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For heterogeneous convection–diffusion systems, dominant convection screens the training Gram matrix so that the slowest VPINN training mode concentrates in the diffusive block, at a rate given by S_DD(K/K_CC)S_DD^H; the Péclet-squared slow

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2026-08-01 07:41 UTC pith:AOBC7JMG

load-bearing objection Solid new screening theorem for VPINN training; the NTK-regime caveat is real but acknowledged, and the Adam part is honestly labeled empirical. the 3 major comments →

arxiv 2607.21352 v1 pith:AOBC7JMG submitted 2026-07-23 math.AP

Double screening in the training dynamics of variational physics-informed neural networks for heterogeneous coupled parabolic systems

Ali Ouattara Kouma , Gossrin Jean-Marc Bomisso This is my paper
classification math.AP MSC 65M1265K1068T0735K4035K57
keywords variational physics-informed neural networksneural tangent kernelcoupled parabolic systemsdouble screeningSchur complementPéclet numbercondition numberAdam optimizer
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.

This paper analyzes why variational physics-informed neural networks (VPINNs) struggle to train on coupled convection–diffusion–reaction systems in which only some components are advected. In the neural tangent kernel regime, training reduces to a linear residual dynamics governed by a Gram matrix built from the system symbol and the kernel. The main result, a double screening theorem, shows that under dominant convection the effective operator for the slow mode is simply the diffusive block of the symbol times the kernel's Schur complement — all inter-component coupling and convective kernel correlations vanish to leading order. From this, the paper derives that the slowest mode localizes in the diffusive components, the condition number grows like the square of the Péclet number, and that this growth slows down shared-step gradient descent even though the continuous flow is unaffected. The authors conclude that the training difficulty is an optimizer artifact and show numerically that Adam with per-component disjoint subnetworks neutralizes it, validating the predictions on finite-width networks.

Core claim

The central claim, Theorem 4.1, is algebraic and unconditional once the Gram matrix is defined. Let bG = bA bK bA^H be the frequency-localized Gram matrix, with symbol bA = S + ibξP, P the projector onto convective components, and bK the Hermitian positive definite tangent kernel. Then as b→∞, the Schur complement of bG with respect to the convective block satisfies bG/bGCC = S_DD (bK/bKCC) S_DD^H + O(b^{-1}). The limit depends neither on the system's coupling blocks nor on the kernel's correlation between convective and diffusive components except through the kernel's Schur complement. Consequently the slowest training mode concentrates in the diffusive block, its rate is λ_min(S_DD (bK/bKC

What carries the argument

The carrying object is the space-time Gram matrix bG = bA bK bA^H, which governs the residual dynamics ṙ = −Gr in the tangent kernel regime. The proof applies the variational characterization of the Schur complement twice: first at the level of bG, where the convective variables act as an O(b^{-1}) lever that freely places the kernel variables; then at the level of bK, where the optimal choice of the convective direction sacrifices the kernel's correlation with the diffusive block, leaving only the Schur complement bK/bKCC. This double minimization — screening first the system coupling, then the kernel correlation — is the mechanism that isolates the diffusive block in the limit.

Load-bearing premise

The entire training-dynamics prediction rests on the assumption that the tangent kernel stays frozen at its infinite-width limit during training; the paper notes this is rigorously established only for one-hidden-layer networks with PINN losses, and if the kernel evolves the spectral analysis no longer describes the actual trajectory.

What would settle it

Integrate the actual gradient flow of a finite-width VPINN on the paper's three-component system and measure the slow-mode decay rate as b increases: if λ_min of the empirical Gram matrix does not saturate to λ_min(S_DD(bK/bKCC)S_DD^H) within finite-width corrections, the double-screening claim fails. A simpler check: compute κ(G) from the assembled Gram matrix in the discrete setting and verify the Pe² slope beyond the threshold; any deviation from that slope in a regime where the kernel is frozen would refute Corollary 5.3.

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

If this is right

  • The slowest mode of the VPINN training dynamics localizes asymptotically in the diffusive block, with rate λ_min(S_DD(bK/bKCC)S_DD^H) independent of convection intensity.
  • The condition number of the Gram matrix grows like Pe², bounded by Pe²/(1−ρ²), which makes shared-step gradient descent take Pe² log(1/τ) iterations to reduce the diffusive residual by a factor τ.
  • Continuous gradient flow suffers no convection slowdown; the training difficulty is attributable to the optimizer's shared step, not to the variational approximation.
  • A fixed diagonal weighting of the residual by component removes the Pe² growth for shared-step descent, with a computable reference frequency.
  • An architecture with disjoint subnetworks for convective and diffusive components removes the kernel correlation and, under Adam, keeps training cost nearly stationary beyond the Péclet threshold.

Where Pith is reading between the lines

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

  • The theorem's algebraic character suggests the double-screening limit holds for any linear system whose symbol has the form S + ibξP, so the same spectral diagnosis should apply to other operator-learning settings, such as Fokker–Planck or linearized transport problems with a projected advection term; this is an extrapolation the paper does not make.
  • The Pe² cost is stated for the residual dynamics; an analogous argument applied to parameter dynamics would suggest that per-block learning-rate schedules on the diffusive parameters could mimic the selective weighting without needing a reference frequency — a testable design for adaptive optimizers.
  • If the frozen-kernel assumption is relaxed, the screening limit can be read as the zero-order term of a perturbation series in the kernel's evolution; tracking λ_min(G(t)) during training would show whether the slow-mode rate drifts, which the paper leaves as an open direction.
  • The rank-collapse result for long-correlation kernels suggests that the test-function support and kernel correlation length should be matched (h ∼ ℓ_K); an extension would verify that this calibration also controls the double-screening rate in spatially distributed domains.

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

3 major / 4 minor

Summary. The manuscript analyzes the gradient-flow training dynamics of variational physics-informed neural networks (VPINNs) applied to heterogeneous coupled linear parabolic convection–diffusion–reaction systems. Restricting to the neural tangent kernel (NTK) regime, it reduces the residual dynamics to a constant-coefficient linear system whose Gram matrix, under frequency localization, takes the form bG = bA bK bA^H with bA = S + i b ξ P. The central algebraic result, Theorem 4.1, identifies the limit of the Schur complement of bG with respect to the convective block as S_DD (bK/bK_CC) S_DD^H + O(b^{-1}). From this the authors derive four consequences: an exact coupling-gap identity, a canonical-correlation degradation law, a Péclet-squared condition-number bound, and the non-participation of temporal frequencies. They then attribute the training slowdown to the shared optimizer step rather than to the continuous flow, propose selective weighting and a disjoint-subnetwork architecture, and report that Adam mitigates the difficulty when the architecture is separated. Numerical validation includes machine-precision checks on the exact symbol, a non-translation-invariant B-spline assembly, random dense systems, a two-dimensional exchanger, and full training of one-hidden-layer networks.

Significance. The algebraic core is clean and gives specific, falsifiable predictions: the double-screening limit is parameter-free once the Gram matrix is defined; the Pe^2 condition-number growth, the diffusive localization of the slow mode, the 1−ρ^2 canonical-correlation factor, and the temporal-frequency escape are concrete and testable. The paper is unusually careful in checking the derived formulas against direct evaluation of the exact matrices, reaching machine precision, and the public code is a clear asset. The authors also explicitly label the Adam/architecture result as empirical rather than proved, which is commendable. The main weakness is that the translation from the algebraic theorem to the paper's advertised conclusions about actual VPINN training depends on Assumption 3.1, which the paper itself states is rigorously available for PINN-type losses only in the one-hidden-layer case; the finite-width validation does not measure kernel drift during training. Thus the claims about deep-network training dynamics are conditional, and the manuscript needs to make that conditionality fully explicit or provide further evidence.

major comments (3)
  1. [§3.3, Assumption 3.1; §9.9] This is the load-bearing bridge between the algebraic theorem and the advertised training-dynamics conclusions. Assumption 3.1 asserts that the tangent kernel is frozen and deterministic, so that r-dot = −G r with constant G factorizing as (3.11). The paper itself notes (p. 5) that for losses containing a differential operator this is 'verified empirically but is established rigorously only for one hidden layer [23]'. The finite-width experiments in §9.9 use single-hidden-layer networks and do not monitor the evolution of Θ_θ(t) during training. If the kernel drifts materially, λ_min(G_0), κ(G_0), and the disjoint-subnetwork prescription describe only the initialization, not the trajectory. This does not weaken Theorem 4.1 as a statement about the fixed matrix bG, but it makes the central claim about real VPINN training conjectural for deeper networks. I recommend either explicitly frami
  2. [§7; Abstract; Table 4] The Adam mitigation claim is advertised in the abstract as 'We finally show that the Adam optimizer ... mitigates this difficulty', yet Section 7 states correctly that this is 'an empirical observation motivated by the theory, not a theorem'. The numerical support is restricted to one-dimensional, one-hidden-layer networks with separately tuned learning rates; no analysis is provided of when Adam's per-parameter scaling actually becomes a per-block scaling. If this is intended as a contribution, it should be either proved under explicit assumptions or the abstract and conclusion should use hedged language consistent with Section 7. As written, the abstract overstates the epistemic status of the Adam/architecture conclusion.
  3. [§3.4, Assumption 3.2; Eq. (3.11)] The reduction from the global Gram matrix (3.8) to the per-frequency block form (3.11) is stated through the informal phrase that test functions are 'concentrated near a frequency (ω,ξ)'. No precise theorem is given for the block-diagonalization of the full operator on a bounded domain with compactly supported test functions. Proposition 8.1 provides a decay estimate for off-diagonal entries but does not prove the spectral block-diagonalization used in the main derivation. The discrete validation in §9.2 checks the integrated scaling, not the per-frequency identity. For a mathematical paper, this reduction is load-bearing: it is the step that turns Theorem 4.1 into a statement about the spectrum of the global training operator G. I would ask for a precise statement (or an explicit approximation theorem with error rates) for this localization step, or for a clear statement that (3.11) is
minor comments (4)
  1. [Abstract / §7] The word 'show' in the abstract for the Adam result should be aligned with the 'empirical observation' language of Section 7. The conclusion also says 'We have established' for the training-dynamics claims; consider adding 'within the NTK regime' or 'under Assumption 3.1' at that point.
  2. [Table 3] The raw iteration counts are formatted ambiguously: '2 755 753' presumably means 2755 and 753, but the spacing makes it look like a single number. Use explicit thousand separators or separate columns.
  3. [§2] P is called 'the orthogonal projector' in Theorem 4.1 but earlier defined as a diagonal projector P = diag(1_{i∈C}). The coordinate projector is indeed an orthogonal projector, but the terminology should be consistent: 'coordinate projector' or 'orthogonal projector onto the coordinate subspace'.
  4. [§9.9 / §9.10] The finite-width networks used for the Adam architecture comparison are described only as 'single-hidden-layer perceptrons of width M' in §9.9; specify whether the same one-hidden-layer setting is used in §9.10, since the architectural prescription is meant to apply to deeper networks.

Circularity Check

0 steps flagged

No circular derivation: Theorem 4.1 is a parameter-free Schur-complement asymptotic derived from stated assumptions; the only self-citation is a reproducibility code repository, not load-bearing.

full rationale

The central claim, Theorem 4.1, is an unconditional algebraic statement about bG = bA bK bA^H with bA = S + ibξP. It is proved by applying the variational characterization of Schur complements twice; no parameter is fitted and no target prediction is assumed in the hypotheses. The spectral consequences (Corollaries 4.3, 5.1–5.4, 6.1–6.2) follow by direct inequalities and are validated against exact numerical evaluation of the matrices, not used to set constants. The NTK reduction is explicitly stated as Assumption 3.1 and is supported by an external citation [23], with the paper itself acknowledging that rigorous justification for PINN losses exists only for one hidden layer; this is a stated model limitation, not a circularity. The Adam claim is honestly labeled 'an empirical observation motivated by the theory, not a theorem' (Section 7), and the selective-weighting frequency ξ0 is chosen by a priori reasoning from the fundamental frequency, not calibrated to the predicted rate. The only self-citation is the code availability reference [14], which is for reproducibility and carries no load in the derivation. Accordingly, no specific circular step is present; the minor self-citation alone does not make the argument circular.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central theorem itself is parameter-free. The listed free parameters are design choices in the proposed remedies and empirical comparisons, not fitted constants used to derive the main results. The axioms are dominated by the two stated statistical/physical assumptions (NTK regime, frequency localization) plus standard linear algebra. No new entities are introduced: 'double screening' and 'screened coupling energy' are names for derived mathematical quantities.

free parameters (2)
  • reference frequency ξ0 in selective weighting = π (fundamental frequency on (0,1))
    The weighting (6.2) uses ξ0 chosen in Remark 6.3 as the fundamental frequency of the Dirichlet Laplacian. This is a heuristic design choice for the remedy, not fitted to the numerical predictions being validated.
  • per-architecture learning rate in Adam experiments = optimized separately for each architecture (not reported numerically)
    Section 9.10 tunes the learning rate for shared-trunk and disjoint-subnetwork architectures to enable a fair empirical comparison; it is a hyperparameter, not used to derive the theoretical claims.
axioms (5)
  • domain assumption In the infinite-width limit, the network's tangent kernel Θ_θ converges to a deterministic training-invariant kernel Θ∞, so the residual dynamics is the constant-coefficient linear ODE ṙ = −G r (Assumption 3.1).
    Invoked in Section 3.3 to justify the linear spectral analysis of training. The paper notes it is established rigorously for PINN losses only for one hidden layer [23], and treats it as a model assumption.
  • domain assumption The test functions are compactly supported and concentrated near a single frequency (ω,ξ); the tangent kernel is translation-invariant with Hermitian positive-definite Fourier transform bK(ω,ξ) (Assumption 3.2).
    Justifies the block-diagonalization (3.11) of the Gram matrix into frequency blocks; Section 8 relaxes this via localization estimates.
  • standard math Schur complement variational characterization: for Hermitian M with M_CC ≻ 0, x_D^H(M/M_CC)x_D = min_{x_C} (x_C,x_D)^H M (x_C,x_D) (Eq. 2.3).
    Core tool used in the proof of Theorem 4.1.
  • standard math Weyl's eigenvalue perturbation theorem and monotonicity of λ_min under the Loewner order.
    Used in Corollaries 4.3, 5.2, 5.3.
  • domain assumption The test functions v_k have compact support in Q and the spatial domain is bounded Lipschitz, so boundary/initial terms can be lifted out and divergence terms integrate to zero (Assumption 3.2 and Section 8.2).
    Used to derive (8.3)–(8.4); standard for variational formulations.

pith-pipeline@v1.3.0-alltime-deepseek · 20174 in / 17575 out tokens · 161024 ms · 2026-08-01T07:41:23.394514+00:00 · methodology

0 comments
read the original abstract

We analyze the training dynamics of variational physics-informed neural networks applied to linear coupled parabolic convection--diffusion--reaction systems, called heterogeneous when only a subset of the components undergoes convective transport. In the neural tangent kernel regime, the gradient flow on the space-time variational residuals reduces to a linear differential system whose operator is a Gram matrix built from the space-time symbol of the system operator and the matrix tangent kernel. The main result is a double screening theorem. Under dominant convection, the Schur complement of this matrix relative to the block of convective components converges to an expression that involves only the diffusive block of the symbol, with no coupling term, together with the Schur complement of the tangent kernel. From this we derive four consequences, namely an exact identity quantifying the screened coupling energy, a degradation law for the training rate governed by the canonical correlations of the kernel, a bound on the condition number in terms of the P\'eclet number, and the non-participation of temporal frequencies in the screening mechanism. The growth of the condition number slows down shared-step gradient descent, whereas the continuous flow suffers no slowdown, which makes the training difficulty attributable to the optimizer rather than to the approximation. We finally show that the Adam optimizer, through its adaptive scaling, mitigates this difficulty provided that the architecture separates the parameters associated with the convective and diffusive components, confirming the architectural prescription that follows from the second screening. The predictions are validated numerically down to machine precision, on a two-dimensional exchanger, and by the full training of finite-width networks.

Figures

Figures reproduced from arXiv: 2607.21352 by Ali Ouattara Kouma, Gossrin Jean-Marc Bomisso.

Figure 1
Figure 1. Figure 1: shows λmin(Gb), λmax(Gb) and κ(Gb) as functions of Pe for the system (9.1) with Kb = I. The predictions of Corollaries 4.3 and 5.3 are verified: λmin(Gb) saturates toward λmin(S DDS T DD) = 0.225555, while λmax(Gb) follows the asymptote b 2 ξ 2 . The condition number exhibits quadratic growth in Pe beyond the threshold (5.5), materialized by the kink observed on the two branches. Finally, the eigenvector a… view at source ↗
Figure 2
Figure 2. Figure 2: Discrete setting. Left: condition number of [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: illustrates a solution of this system, obtained by finite differences. The convective component u1 exhibits a marked asymmetry in the transport direction β, while the diffusive component u2, coupled to u1 only through the exchange term α, remains smooth and symmetric. This contrast between an advected component and a diffusive component is the structural signature of the heterogeneous systems under study. … view at source ↗
Figure 4
Figure 4. Figure 4: Two-dimensional exchanger on (0, 1)2 , exact symbol: mean condition number of Gb over the 64 Fourier modes as a function of Pe, with the slope Pe2 (dots). 9.4. The double screening theorem for n components For (n, m) ∈ {(3, 1),(4, 1),(5, 2),(6, 3)}, we build S = R + 2.5 In, where R has independent Gaussian entries, and Kb = BBT + 0.3 In, with B Gaussian. These choices produce dense couplings and kernels de… view at source ↗
Figure 5
Figure 5. Figure 5: shows the relative spectral error between Gb/GbCC and the limit S DD(Kb/ KbCC)S T DD as a function of b. The observed convergence is in O(b −2 ), in accordance with Remark 4.2, and continues down to the round-off￾dominated regime for b = 107 . 10 1 10 2 10 3 10 4 10 5 10 6 10 7 b 10 15 10 13 10 11 10 9 10 7 10 5 10 3 10 1 relative spectral error n=3, m =1 n=4, m =1 n=5, m =2 n=6, m =3 slope b 2 [PITH_FULL… view at source ↗
Figure 6
Figure 6. Figure 6: compares, for ρ ∈ [0, 0.995] and b = 106 , the value of λmin(Gb), the exact limit given by Theorem 4.1, and the lower bound 1 − ρ 2 of Corollary 5.2. The relative gap between the computed value and the exact limit remains below 6.1 × 10−14. The slowdown factor (1 − ρ 2 ) −1 reaches about 5.3 for ρ = 0.9 and 50.3 for ρ = 0.99. 0.0 0.2 0.4 0.6 0.8 1.0 canonical correlation 0.00 0.05 0.10 0.15 0.20 0.25 min m… view at source ↗
Figure 7
Figure 7. Figure 7: Training cost of the diffusive components as a function of Pe, system (9.1): raw dynamics, of slope Pe2 , and selective weighting, with saturation. 9.8. Localization [PITH_FULL_IMAGE:figures/full_fig_p019_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Localization. Left: off-diagonal decay of the entries of G as a function of the separation of the supports, with predicted Gaussian envelopes (dots). Right: normalized spectrum of G for increasing ℓK, with collapse toward the rank of the coupling matrix. 9.9. Finite-width networks We verify that the predictions persist for finite-width networks actually trained: single-hidden-layer perceptrons of width M, … view at source ↗
Figure 9
Figure 9. Figure 9: Finite-width networks with three outputs. Left: slices [PITH_FULL_IMAGE:figures/full_fig_p020_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Full training of a network of width M = 200 with three outputs, system (9.1). Left: without weighting, with stratification and capping by Pe. Right: with selective weighting, with superposition of the curves and convergence independent of Pe. 9.10. Adam optimizer and architecture We finally compare the training by Adam of two architectures of identical total width: a shared trunk with three heads and thre… view at source ↗
Figure 11
Figure 11. Figure 11: Training by Adam: iterations to reach ∥rD∥ / [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗

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Reference graph

Works this paper leans on

23 extracted references · 1 canonical work pages

  1. [1]

    Berrone, C

    S. Berrone, C. Canuto, M. Pintore, Variational physics informed neural networks: The role of quadratures and test functions, J. Sci. Comput. 92 (2022) 100.https://doi.org/10.1007/s10915-022-01950-4

    doi:10.1007/s10915-022-01950-4 2022

  2. [2]

    Berrone, M

    S. Berrone, M. Pintore, Meshfree variational physics informed neural networks: An adaptive training strategy, Algorithms 17 (2024) 415.https://doi.org/10.3390/a17090415

    doi:10.3390/a17090415 2024

  3. [3]

    Chizat, E

    L. Chizat, E. Oyallon, F. Bach, On lazy training in differentiable programming, Adv. Neural Inf. Process. Syst. 32 (2019)

    2019

  4. [4]

    De Ryck, S

    T. De Ryck, S. Mishra, Error analysis for physics-informed neural networks approximating Kolmogorov PDEs, Adv. Comput. Math. 48 (2022) 79.https://doi.org/10.1007/s10444-022-09985-9

    doi:10.1007/s10444-022-09985-9 2022

  5. [5]

    Heinlein, A.A

    A. Heinlein, A.A. Howard, D. Beecroft, P. Stinis, Multifidelity domain decomposition-based physics-informed neural networks and operators for time-dependent problems, arXiv:2401.07888 (2022)

    Pith/arXiv arXiv 2022

  6. [6]

    R. Hu, Q. Lin, A. Raydan, S. Tang, Higher-order error estimates for physics-informed neural networks approximating the primitive equations, Partial Differ. Equ. Appl. 4 (2023) 34. https://doi.org/10.1007/ s42985-023-00254-y

    2023

  7. [7]

    Jacot, F

    A. Jacot, F. Gabriel, C. Hongler, Neural tangent kernel: Convergence and generalization in neural networks, Adv. Neural Inf. Process. Syst. 31 (2018)

    2018

  8. [8]

    Jagtap, E

    A.D. Jagtap, E. Kharazmi, G.E. Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems, Comput. Methods Appl. Mech. Engrg. 365 (2020) 113028.https://doi.org/10.1016/j.cma.2020.113028

    arXiv 2020

  9. [9]

    V . John, J. Novo, Error analysis of the SUPG finite element discretization of evolutionary convection–diffusion– reaction equations, SIAM J. Numer. Anal. 49 (2011) 1149–1176. https://doi.org/10.1137/100789002

    work page doi:10.1137/100789002 2011

  10. [10]

    Kharazmi, Z

    E. Kharazmi, Z. Zhang, G.E. Karniadakis, Variational physics-informed neural networks for solving partial differential equations, arXiv:1912.00873 (2019)

    Pith/arXiv arXiv 1912

  11. [11]

    Kharazmi, Z

    E. Kharazmi, Z. Zhang, G.E. Karniadakis, hp-VPINNs: Variational physics-informed neural networks with domain decomposition, Comput. Methods Appl. Mech. Engrg. 374 (2021) 113547. https://doi.org/10. 1016/j.cma.2020.113547 23

    arXiv 2021

  12. [12]

    Kingma, J

    D.P. Kingma, J. Ba, Adam: A method for stochastic optimization, Int. Conf. Learn. Represent. (2015)

    2015

  13. [13]

    Krishnapriyan, A

    A. Krishnapriyan, A. Gholami, S. Zhe, R. Kirby, M.W. Mahoney, Characterizing possible failure modes in physics-informed neural networks, Adv. Neural Inf. Process. Syst. 34 (2021)

    2021

  14. [14]

    Double screening in the training dynamics of variational physics- informed neural networks

    A.O. Kouma, G.J.-M. Bomisso, Code for “Double screening in the training dynamics of variational physics- informed neural networks”, Zenodo, 2026

    2026

  15. [15]

    J. Lee, Y . Bahri, R. Novak, S.S. Schoenholz, J. Pennington, J. Sohl-Dickstein, Deep neural networks as Gaussian processes, Int. Conf. Learn. Represent. (2018)

    2018

  16. [16]

    Mishra, R

    S. Mishra, R. Molinaro, Estimates on the generalization error of physics-informed neural networks for approximating PDEs, IMA J. Numer. Anal. 42 (2022) 981–1022. https://doi.org/10.1093/imanum/ drab032

    doi:10.1093/imanum/ 2022

  17. [17]

    Moseley, A

    B. Moseley, A. Markham, T. Nissen-Meyer, Finite basis physics-informed neural networks (FBPINNs): A scalable domain decomposition approach for solving differential equations, Adv. Comput. Math. 49 (2023) 62. https://doi.org/10.1007/s10444-023-10065-9

    doi:10.1007/s10444-023-10065-9 2023

  18. [18]

    Neal, Bayesian Learning for Neural Networks, Lecture Notes in Statistics 118, Springer, New York, 1996.https://doi.org/10.1007/978-1-4612-0745-0

    R.M. Neal, Bayesian Learning for Neural Networks, Lecture Notes in Statistics 118, Springer, New York, 1996.https://doi.org/10.1007/978-1-4612-0745-0

    doi:10.1007/978-1-4612-0745-0 1996

  19. [19]

    Raissi, P

    M. Raissi, P. Perdikaris, G.E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378 (2019) 686–707.https://doi.org/10.1016/j.jcp.2018.10.045

    doi:10.1016/j.jcp.2018.10.045 2019

  20. [20]

    H.-G. Roos, M. Stynes, L. Tobiska, Robust Numerical Methods for Singularly Perturbed Differential Equations, 2nd ed., Springer, Berlin, 2008.https://doi.org/10.1007/978-3-540-34467-4

    doi:10.1007/978-3-540-34467-4 2008

  21. [21]

    Y . Shin, J. Darbon, G.E. Karniadakis, On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs, Commun. Comput. Phys. 28 (2020) 2042–2074. https: //doi.org/10.4208/cicp.OA-2020-0193

    doi:10.4208/cicp.oa-2020-0193 2020

  22. [22]

    S. Wang, Y . Teng, P. Perdikaris, Understanding and mitigating gradient flow pathologies in physics- informed neural networks, SIAM J. Sci. Comput. 43 (2021) A3055–A3081. https://doi.org/10.1137/ 20M1318043

    2021

  23. [23]

    S. Wang, X. Yu, P. Perdikaris, When and why PINNs fail to train: A neural tangent kernel perspective, J. Comput. Phys. 449 (2022) 110768.https://doi.org/10.1016/j.jcp.2021.110768 24

    arXiv 2022