation updates. These additions will be based on the simulations already performed. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical timing comparison of hybrid solver

full rationale

The manuscript presents an engineering implementation of a hybrid solver (IGA-ADS explicit saturation updates + CRVPINN pressure solves with pretraining plus 100 Adam steps per timestep) and reports wall-clock speedup versus IGA-ADS+MUMPS on a single node. No derivation chain exists; the 3x speedup is stated as a measured outcome of the implementation, not a mathematical prediction or first-principles result. No equation reduces to its own inputs by construction, no fitted parameter is relabeled as a prediction, and no self-citation supplies a uniqueness theorem or ansatz that the central claim depends upon. The CRVPINN iteration budget is an empirical design choice whose long-term accuracy is an assumption (as noted by the skeptic), but that assumption is not smuggled in via circular definition or self-referential citation. The paper is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information in the abstract to enumerate free parameters, axioms, or invented entities; the method relies on standard Darcy-law assumptions and neural-network training but no explicit list is supplied.

pith-pipeline@v0.9.0 · 5533 in / 1053 out tokens · 15311 ms · 2026-05-09T23:31:01.824817+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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A hybrid method for inversion of 3d dc resistivity logging mea- surements.Natural computing, 14(3):355–374

Ewa Gajda-Zagórska, Robert Schaefer, Maciej Smołka, Maciej Paszyński, and David Pardo. A hybrid method for inversion of 3d dc resistivity logging mea- surements.Natural computing, 14(3):355–374. 30 (a) Configuration K1 (b) Configuration K2 (c) Configuration K3 Figure 16: CRVPINN loss for uniform permeabil- ity field (a) Configuration K1 (b) Configuration ...

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