Pith. sign in

REVIEW 2 major objections 6 minor 58 references

This paper claims that the overlapping Schwarz alternating method can couple subdomain-local data-driven reduced-order models, built by operator inference, with each other and with full-order finite element models, achieving accurate soluti

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · deepseek-v4-flash

2026-08-03 21:03 UTC pith:RMKRNUVE

load-bearing objection Solid, honest engineering demonstration of Schwarz-coupled OpInf ROMs in 3D solid dynamics, but the headline accuracy and speedup claims rest on a non-converged FOM-FOM reference. the 2 major comments →

arxiv 2511.20687 v4 pith:RMKRNUVE submitted 2025-11-20 math.NA cs.AIcs.NAmath-phmath.MP

Hybrid coupling with operator inference and the overlapping Schwarz alternating method

Irina Tezaur , Eric Parish , Anthony Gruber , Ian Moore , Christopher Wentland , Alejandro Mota This is my paper
classification math.NA cs.AIcs.NAmath-phmath.MP MSC 65M5565M6074S0574B20
keywords operator inferenceoverlapping Schwarz alternating methoddomain decompositionreduced order modelsolid dynamicshyperelasticityhybrid FOM-ROM couplingnon-intrusive model reduction
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the overlapping Schwarz alternating method (O-SAM) can stitch together subdomain-local reduced-order models built by operator inference — learned from data without modifying the full-order code — and couple them to each other or to full-order finite-element models, even when subdomains use different meshes, element types, and time integrators. The central demonstration is on fully nonlinear three-dimensional solid dynamics problems: bolted joints, torsion, and tension specimens, where the paper reports speedups up to about 106× relative to a pure FOM-FOM Schwarz coupling, with relative errors typically in the 0.1–5% range. The authors position this as the first application of Schwarz-iterated operator-inference coupling to realistic 3D nonlinear solid mechanics, and argue it enables workflows in which only a few physically motivated subdomains need to be meshed or modeled at high fidelity.

Core claim

The paper's central claim is that converged O-SAM iterations with subdomain-local OpInf ROMs — linear, quadratic, or cubic — reproduce the accuracy of FOM-FOM Schwarz solutions while costing far less online CPU time. The claim is supported by four test problems: a 1D elastic wave with mixed time integrators, a 3D bolted joint with cubic Saint Venant–Kirchhoff nonlinearities, a 3D torsion bar with generic hyperelastic nonlinearities modeled by quadratic OpInf, and a 3D tension specimen where quadratic OpInf models coupled to each other are reported to match the FOM-FOM reference to roughly 0.01–5% error while running up to about 106× faster. A key reported observation is that the hybrid coupl

What carries the argument

The central mechanism is the overlapping Schwarz alternating method (O-SAM) itself, which exchanges Dirichlet boundary data between subdomain solves and iterates until displacement/velocity convergence; the paper's novel twist is to let some (or all) subdomain solvers be OpInf ROMs. OpInf (operator inference) learns reduced linear/quadratic/cubic operators from snapshots of a FOM-FOM Schwarz simulation, using a least-squares fit with regularization. Two enabling components are introduced: boundary POD bases (Φ∂S, Φ∂φ) that compress the boundary forcing term so the learned boundary operator does not scale with interface DoF count, and a subdomain-local regularization sweep that chooses λ by a

Load-bearing premise

The accuracy and speedup numbers rest on treating the FOM-FOM Schwarz solution as the ground truth, but in the headline tension-specimen case that reference never reached the Schwarz convergence tolerances within the 32-iteration cap, so reported errors are relative to an unconverged baseline.

What would settle it

Re-run the tension-specimen case allowing the FOM-FOM O-SAM to iterate until the stated tolerances are met (or obtain a monolithic reference with a compatible mesh) and count the offline training cost; if the converged reference differs from the 32-iteration reference by more than the claimed hybrid-model errors, or if the end-to-end time including training exceeds the FOM-FOM time for a single query, the reported accuracy and 106× speedup are not validated.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Multi-query analyses (design, uncertainty quantification) on complex geometries can replace one subdomain's high-fidelity solver with a learned surrogate, cutting online solve time while preserving Schwarz-level accuracy.
  • Different meshes, element types, and time integrators within one simulation become routine: ROM subdomains can use implicit time stepping with coarse meshes while FOM subdomains run explicit, as demonstrated in the torsion and wave cases.
  • Because hybrid couplings converge in fewer Schwarz iterations than FOM-FOM, the speedup is not solely from reduced DoFs but also from faster coupling convergence, making ROM placement in 'simpler' subdomains an attractive accuracy-preserving acceleration strategy.
  • Higher-order polynomial OpInf (quadratic and cubic) is sufficient to reproduce the global response of fully non-polynomial hyperelastic models once localized via Schwarz, suggesting the polynomial structural assumption is not an obstacle in the tested regime.
  • A single Schwarz iteration per time step (an additive-Schwarz-like shortcut) can converge to the wrong solution in nonlinear problems; iterating to the tolerance is necessary — a direct caution for cheaper Schwarz variants.

Where Pith is reading between the lines

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

  • The 106× speedup is an online-only figure; it excludes the offline FOM-FOM training runs and OpInf operator learning. In a one-off analysis the method would be slower end-to-end, so the practical benefit is in multi-query settings where training costs amortize.
  • In the tension-specimen benchmark the FOM-FOM reference run consumed all 32 allowed Schwarz iterations without meeting the tolerance (mean/max 32.0/32), so the reported errors and speedup are measured against a reference that may not be fully converged; an independently converged reference could change those numbers.
  • The observation that ROM-involved couplings converge in fewer Schwarz iterations suggests a testable extension: using an intentionally coarsened or smoothed model in selected subdomains may accelerate Schwarz convergence even without formal ROM accuracy, as long as the final iterate is checked against tolerances.
  • Because the coupling is formulated at the algebraic level and the paper frames O-SAM as PDE-agnostic, the same recipe should transfer to other nonlinear PDEs (e.g., thermo-mechanics or plasticity) provided the polynomial OpInf form is replaced or augmented appropriately.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper proposes a hybrid coupling method for subdomain-local non-intrusive Operator Inference (OpInf) reduced order models (ROMs) and full order models (FOMs) using the overlapping Schwarz alternating method (O-SAM). The offline stage builds subdomain-local POD bases and infers cubic (or lower-order) polynomial ROM operators, including a new boundary-POD compression and a regularization-parameter selection procedure. The online stage couples these ROMs with each other or with FOMs using O-SAM, allowing disparate meshes, element types, and time integrators. Numerical experiments on 1D wave propagation, a 3D bolted joint, a 3D torsion specimen, and a 3D tension specimen demonstrate, in most cases, good accuracy relative to FOM-FOM O-SAM references and substantial online speedups, with the largest reported speedup being 106x in the tension specimen. The paper emphasizes a 'top-down' training strategy and evaluates both reproductive and predictive scenarios.

Significance. If the central claims hold, the paper provides a practical, plug-and-play framework for coupling non-intrusive ROMs with FOMs in nonlinear solid dynamics, addressing a long-standing barrier in multiscale simulation. The methodology is well-motivated and the exposition is generally clear. Notable strengths include: (i) a reproducible implementation with commit-hashed code and input decks; (ii) predictive test cases that use parameter/initial-condition combinations not in the training set; (iii) a demonstration that quadratic/cubic polynomial ROMs can serve as local surrogates even for non-polynomial hyperelastic models; and (iv) a thorough comparison with related work. The main caveat is that the flagship tension-specimen accuracy and speedup claims rest on a FOM-FOM reference that did not meet the stated Schwarz convergence tolerances, which undermines the headline numbers.

major comments (2)
  1. [Section 5.4, Tables 7-8] The FOM-FOM reference run in the tension specimen hits the 32-iteration cap with mean/max Schwarz iterations of 32.0/32, i.e., it does not satisfy the stated tolerances (δrel=1e-8, δabs=1e-6). Since the reported hybrid errors are computed via Eq. (28) relative to this FOM-FOM solution, the O(1e-4)-O(1e-3) errors and the 106x speedup are not validated against a converged coupled solution. Please either provide a converged FOM-FOM reference (or a monolithic reference), or explicitly characterize the errors as being relative to an unconverged iterate and perform a sensitivity study with respect to the Schwarz iteration cap.
  2. [Section 5 (above §5.1) and §5.4, Tables 7-8] The reported speedups are online-only. The 106x figure compares online CPU times (4m 42.1s vs. 8h 19m 29.5s) and excludes the offline FOM-FOM runs required for 'top-down' training as well as the OpInf operator-inference time. For a single query, the end-to-end speedup is therefore not established. Please state this limitation explicitly and, if appropriate, provide an amortized cost analysis for multi-query scenarios.
minor comments (6)
  1. [Section 2 (text near Eq. (8))] Typo: 'denots' should be 'denotes'.
  2. [Eq. (22)] The notation for the convergence check is hard to parse; consider simplifying the subscript/superscript structure or adding a short explanation after the equation.
  3. [Table 7, QOpInf-QOpInf 1-iter row] CPU time is listed as '1m 43.5' without a unit; add 's' for consistency.
  4. [§5.3-5.4] It would help the reader if the text explicitly states that QOpInf is chosen as a quadratic surrogate for the non-polynomial Neohookean model, rather than leaving the reader to infer this from the model assignment.
  5. [§5.1.3] Scenario 3 is predictive in initial condition but not in material parameters; the distinction between state-predictive and parameter-predictive should be stated more clearly to avoid overstating the predictive scope.
  6. [§1.2] The claim of being 'first' to apply SAM-based OpInf coupling to realistic 3D nonlinear solid dynamics could be softened given the closely related works [16] and [19]; consider phrasing as 'among the first'.

Circularity Check

0 steps flagged

No significant circularity: the headline predictive evaluations use out-of-sample parameters/initial conditions, and the reference solutions are independent FOM-FOM O-SAM runs (plus an exact solution in 1D).

full rationale

The paper's derivation chain is not circular. OpInf operators are learned by solving the regularized least-squares problem (25) from FOM-FOM O-SAM training snapshots, and the online O-SAM couplings (Section 4.2.2 and 4.2.3) then integrate these learned operators with Schwarz boundary transfer. The key predictive claims use parameter/initial-condition combinations that are not present in the training set: Scenario 3 in Section 5.1 trains on the Rounded Square initial condition and predicts the Symmetric Gaussian; the bolted joint predictive case trains on (a1,a2,a3) = (2,0,0), (0,0,2), (2,2,0) and predicts (2,0,2); the torsion predictive case trains on b1,b2 in {500,1000,5000,8000} and predicts 5500; and the tension predictive case trains on alpha in {0.003,0.006} and predicts alpha = 0.005. Thus the reported predictive errors are not restatements of the training data. The self-citations to [36,37,47,35,34] are background/prior-work support for FOM-FOM O-SAM convergence and earlier coupling frameworks; they are not load-bearing for the new hybrid method's accuracy, which is assessed against FOM-FOM O-SAM references and, in the 1D case, against the exact analytical solution (31). The paper does have validation limitations - the FOM-FOM tension-specimen reference hits the 32-iteration cap (Tables 7-8) and no monolithic reference is computed - but these are ground-truth/correctness concerns, not circularity. No step in the derivation reduces by construction to its own inputs.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The method itself introduces no new physical entities. Its free parameters are standard OpInf/ROM hyperparameters (basis sizes, regularization, tolerances) plus explicit modeling simplifications that trade fidelity for tractability. The most consequential assumptions are the polynomial surrogate ansatz and the validity of FOM-FOM O-SAM as ground truth.

free parameters (3)
  • Subdomain-local OpInf regularization parameters lambda_i = 10^-11 to 10^-4 depending on test/subdomain (Tables 3-8)
    Selected by grid search that minimizes trajectory error against training data (Section 4.2.4); these are fit on data and affect all reported accuracies.
  • POD basis dimensions (interior and boundary) = e.g., r=2-56; boundary bases 1-24
    Chosen to retain 99.999%-99.9999% snapshot energy (Section 5); not fit to a target error but a model-selection choice.
  • Schwarz tolerances and controller time-step = delta_abs=1e-8..1e-4, delta_rel=1e-12..1e-6; controller step per problem (e.g., 1e-7 s in 1D, 2e-6 s in torsion)
    User-specified convergence/accuracy controls that determine iteration counts and errors; chosen by hand.
axioms (6)
  • domain assumption O-SAM converges for nonlinear FOM-FOM solid-dynamics problems (cited [36,37]); the same convergence behavior is assumed to transfer to FOM-OpInf and OpInf-OpInf couplings.
    Invoked in Section 4 and 5 as justification for iterating to convergence; no convergence proof is given for the ROM-coupled case.
  • ad hoc to paper A polynomial (cubic) ROM structure (11)/(17) adequately represents the dynamics even when the underlying material is non-polynomial (Neohookean, Sections 5.3-5.4).
    Stated in Section 2 and Remark 1 that (9) is a surrogate; validated only empirically.
  • ad hoc to paper The terms h(H,C,chi_h) and M_uc chi_doubledot_h can be neglected in going from (10) to (11).
    Explicit modeling decisions in Section 2; no quantitative error analysis is provided.
  • domain assumption Top-down training is possible: a FOM-FOM O-SAM solution for the full physical domain can be obtained and its snapshots are adequate for subdomain ROM training.
    Stated in Section 4.2 and Remark 7; the method's whole offline stage depends on it.
  • ad hoc to paper FOM-FOM O-SAM can serve as a reference solution even when it fails to meet the Schwarz tolerance within the allowed iterations (tension specimen).
    Section 5.4 and Tables 7-8 use the non-converged FOM-FOM run as ground truth; this is the weakest reference assumption.
  • standard math OpInf least-squares recovers the projected operators in the limit of sufficient data (Eq. 19, cited from [44]).
    Used to justify the learned operators; not re-derived.

pith-pipeline@v1.3.0-alltime-deepseek · 40698 in / 17739 out tokens · 184254 ms · 2026-08-03T21:03:17.556265+00:00 · methodology

0 comments
read the original abstract

This paper presents a novel hybrid approach for coupling subdomain-local non-intrusive Operator Inference (OpInf) reduced order models (ROMs) with each other and with subdomain-local high-fidelity full order models (FOMs) with using the overlapping Schwarz alternating method (O-SAM). The proposed methodology addresses significant challenges in multiscale modeling and simulation, particularly the long runtime and complex mesh generation requirements associated with traditional high-fidelity simulations. By leveraging the flexibility of O-SAM, we enable the seamless integration of disparate models, meshes, and time integration schemes, enhancing computational efficiency while maintaining high accuracy. Our approach is demonstrated through a series of numerical experiments on complex three-dimensional (3D) solid dynamics problems, showcasing speedups of up to 106x compared to conventional FOM-FOM couplings. This work paves the way for more efficient simulation workflows in engineering applications, with potential extensions to a wide range of partial differential equations.

Figures

Figures reproduced from arXiv: 2511.20687 by Alejandro Mota, Anthony Gruber, Christopher Wentland, Eric Parish, Ian Moore, Irina Tezaur.

Figure 1
Figure 1. Figure 1: Illustration showing an domain decomposition of a 2D domain Ω into two overlapping subdomains, [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: 6To simplify the presentation, we have suppressed the dependence of stiffness matrices Ki on the Dirichlet and Schwarz boundary conditions. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: Field transfer in the O-SAM algorithm within one controller time-step [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: 1D linear elastic wave propagation problem: singular value decay (a) and projection errors [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, reproductive [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, reproductive [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, reproductive [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: 1D linear elastic wave propagation problem, Rounded Square initial condition, reproductive [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: 1D linear elastic wave propagation problem, Rounded Square initial condition, reproductive [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: 1D linear elastic wave propagation problem, Rounded Square initial condition, reproductive [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: 1D linear elastic wave propagation problem, Rounded Square initial condition, reproductive [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, predictive [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, predictive [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, predictive [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: 1D linear elastic wave propagation problem, Symmetric Gaussian initial condition, predictive [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: 3D nonlinear hyperelastic bolted joint problem: geometry and meshes. [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: 3D nonlinear hyperelastic bolted joint problem: POD singular value decay (1 [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: 3D nonlinear hyperelastic bolted joint problem, reproductive regime: relative errors in [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: 3D nonlinear hyperelastic bolted joint problem, predictive regime: plots of the displacement [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: 3D nonlinear hyperelastic bolted joint problem, predictive regime: plots of the average von [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: 3D nonlinear hyperelastic torsion problem: domain decomposition and meshes. Ω [PITH_FULL_IMAGE:figures/full_fig_p034_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: 3D nonlinear hyperelastic torsion problem: computed displacement solution magnitudes in Ω [PITH_FULL_IMAGE:figures/full_fig_p035_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: 3D nonlinear hyperelastic torsion problem: computed velocity solution magnitudes in Ω [PITH_FULL_IMAGE:figures/full_fig_p035_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: 3D nonlinear hyperelastic tension-specimen problem: geometry (a) and domain decomposition [PITH_FULL_IMAGE:figures/full_fig_p036_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: 3D nonlinear hyperelastic tension specimen problem, reproductive regime: plots of the [PITH_FULL_IMAGE:figures/full_fig_p038_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: 3D nonlinear hyperelastic tension specimen problem, predictive regime: plots of the average [PITH_FULL_IMAGE:figures/full_fig_p039_25.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

58 extracted references · 7 canonical work pages

  1. [1]

    Quadratic approximation manifold for mitigating the Kol- mogorov barrier in nonlinear projection-based model order reduction

    Joshua Barnett and Charbel Farhat. “Quadratic approximation manifold for mitigating the Kol- mogorov barrier in nonlinear projection-based model order reduction”. In:Journal of Computational Physics464 (2022), p. 111348.doi:10.1016/j.jcp.2022.111348

    arXiv 2022

  2. [2]

    Neural-network-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility

    Joshua Barnett, Charbel Farhat, and Yvon Maday. “Neural-network-augmented projection-based model order reduction for mitigating the Kolmogorov barrier to reducibility”. In:Journal of Computational Physics492 (2023), p. 112420.doi:10.1016/j.jcp.2023.112420. 41

    arXiv 2023

  3. [3]

    ArXiv pre-print

    Joshua Barnett, Irina Tezaur, and Alejandro Mota.The Schwarz alternating method for the seamless coupling of nonlinear reduced order models and full order models. ArXiv pre-print. 2022.url: https://arxiv.org/abs/2210.12551

    Pith/arXiv arXiv 2022

  4. [4]

    A zonal Galerkin-free POD model for incompressible flows

    Michel Bergmann et al. “A zonal Galerkin-free POD model for incompressible flows”. In:Journal of Computational Physics352 (2018), pp. 301–325.doi:10.1016/j.jcp.2017.10.001

    doi:10.1016/j.jcp.2017.10.001 2018

  5. [5]

    Boresi and Richard J

    Arthur P. Boresi and Richard J. Schmidt.Advanced Mechanics of Materials. John Wiley & Sons, 2013

    2013

  6. [6]

    Iterative methods for model reduction by domain decomposition

    Marcelo Buffoni, Haysam Telib, and Angelo Iollo. “Iterative methods for model reduction by domain decomposition”. In:Computers & Fluids38.6 (2009), pp. 1160–1167.doi: 10.1016/j.compfluid. 2008.11.008

    doi:10.1016/j.compfluid 2009

  7. [7]

    Youngsoo Choi et al.Defining Foundation Models for Computational Science: A Call for Clarity and Rigor. 2025. arXiv:2505.22904 [cs.LG].url:https://arxiv.org/abs/2505.22904

    Pith/arXiv arXiv 2025

  8. [8]

    Train small, model big: Scalable physics simulators via reduced order modeling and domain decomposition

    Seung Whan Chung et al. “Train small, model big: Scalable physics simulators via reduced order modeling and domain decomposition”. In:Computer Methods in Applied Mechanics and Engineering 427 (2024), p. 117041.doi:10.1016/j.cma.2024.117041

    arXiv 2024

  9. [9]

    A hybrid method based on POD and domain decomposition to compute the 2-D aerodynamic flow field - incompressible validation

    Davide Cinquegrana, Raffaele Donelli, and Antonio Viviani. “A hybrid method based on POD and domain decomposition to compute the 2-D aerodynamic flow field - incompressible validation”. In: XX Congresso dell’Associazione Italiana di Meccanica Teorica e Applicata, Bologna, ITA. 2011, pp. 1–10

    2011

  10. [10]

    A domain decomposition framework for fluid-structure interaction problems

    Simone Deparis, Marco Discacciati, and Alfio Quarteroni. “A domain decomposition framework for fluid-structure interaction problems”. In:Computational Fluid Dynamics 2004: Proceedings of the Third International Conference on Computational Fluid Dynamics, ICCFD3, Toronto, 12–16 July

    2004

  11. [11]

    Alejandro N Diaz et al.Interpretable and flexible non-intrusive reduced-order models using repro- ducing kernel Hilbert spaces. 2025. arXiv: 2506.10224 [cs.CE].url: https://arxiv.org/abs/ 2506.10224

    arXiv 2025

  12. [12]

    Diaz et al.Kernel manifolds: nonlinear-augmentation dimensionality reduction using reproducing kernel Hilbert spaces

    Alejandro N. Diaz et al.Kernel manifolds: nonlinear-augmentation dimensionality reduction using reproducing kernel Hilbert spaces. 2025. arXiv: 2509.00224 [cs.CE].url: https://arxiv.org/ abs/2509.00224

    Pith/arXiv arXiv 2025

  13. [13]

    An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition

    Marco Discacciati, Ben J. Evans, and Matteo Giacomini. “An overlapping domain decomposition method for the solution of parametric elliptic problems via proper generalized decomposition”. In: Computer Methods in Applied Mechanics and Engineering418 (2024), p. 116484.doi: 10.1016/j. cma.2023.116484

    arXiv 2024

  14. [14]

    Evans, and Matteo Giacomini.PGD-based local surrogate models via over- lapping domain decomposition: a computational comparison

    Marco Discacciati, Ben J. Evans, and Matteo Giacomini.PGD-based local surrogate models via over- lapping domain decomposition: a computational comparison. 2025. arXiv: 2508.01313 [math.NA]. url:https://arxiv.org/abs/2508.01313

    arXiv 2025

  15. [15]

    Model reduction of coupled systems based on non- intrusive approximations of the boundary response maps

    Niccol` o Discacciati and Jan S. Hesthaven. “Model reduction of coupled systems based on non- intrusive approximations of the boundary response maps”. In:Computer Methods in Applied Mechanics and Engineering420 (2024), p. 116770.doi:10.1016/j.cma.2024.116770

    arXiv 2024

  16. [16]

    Domain Decomposition for Data-Driven Reduced Modeling of Large- Scale Systems

    Ionut-Gabriel Farcas et al. “Domain Decomposition for Data-Driven Reduced Modeling of Large- Scale Systems”. In:AIAA Journal62.11 (2024), pp. 1–16.doi:10.2514/1.J063715

    doi:10.2514/1.j063715 2024

  17. [17]

    Schwarz methods over the course of time

    Martin J. Gander. “Schwarz methods over the course of time”. In:Electronic Transactions on Numerical Analysis31 (2008), pp. 228–255

    2008

  18. [18]

    Operator inference for non-intrusive model reduction with quadratic manifolds

    Rudy Geelen, Stephen Wright, and Karen Willcox. “Operator inference for non-intrusive model reduction with quadratic manifolds”. In:Computer Methods in Applied Mechanics and Engineering 403 (2023), p. 115717.doi:10.1016/j.cma.2022.115717

    arXiv 2023

  19. [19]

    Non-intrusive reduced-order modeling for dynamical systems with spatially localized features

    Leonidas Gkimisis et al. “Non-intrusive reduced-order modeling for dynamical systems with spatially localized features”. In:Computer Methods in Applied Mechanics and Engineering444 (2025), p. 118115.doi:10.1016/j.cma.2025.118115

    arXiv 2025

  20. [20]

    Variationally Consistent Hamiltonian Model Reduction

    Anthony Gruber and Irina Tezaur. “Variationally Consistent Hamiltonian Model Reduction”. In: SIAM Journal on Applied Dynamical Systems24.1 (2025), pp. 376–414.doi: 10.1137/24M1652490

    work page doi:10.1137/24m1652490 2025

  21. [21]

    Lumley, and Gahl Berkooz.Turbulence, Coherent Structures, Dynamical Systems and Symmetry

    Philip Holmes, John L. Lumley, and Gahl Berkooz.Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, 1996. 42

    1996

  22. [22]

    Holzapfel.Nonlinear Solid Mechanics: A Continuum Approach for Engineering

    Gerhard A. Holzapfel.Nonlinear Solid Mechanics: A Continuum Approach for Engineering. English. Wiley, 2000, p. 455.isbn: 9780471823049

    2000

  23. [23]

    Galerkin proper orthogonal decomposition methods for parabolic problems

    K. Kunisch and S. Volkwein. “Galerkin proper orthogonal decomposition methods for parabolic problems”. In:Numerische Mathematik90.1 (2001), pp. 117–148.doi:10.1007/s002110100282

    doi:10.1007/s002110100282 2001

  24. [24]

    Accessed: 2025-10-31

    Sandia National Laboratories.Automating complex 3D modeling. Accessed: 2025-10-31. 2020.url: https://www.sandia.gov/labnews/2020/04/23/vorocrust/

    2025

  25. [25]

    Accessed: 2025-10-23

    Sandia National Laboratories.CUBIT Mesher. Accessed: 2025-10-23. 2023.url: https://cubit. sandia.gov/

    2025

  26. [26]

    Accessed: 2025-10-23

    Sandia National Laboratories.norma-opinf. Accessed: 2025-10-23. 2023.url: https://github. com/sandialabs/norma-opinf

    2025

  27. [27]

    https: //github.com/sandialabs/Norma.jl

    Sandia National Laboratories.Norma.jl: A Julia testbed for coupling and multiphysics. https: //github.com/sandialabs/Norma.jl. Commit on GitHub: accessed on September 2, 2025. 2025

    2025

  28. [28]

    Accessed: 2025-11-07

    Sandia National Laboratories.SIERRA/SM: A Multi-Physics Simulation Framework. Accessed: 2025-11-07. 2023.url:https://www.sandia.gov/sierra/

    2025

  29. [29]

    Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders

    Kookjin Lee and Kevin T. Carlberg. “Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders”. In:Journal of Computational Physics404 (2020), p. 108973. doi:10.1016/j.jcp.2019.108973

    arXiv 2020

  30. [30]

    D3M: A Deep Domain Decomposition Method for Partial Differential Equations

    Ke Li et al. “D3M: A Deep Domain Decomposition Method for Partial Differential Equations”. In: IEEE Access8 (2020), pp. 5283–5294.doi:10.1109/ACCESS.2019.2957200

    arXiv 2020

  31. [31]

    Deep Domain Decomposition Method: Elliptic Problems

    Wuyang Li, Xueshuang Xiang, and Yingxiang Xu. “Deep Domain Decomposition Method: Elliptic Problems”. In:Proceedings of Machine Learning Research107 (2020), pp. 269–286

    2020

  32. [32]

    Separable physics-informed DeepONet: Breaking the curse of dimensionality in physics-informed machine learning

    Luis Mandl et al. “Separable physics-informed DeepONet: Breaking the curse of dimensionality in physics-informed machine learning”. In:Computer Methods in Applied Mechanics and Engineering 434 (2025), p. 117586.doi:10.1016/j.cma.2024.117586

    arXiv 2025

  33. [33]

    Data-driven reduced-order models via regularised Operator Inference for a single-injector combustion process

    Shane A. McQuarrie, Cheng Huang, and Karen E. Willcox. “Data-driven reduced-order models via regularised Operator Inference for a single-injector combustion process”. In:Journal of the Royal Society of New Zealand51.2 (2021), pp. 194–211.doi:10.1080/03036758.2020.1863237

    arXiv 2021

  34. [34]

    Ian Moore et al.Domain Decomposition-Based Coupling of High-Fidelity Finite Element and Reduced Order Operator Inference Models Using the Schwarz Alternating Method. 2025. arXiv: 2510.05350 [math.NA].url:https://arxiv.org/abs/2510.05350

    arXiv 2025

  35. [35]

    Ian Moore et al.Domain Decomposition-based coupling of Operator Inference reduced order models via the Schwarz alternating method. 2024. arXiv:2409.01433

    Pith/arXiv arXiv 2024

  36. [36]

    The Schwarz alternating method in solid mechanics

    Alejandro Mota, Irina Tezaur, and Coleman Alleman. “The Schwarz alternating method in solid mechanics”. In:Computer Methods in Applied Mechanics and Engineering319 (2017), pp. 19–51. doi:10.1016/j.cma.2017.02.006

    work page doi:10.1016/j.cma.2017.02.006 2017

  37. [37]

    The Schwarz alternating method for dynamic solid mechanics

    Alejandro Mota, Irina Tezaur, and Gregory Phlipot. “The Schwarz alternating method for dynamic solid mechanics”. In:International Journal for Numerical Methods in Engineering(2022), pp. 1–36. doi:10.1002/nme.6982

    work page doi:10.1002/nme.6982 2022

  38. [38]

    A variational, finite-deformation constitutive model for piezoelectric materials

    Alejandro Mota and Jonathan A. Zimmerman. “A variational, finite-deformation constitutive model for piezoelectric materials”. In:International Journal for Numerical Methods in Engineering85.6 (2011), pp. 752–767.doi:10.1002/nme.2993

    work page doi:10.1002/nme.2993 2011

  39. [39]

    A Fundamentally New Coupled Approach to Contact Mechanics via the Dirichlet-Neumann Schwarz Alternating Method

    Alejandro Mota et al. “A Fundamentally New Coupled Approach to Contact Mechanics via the Dirichlet-Neumann Schwarz Alternating Method”. In:International Journal for Numerical Methods in Engineering126.9 (2025), e70039.doi:10.1002/nme.70039

    work page doi:10.1002/nme.70039 2025

  40. [40]

    Finite-element simulation of firearm injury to the human cranium

    Alejandro Mota et al. “Finite-element simulation of firearm injury to the human cranium”. In: Computational Mechanics31 (2003), pp. 115–121.doi:10.1007/s00466-002-0398-8

    work page doi:10.1007/s00466-002-0398-8 2003

  41. [41]

    Parish, A

    E. Parish, A. Gruber, and I. Tezaur.Non-intrusive reduced-order modeling of dynamical systems using structure-preserving composable neural networks. in preparation. 2025

    2025

  42. [42]

    Embedded symmetric positive semi-definite machine-learned elements for reduced-order modeling in finite-element simulations with application to threaded fasteners

    Eric Parish et al. “Embedded symmetric positive semi-definite machine-learned elements for reduced-order modeling in finite-element simulations with application to threaded fasteners”. In: Computational Mechanics74 (2024), pp. 1357–1381.doi:10.1007/s00466-024-02481-5. 43

    doi:10.1007/s00466-024-02481-5 2024

  43. [43]

    Model Reduction for Transport-Dominated Problems via Online Adaptive Bases and Adaptive Sampling

    Benjamin Peherstorfer. “Model Reduction for Transport-Dominated Problems via Online Adaptive Bases and Adaptive Sampling”. In:SIAM Journal on Scientific Computing42.5 (2020), A2803– A2836.doi:10.1137/19M1257275

    doi:10.1137/19m1257275 2020

  44. [44]

    Data-driven operator inference for nonintrusive projection-based model reduction

    Benjamin Peherstorfer and Karen Willcox. “Data-driven operator inference for nonintrusive projection-based model reduction”. In:Computer Methods in Applied Mechanics and Engineering 306 (2016), pp. 196–215

    2016

  45. [45]

    Reduced Operator Inference for Nonlinear Partial Differential Equations

    Elizabeth Qian, Ionut ¸-Gabriel Farca¸ s, and Karen Willcox. “Reduced Operator Inference for Nonlinear Partial Differential Equations”. In:SIAM Journal on Scientific Computing44.4 (2022), A1934– A1959.doi:10.1137/21M1393972

    work page doi:10.1137/21m1393972 2022

  46. [46]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

    Maziar Raissi, Paris Perdikaris, and George E. Karniadakis. “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”. In:Journal of Computational Physics378 (2019), pp. 686–707.doi: 10. 1016/j.jcp.2018.10.045

    2019

  47. [47]

    Cameron Rodriguez et al.Transmission Conditions for the Non-Overlapping Schwarz Coupling of Full Order and Operator Inference Models. 2025. arXiv: 2509.12228 [math.NA].url: https: //arxiv.org/abs/2509.12228

    arXiv 2025

  48. [48]

    Henrik Rosenberger, Benjamin Sanderse, and Giovanni Stabile.Exact operator inference with minimal data. 2025. arXiv:2506.01244 [math.NA].url:https://arxiv.org/abs/2506.01244

    arXiv 2025

  49. [49]

    Z¨ urcher u

    Hermann Amandus Schwarz.Ueber einen Grenz¨ ubergang durch alternirendes Verfahren. Z¨ urcher u. Furrer, 1870

  50. [50]

    Turbulence and the dynamics of coherent structures, part III: dynamics and scaling

    Lawrence Sirovich. “Turbulence and the dynamics of coherent structures, part III: dynamics and scaling”. In:Quarterly of Applied Mathematics45.3 (1987), pp. 583–590

    1987

  51. [51]

    Will Snyder, Irina Tezaur, and Christopher Wentland.Domain decomposition-based coupling of physics-informed neural networks via the Schwarz alternating method. 2023. arXiv: 2311.00224 [math.NA]

    Pith/arXiv arXiv 2023

  52. [52]

    Anderson Acceleration for Fixed-Point Iterations

    Homer F. Walker and Peng Ni. “Anderson Acceleration for Fixed-Point Iterations”. In:SIAM Journal on Numerical Analysis49.4 (2011), pp. 1715–1735.doi:10.1137/10078356X

    doi:10.1137/10078356x 2011

  53. [53]

    Mosaic flows: A transferable deep learning framework for solving PDEs on unseen domains

    Hengjie Wang et al. “Mosaic flows: A transferable deep learning framework for solving PDEs on unseen domains”. In:Computer Methods in Applied Mechanics and Engineering389 (2022), p. 114424.doi:10.1016/j.cma.2021.114424

    arXiv 2022

  54. [54]

    Wei Wang et al.Time Marching Neural Operator FE Coupling: AI Accelerated Physics Modeling

  55. [55]

    Wentland et al.The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models

    Christopher R. Wentland et al.The role of interface boundary conditions and sampling strategies for Schwarz-based coupling of projection-based reduced order models. ArXiv pre-print. 2024.url: https://arxiv.org/abs/2410.04668

    Pith/arXiv arXiv 2024

  56. [56]

    GPTune: A Framework for Automated Hyperparameter Tuning of Machine Learning Models

    Zhiqiang Yuan et al. “GPTune: A Framework for Automated Hyperparameter Tuning of Machine Learning Models”. In:Proceedings of the 2021 IEEE International Conference on Big Data (Big Data). IEEE, 2021, pp. 1234–1242.doi:10.1109/BigData52589.2021.9671740. 44

    arXiv 2021

  57. [2004]

    2006, pp

    Springer. 2006, pp. 41–58

    2006

  58. [2025]

    arXiv:2504.11383 [cs.LG].url:https://arxiv.org/abs/2504.11383

    Pith/arXiv arXiv