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arxiv: 2607.06261 · v1 · pith:ZFVAF5OK · submitted 2026-07-07 · math.NA · cs.NA

Learning Adaptive Coarse Spaces Using Transferable Neural Network Models for Linear and Nonlinear Overlapping Domain Decomposition Methods

Axel Klawonn , Martin Lanser , Janine Weber-Hamacher This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-07-08 11:30 UTCglm-5.2pith:ZFVAF5OKrecord.jsonopen to challenge →

classification math.NA cs.NA MSC 65F1065N3065N5568T0568T07
keywords domain decompositionoverlapping Schwarz methodadaptive coarse spacesneural networksAGDSWmachine learningnumerical linear algebrap-Laplace
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The pith

Neural Networks Trained on Simple Diffusion Replace Expensive Eigenvalue Solves

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

The paper claims that neural networks trained solely on data from linear scalar diffusion problems can predict adaptive coarse basis functions for overlapping Schwarz domain decomposition methods, replacing expensive generalized eigenvalue solves entirely. A two-stage approach is used: regression networks predict the shape of adaptive constraints, while a classification network predicts how many constraints each edge needs. The authors show that these networks, trained only on linear scalar problems, transfer without retraining to linear elasticity and nonlinear p-Laplace equations while maintaining robust convergence. The central mechanism is the AGDSW-slab coarse space, where the network input is restricted to coefficient values in a narrow slab around each subdomain edge, a sign-invariant loss function handles the sign ambiguity of eigenvectors, and a post-processing routine enforces flat plateaus in the predicted constraints that the authors show are critical for convergence.

Core claim

The core object is the LAGDSW coarse space: a learned replacement for the adaptive GDSW coarse space in overlapping Schwarz methods. It is constructed by sampling coefficient values in a slab around each interface edge, feeding them to a classifier that determines the number of adaptive constraints needed, then feeding them to regression networks that predict the constraint shapes. A plateau post-processing step then enforces constant-valued regions in the predictions. The authors demonstrate that this pipeline, trained exclusively on 36,000 linear scalar diffusion instances with coefficient contrast 1e6, produces coarse spaces that keep iteration counts low for linear elasticity (with a rig

What carries the argument

Two-stage neural network pipeline (regression + classification) with sign-invariant MSE loss, slab-based sampling, and plateau post-processing

If this is right

  • If the transferability claim holds broadly, solvers for heterogeneous PDEs could avoid eigenvalue computations entirely, reducing setup time by the roughly 50% that eigenvalue construction currently costs in parallel implementations.
  • The plateau post-processing discovery suggests that the geometric structure of coarse basis functions matters more for convergence than their exact numerical values, which could inform non-machine-learning approximations as well.
  • The classifier-based approach to selecting the number of constraints per edge could be applied to other adaptive domain decomposition methods beyond AGDSW, such as FETI-DP or BDDC.
  • The deterioration of the lifted coarse space for elasticity at extreme coefficient jumps (1e6) identifies a boundary where scalar-to-vector lifting fails, motivating either problem-specific training or hybrid approaches that fall back to eigenvalue solves on difficult edges.

Where Pith is reading between the lines

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

  • The fact that networks trained on striped coefficient patterns generalize to dual-phase steel microstructures suggests that the relevant features for constraint prediction are local geometric features of the coefficient field near edges, not global problem structure.
  • The success of the sign-invariant loss function implies that the coarse space only needs to span the right subspace, not reproduce specific eigenvectors, which raises the question of whether simpler surrogate methods (e.g., geometric heuristics) might partially substitute for neural networks.
  • The failure of the lifting heuristic for elasticity at high contrast suggests that vector-valued problems have coarse modes that cannot be decomposed into scalar modes multiplied by rigid body motions, pointing to a structural limitation of the lifting approach rather than a limitation of the learning.
  • If the plateau structure is truly the load-bearing feature, one could test whether a purely geometric plateau-detection algorithm on the coefficient field, without any neural network, already captures most of the robustness.

Load-bearing premise

The heuristic lifting of scalar learned constraints to the elasticity system assumes that multiplying scalar constraints by rigid body modes yields a robust coarse space, but the paper itself states there is no theory supporting this, and the authors' own experiments show it deteriorates at large coefficient jumps where the eigenvalue-based approach remains robust.

What would settle it

Show that for a broad class of realistic elasticity microstructures with coefficient jumps of 1e6 or higher, the lifted LAGDSW coarse space consistently requires significantly more iterations than AGDSW or fails to converge, demonstrating that the scalar-to-vector transfer is not reliable for extreme contrasts.

Figures

Figures reproduced from arXiv: 2607.06261 by Axel Klawonn, Janine Weber-Hamacher, Martin Lanser.

Figure 1
Figure 1. Figure 1: Visualization of our network models Nl to predict discrete approximations of the AGDSW edge constraint. Dark blue corresponds to a low coefficient and yellow corresponds to a high coefficient. The red samples within slabs are used as input data for a feedforward neural network (left and middle). A plateau post-processing is applied to the approximated edge constraints obtained by the neural network (middle… view at source ↗
Figure 2
Figure 2. Figure 2: Exemplary training data with a smart random coefficient distribution. [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of different loss curves for training and validation loss for the [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Visualization of our SciML model for the prediction of discrete adaptive [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Loss curves for training and validation data as well as confusion matrix for [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Top left: Example of an original AGDSW (slab) constraint with marked plateaus; top right: perturbed constraint adding noise within plateaus; bottom left: perturbed constraint adding noise outside the plateaus; bottom right: perturbed constraint adding noise everywhere. constant. Consequently, the most important task for our prediction models is to accurately predict the plateaus and the right constant valu… view at source ↗
Figure 7
Figure 7. Figure 7: Iteration count versus noise for all three modes: perturbation within [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Three different adaptive constraints (ground truth), the neural network pre [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Pareto front plot: Comparison of different setups (AGDSW-slab versus AGDSW; clean NN outputs versus post-processed ones) for 100 subdomains and 100 different coefficient distributions (randomly generated; see [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Three exemplary edges; from top to bottom: prediction with slab-based [PITH_FULL_IMAGE:figures/full_fig_p017_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Same three edges as in Fig [PITH_FULL_IMAGE:figures/full_fig_p017_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Pareto front plots: coarse space size versus PCG iterations; compar￾ison of different coarse spaces for 100 (top) and 400 (bottom) subdomains. While the x-axis represents the size of the coarse space, the y-axis shows the number of PCG iterations; 100 different coefficient distributions (randomly generated; see [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of scaling behavior of different coarse spaces; scalability from [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of scaling behavior of LAGDSW with classifier; scalability [PITH_FULL_IMAGE:figures/full_fig_p021_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Increasing the subdomain size, that is, increasing [PITH_FULL_IMAGE:figures/full_fig_p022_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Increasing the high coefficient ρmax from 100 to 1e8. We considered 20 dif￾ferent coefficient distributions and solved with GDSW, AGDSW-slab, and LAGDSW￾slab in different variants using a fixed number of three constraints per edge, combining LAGDSW with a classifier, and additionally solving the adaptive eigenvalue problem on edges assigned to class 4 (clf+evp@4). Top: Iteration counts; Bottom: Average nu… view at source ↗
Figure 17
Figure 17. Figure 17: Pareto front plot: Coarse space size versus PCG iterations; comparison of different coarse spaces for linear elasticity and 400 subdomains; Elow = 210 and Ehigh = 2.1e8. While the x-axis represents the size of the coarse space, the y-axis shows the number of PCG iterations; 10 different randomly chosen parts of a micro￾section of a realistic dual-phase steel are considered (see left image) and solved with… view at source ↗
Figure 18
Figure 18. Figure 18: Pareto front plots: Coarse space size versus PCG iterations; compar￾ison of different coarse spaces for 100 subdomains for an elasticity problem. Top: Elow = 210 and Ehigh = 210 000; Bottom: Elow = 210 and Ehigh = 2.1e8. While the x-axis represents the size of the coarse space, the y-axis shows the number of PCG iterations; 100 different coefficient distributions (randomly generated; see [PITH_FULL_IMAGE… view at source ↗
Figure 19
Figure 19. Figure 19: Pareto front plot: Comparison of different coarse spaces for 100 sub￾domains for an elasticity problem; Elow = 210 and Ehigh = 2.1e8. While the x-axis represents the size of the coarse space, the y-axis shows the number of PCG itera￾tions; 100 different coefficient distributions (randomly generated; see [PITH_FULL_IMAGE:figures/full_fig_p027_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Comparison of Newton-Krylov-Schwarz and nonlinear Schwarz methods [PITH_FULL_IMAGE:figures/full_fig_p030_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Comparison of Newton-Krylov-Schwarz and nonlinear Schwarz methods [PITH_FULL_IMAGE:figures/full_fig_p031_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Comparison of Newton-Krylov-Schwarz and nonlinear Schwarz methods [PITH_FULL_IMAGE:figures/full_fig_p032_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Comparison of Newton-Krylov-Schwarz and nonlinear Schwarz methods [PITH_FULL_IMAGE:figures/full_fig_p033_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Left: Training data generated on a mesh with H/h = 10 and mapped to H/h = 20. Right: Same data with randomly generated refinement at the boundary of high coefficient areas [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Left: Part of the original DP steel microsection. Right: Coarsening using a majority voting on blocks of 2 × 2 pixels is applied to match the resolution of non-refined smart training data. and the refined one, perform equally well; see [PITH_FULL_IMAGE:figures/full_fig_p034_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Pareto front plot (stationary diffusion): Coarse space size versus PCG iterations; comparison of different coarse spaces for 10 different subsections of a dual-phase steel microstructure (see left image); stationary diffusion with 400 subdomains each; ρlow = 1 and ρhigh = 1e6. While the x-axis represents the size of the coarse space, the y-axis shows the number of PCG iterations; coarse spaces: robust AGD… view at source ↗
Figure 27
Figure 27. Figure 27: Pareto front plot (linear elasticity): Coarse space size versus PCG iterations; comparison of different coarse spaces for 10 different subsections of a dual￾phase steel microstructure (see left image); linear elasticity with 400 subdomains each; Elow = 210.0 and Ehigh = 2.1e8. While the x-axis represents the size of the coarse space, the y-axis shows the number of PCG iterations; coarse spaces: robust AGD… view at source ↗
Figure 28
Figure 28. Figure 28: Pareto front plot (linear elasticity): Coarse space size versus PCG iterations; comparison of different coarse spaces for 10 different subsections of a dual￾phase steel microstructure, which is coarsened by a 2×2 pixel majority voting (see left image); linear elasticity with 400 subdomains each; Elow = 210.0 and Ehigh = 2.1e8. While the x-axis represents the size of the coarse space, the y-axis shows the … view at source ↗
read the original abstract

Domain decomposition methods have been established as efficient and parallel scalable iterative solvers and preconditioners for the solution of large-scale systems arising from the discretization of partial differential equations. In particular, overlapping Schwarz methods have been successfully applied to a wide range of linear and nonlinear problems. However, for problems with highly heterogeneous coefficients, standard domain decomposition methods typically suffer from deteriorating convergence rates. Robustness with respect to the coefficient contrast can be achieved by enriching the coarse space with adaptively selected constraints obtained from local generalized eigenvalue problems. The construction of these adaptive coarse spaces, however, can account for a significant part of the overall computing time. In the present work, machine learning techniques are employed to reduce this part of the computing time in the context of the adaptive Generalized Dryja-Smith-Widlund (AGDSW) coarse space. A two-stage approach is proposed in which regression neural networks are used to predict the adaptive coarse basis functions, while a classification neural network is employed to predict the number of basis functions required to ensure robustness. As a consequence, adaptive coarse spaces can be set up in the online phase without solving any eigenvalue problem. Particular attention is paid to problem-specific aspects, including sign-invariant loss functions and post-processing strategies to significantly improve the predicted constraints. The proposed approach is first investigated for scalar diffusion problems with high coefficient contrasts and is subsequently transferred, without retraining, to problems of linear elasticity and to nonlinear $p$-Laplace problems, also within a nonlinear Schwarz framework.

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.

Referee Report

2 major / 8 minor

Summary. The paper proposes a two-stage machine learning approach to replace eigenvalue solves in the construction of AGDSW-type adaptive coarse spaces for overlapping Schwarz methods. Regression neural networks predict adaptive coarse basis functions, and a classification network predicts the number of constraints needed per edge. The approach is trained exclusively on linear scalar diffusion data and then transferred without retraining to linear elasticity (via a lifting heuristic) and nonlinear p-Laplace problems (within both Newton-Krylov-Schwarz and nonlinear Schwarz frameworks). The numerical experiments are extensive, covering scaling in subdomain count, subdomain size, coefficient contrast, and different PDE types, and include honest reporting of failure cases.

Significance. The paper makes a substantial contribution to the intersection of domain decomposition methods and machine learning. The two-stage regression-plus-classification approach that fully eliminates eigenvalue solves in the online phase is a clear advance over prior work. The sign-invariant MSE loss and plateau post-processing are well-motivated, problem-aware design choices. The transferability to nonlinear Schwarz methods (p-Laplace) without retraining is a genuinely new and valuable result. The extensive numerical studies, including Pareto front plots of coarse space size versus iteration count, provide a thorough and honest assessment. The paper also ships a clear, reproducible description of all hyperparameters and training data generation procedures.

major comments (2)
  1. The headline transferability claim to linear elasticity is stated without sufficient qualification. The abstract states that the networks are 'transferred, without retraining, to problems of linear elasticity,' but Section 5.2 explicitly notes 'there is no theory that supports the assumption that a lifted adaptive coarse space is robust for elasticity problems,' and Fig. 18 (bottom, E_high=2.1e8) shows LAGDSW deteriorating significantly while AGDSW-slab remains robust. Crucially, Fig. 19 demonstrates that this is not a neural network quality issue: even exact scalar AGDSW-slab constraints, when lifted, fail to span the same coarse space as directly computed elasticity AGDSW-slab. This means the limitation is fundamental to the lifting approach, not to the NN. The abstract and introduction should explicitly qualify the elasticity transferability claim as conditional on 'realistic' (moder-
  2. Section 5.2, Remark: The lifting heuristic multiplies scalar learned constraints by rigid body modes [1,0], [0,1], and [-y,x]. The paper is transparent that this lacks theoretical support and shows it fails for extreme contrasts. However, the central claim of transferability to elasticity depends on this heuristic. The paper should more clearly delineate which elasticity modes cannot be represented as scalar constraints times rigid body modes (the 'important modes' mentioned in the Remark), and whether these are modes involving coupling between displacement components or modes with spatial structure not capturable by any scalar function. A brief discussion of what the elasticity eigenvalue problem captures that the lifted scalar problem cannot would strengthen the paper's honesty and utility.
minor comments (8)
  1. Section 5.2: The rigid body mode for rotation is written as [-x, y] in one place and [-y, x] in another. Please use a consistent convention throughout.
  2. Section 4.1: 'epsilon_abs = 0.024 for the second constraint, epsilon_abs = 0.034 for the third one, and finally epsilon_abs = 0.071 for the third.' The last value should presumably be for the fourth constraint, not the third.
  3. Section 4.3.2: The median of 123.75 for 1600 subdomains is reported, but it is unclear if this refers to PCG iterations or coarse constraints per subdomain. Please clarify.
  4. Section 2.1: The sentence beginning 'The problem of stationary diffusion, is' has a formatting issue (missing space).
  5. Figures 9, 12, 15, 16, 18, 19, 26, 27, 28: The Pareto front plots would benefit from larger axis labels and markers for readability in the current format.
  6. Section 3.2: The training data generation uses H/h=10 mapped to H/h=20, but the sampling grid has 40 points per row. The relationship between these resolutions should be stated more explicitly.
  7. The paper would benefit from a brief discussion of computational timing: how much faster is the NN-based coarse space construction compared to solving the eigenvalue problems? This is the primary motivation but no runtime comparison is provided in the numerical results.
  8. Section 6.2: The statement that LAGDSW with k=3 'often fails to converge' for nonlinear Schwarz is attributed to the coarse space being too large. This claim would be stronger with supporting evidence or a reference to the specific failure cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive assessment. Both major comments concern the transferability claim to linear elasticity and the lifting heuristic. We agree that the abstract and introduction overstate the elasticity results and will revise accordingly. We also agree that a more detailed discussion of which elasticity modes cannot be represented by lifted scalar constraints would strengthen the paper and will add it.

read point-by-point responses

  1. Referee: The headline transferability claim to linear elasticity is stated without sufficient qualification. The abstract states that the networks are 'transferred, without retraining, to problems of linear elasticity,' but Section 5.2 explicitly notes 'there is no theory that supports the assumption that a lifted adaptive coarse space is robust for elasticity problems,' and Fig. 18 (bottom, E_high=2.1e8) shows LAGDSW deteriorating significantly while AGDSW-slab remains robust. Crucially, Fig. 19 demonstrates that this is not a neural network quality issue: even exact scalar AGDSW-slab constraints, when lifted, fail to span the same coarse space as directly computed elasticity AGDSW-slab. This means the limitation is fundamental to the lifting approach, not to the NN. The abstract and introduction should explicitly qualify the elasticity transferability claim as conditional on 'realistic' (moder[

    Authors: The referee is correct on all points. The abstract and introduction currently overstate the elasticity transferability claim by omitting the qualification that the body of the paper makes clear. As the referee observes, and as our own Figure 19 demonstrates, the limitation is fundamental to the lifting heuristic itself, not to the neural network predictions: even exact scalar AGDSW-slab constraints, when lifted via rigid body modes, fail to span the same coarse space as directly computed elasticity AGDSW-slab. We will revise the abstract to explicitly qualify the elasticity claim as conditional on realistic coefficient contrasts and to note that the lifting approach is a heuristic without theoretical guarantees. Specifically, we will change the relevant sentence to indicate that the networks are transferred to linear elasticity 'using a lifting heuristic that is effective for realistic coefficient contrasts but lacks theoretical guarantees and deteriorates for extreme contrasts.' We will make a corresponding revision in the introduction, replacing the current unqualified statement that the same networks 'can also be used for the system of linear elasticity with almost no loss in efficiency' with a statement that accurately reflects the Section 5.3 findings: the approach works well for physically realistic jumps (e.g., E_high/E_low = 1000, as in dual-phase steels) but deteriorates for extreme contrasts, and that this limitation is inherent to the lifting approach rather than the learned predictions. revision: yes

  2. Referee: Section 5.2, Remark: The lifting heuristic multiplies scalar learned constraints by rigid body modes [1,0], [0,1], and [-y,x]. The paper is transparent that this lacks theoretical support and shows it fails for extreme contrasts. However, the central claim of transferability to elasticity depends on this heuristic. The paper should more clearly delineate which elasticity modes cannot be represented as scalar constraints times rigid body modes (the 'important modes' mentioned in the Remark), and whether these are modes involving coupling between displacement components or modes with spatial structure not capturable by any scalar function. A brief discussion of what the elasticity eigenvalue problem captures that the lifted scalar problem cannot would strengthen the paper's honesty and utility.

    Authors: We agree that this discussion would strengthen the paper and will add it. The key distinction is as follows. The lifted scalar approach constructs, from each scalar edge constraint l, three vector-valued constraints: l*[1,0], l*[0,1], and l*[-y,x]. This means the spatial profile of each displacement component is always proportional to the same scalar function l (up to the rigid body mode factor). The elasticity eigenvalue problem, by contrast, solves a generalized eigenvalue problem on the full vector-valued stiffness matrix and can produce eigenvectors where the x- and y-displacement components have genuinely different spatial profiles — i.e., modes with coupling between displacement components that cannot be decomposed as a single scalar function times a rigid body mode. For moderate coefficient contrasts, the dominant adaptive modes are well-approximated by lifted scalar constraints, which is why the heuristic works. For extreme contrasts with complicated microstructures, the elasticity eigenvalue problem captures modes where the two displacement components respond differently to the heterogeneous coefficient structure, and these cannot be represented by any scalar function multiplied by a rigid body mode. We will add a paragraph to Section 5.2 (Remark) making this explicit, clarifying that the unrepresentable modes are precisely those involving component-wise coupling with spatial structure not capturable by any single scalar function. revision: yes

Circularity Check

0 steps flagged

No significant circularity found. The derivation is self-contained against external benchmarks.

full rationale

The paper's central claim—that neural networks trained on scalar diffusion eigenvalue data can predict adaptive coarse basis functions replacing eigenvalue solves—is validated against external benchmarks: PCG iteration counts and condition numbers on independently generated coefficient distributions, compared against the exact AGDSW/AGDSW-slab method. The training data (36,000 eigenvalue problem solutions on smart-random coefficient fields) is generated separately from the test data (different random distributions, different subdomain counts, different PDEs). The predictions are not tautologically forced by the training: the networks predict constraint shapes, which are then post-processed, lifted (for elasticity), and embedded into a Schwarz preconditioner whose convergence is measured independently. The self-citations to [14, 22, 23, 38] establish prior ML frameworks for FETI-DP/BDDC but are not load-bearing for the present paper's claims, which are validated by the numerical experiments within this paper. The Section 5.2 lifting heuristic for elasticity is explicitly acknowledged as unsupported by theory and shown by the paper's own data (Fig. 18–19) to fail for extreme contrasts—this is a correctness limitation, not circularity. The condition number bounds in Section 2.4 cite [11, 12, 29] (co-authored by Klawonn), but these are standard AGDSW theory results used as background, not as the basis for the paper's novel claims. No step in the derivation chain reduces to its inputs by construction.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 2 invented entities

The axiom ledger reveals that the paper's central transferability claim rests on an ad hoc axiom (lifting scalar constraints to elasticity) that the authors themselves flag as theoretically unsupported. The free parameters are standard for ML approaches and are tuned on training data, not test data. The invented entities (LAGDSW, sign-inv-MSE) are validated against external benchmarks.

free parameters (4)
  • k (number of regression networks) = 3
    Chosen a priori as the number of adaptive coarse basis functions to predict; problem-dependent but fixed for the class of problems considered.
  • epsilon_abs (post-processing threshold) = 0.024, 0.034, 0.071
    Tuned separately for each regression network N1, N2, N3 on training data to ensure 100% plateau coverage. Values tested between 0.01 and 0.1.
  • tol (eigenvalue tolerance) = 0.01
    User-chosen tolerance for selecting eigenvectors in the AGDSW eigenvalue problem; determines the ground truth training labels.
  • Neural network hyperparameters = 3 hidden layers, {150,100,80} neurons, 5% dropout, GeLU, Adam, lr=0.001
    Optimized via grid search with cross-validation on training/validation data.
axioms (4)
  • domain assumption The AGDSW condition number bound kappa <= C(1 + 1/tol)^2 holds for the problems considered.
    Invoked in Section 2.4; standard result from [11,12,29] assuming the eigenvalue problems are solved exactly.
  • domain assumption Mirrored (sign-flipped) adaptive constraints are equally robust for the Schwarz method.
    Invoked in Section 3.1 to justify the sign-invariant MSE loss function; based on empirical experience, not formal theory.
  • ad hoc to paper Noise in plateaus of adaptive constraints is more detrimental to convergence than noise outside plateaus.
    Invoked in Section 4.1 to motivate the plateau post-processing; supported by a numerical noise study (Fig. 7) but without formal proof.
  • ad hoc to paper Scalar adaptive constraints can be lifted to elasticity by multiplication with rigid body modes.
    Invoked in Section 5.2; the paper explicitly states 'there is no theory that supports this assumption.' It is a heuristic extension of the GDSW lifting procedure.
invented entities (2)
  • LAGDSW coarse space independent evidence
    purpose: Learned AGDSW coarse space where adaptive constraints are predicted by neural networks instead of eigenvalue problems.
    Validated externally by comparing PCG/GMRES iteration counts and coarse space sizes against GDSW and AGDSW-slab on multiple test problems.
  • Sign-invariant MSE loss function independent evidence
    purpose: Loss function that takes the minimum of MSE(y, y_pred) and MSE(y, -y_pred) to handle sign ambiguity in eigenvectors.
    Falsifiable: if sign-flipped constraints were not robust, the resulting LAGDSW coarse spaces would show deteriorated convergence, which they do not in the reported experiments.

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Reference graph

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