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Sequential data assimilation for PDEs using shape-morphing solutions

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Pith's one-line read Sequential data assimilation corrects shape-morphing PDE solutions toward the true state, with a uniform convergence guarantee under enough sensors.

desk verdict First DA method for shape-morphing solutions, with a clean conditional convergence proof and convincing numerics on three PDEs; the main gap is that the theorem assumes away the Newton convergence the algorithm needs. read the letter →

arxiv 2411.16593 v2 pith:4ESVWTAF submitted 2024-11-25 math.NA cs.NAmath.DSnlin.CD

Zachary T. Hilliard , Mohammad Farazmand This is my paper · ORCID
classification math.NAcs.NAmath.DSnlin.CD
keywords dataassimilationshape-morphingsolutionsevolutionaldeepneuralnetworksPDEapproximationNewton-likeiterationsuniformconvergencesensorplacementpredictor-corrector
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces data-assimilated shape-morphing solutions (DA-SMS), a predictor-corrector scheme for incorporating observational data into shape-morphing approximations of time-dependent PDEs. Between observation times the SMS equations evolve the parameters, and at observation times Newton-like iterations correct the parameters using sensor measurements. The paper proves that under certain conditions, if the sensors are dense enough, the corrected SMS converges uniformly in space to the true solution. Numerical experiments on the nonlinear Schrödinger equation, the Kuramoto–Sivashinsky equation, and a two-dimensional advection-diffusion equation show that relatively sparse, noisy observations with a single Newton iteration can keep errors low through the forecast window.

What carries the argument

The shape-morphing ansatz, û(x, θ(t)) = Σ_i a_i(t) φ_i(x, β_i(t)), is evolved by the shape-morphing equation M(θ)θ̇ = f(θ), or its regularized collocation version, where M is the metric tensor of the parameter-to-solution map. The correction step solves C(θ + δθ) = y through the regularized Newton-like iteration θ^(k+1) = θ^(k) + J̃_r(θ^(k))⁺(y − ŷ^(k)). The convergence proof rests on the sensor-spacing parameter Δ, defined as the maximum distance from any point in Ω to its nearest sensor, and on Lemma 1, which uses Lipschitz continuity of u and û to convert convergence at the sensors into uniform convergence over the domain.

What would settle it

Run the Kuramoto–Sivashinsky example with a fixed tolerance ε, choose sensors with spacing Δ just below ε/(2(Lu + L_û)), and iterate the Newton corrections several times at one observation time; if the sensor residual stops decreasing before reaching ε/2, for instance because the Jacobian loses rank or the initial parameter guess lies outside the convergence basin, then the premise of Assumption 1 fails and the claimed uniform convergence is not observed.

Watch

Extended reading notes

Core claim

The central claim is that DA-SMS converges uniformly to the true state u on the whole spatial domain, not just at the sensor locations. The proof combines a triangle-inequality lemma bounding the pointwise error by (Lu + L_û)Δ plus the error at the nearest sensor, with an assumption that the Newton-like iterations drive the sensor error to zero. If the sensor spacing satisfies Δ < ε/(2(Lu + L_û)), then for large iteration number k the uniform error sup_{x∈Ω} |û(x, θ^(k)) − u(x)| is below ε. The paper also derives a continuous-time variant that enforces matching of the observed time derivatives, and reports that clean or 5%-noisy data extend the predictability horizon in all three examples.

Load-bearing premise

The proof relies on the Newton-like corrections actually driving the approximate solution to match the true data at every sensor in the limit, and on a uniform bound on how fast the approximate solution can change in space holding along the entire parameter trajectory.

Editorial extensions

If this is right

  • DA-SMS should extend the predictability horizon of shape-morphing PDE solvers, particularly for chaotic systems where unassisted SMS errors grow rapidly.
  • In practice a single Newton-like iteration per observation time appears sufficient, which makes the scheme cheap enough for sequential real-time forecasting.
  • Increasing the number of sensors and reducing the spacing Δ decreases the uniform error bound, though the bound is pessimistic compared with observed behavior.
  • With Gaussian observational noise of variance σ², the expected uniform error retains a floor proportional to σ, so the method cannot be expected to converge beyond the noise level.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: the convergence guarantee is conditional on the Newton-like iterations actually reaching a parameter vector that matches the observations at all sensors; the paper does not prove that this happens for its SMS ansatz, so the local convergence basin of the underdetermined iterations is the main point to test in practice.
  • Editorial inference: the pessimistic sensor-density bound suggests that adaptive sensor placement, rather than uniform spacing, could achieve the same uniform error with fewer observations.
  • Editorial inference: the network construction that exactly enforces mixed Dirichlet–Neumann boundary conditions could be reused as a building block for other PDE solvers that use shape-morphing or neural ansatzes.
  • Editorial inference: although the proof is written for pointwise observations, the NLS example shows that nonlinear observation operators (such as the wave envelope modulus) can be assimilated; a natural extension is to carry the uniform-convergence argument over to general bounded observation operators.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper introduces DA-SMS, a sequential data assimilation method for shape-morphing solutions (SMS) of PDEs. The method alternates between evolving the SMS parameters with the SMS ODEs between observation times and correcting the parameters at observation times via Newton-like iterations on the observation residual. The main theoretical result, Theorem 1 in Section 3.2, states that if Assumption 1 holds (Lipschitz continuity of the truth and the SMS, and existence of a parameter sequence that asymptotically matches the observations at all sensor locations), then for sufficiently many sensors the SMS converges uniformly to the true solution at a fixed time. The paper also derives a continuous-time constrained variant and demonstrates the discrete method on the nonlinear Schrödinger, Kuramoto–Sivashinsky, and two-dimensional advection-diffusion equations, including a novel neural-network construction that enforces mixed Dirichlet–Neumann boundary conditions exactly.

Significance. The paper addresses a timely and practically relevant question: how to assimilate sparse observations into nonlinear shape-morphing (neural-Galerkin) PDE solvers. Its strengths are the clean formulation of the predictor-corrector scheme, a transparent derivation of the uniform-in-space error bound via the triangle inequality (Lemma 1 and Theorem 1), and three nontrivial numerical examples, one of which (KS) is chaotic. The boundary-condition construction in Appendix B is a useful byproduct. However, the advertised convergence guarantee is substantially narrower than stated: Theorem 1 is conditional on an assumption about the existence of a sensor-matching parameter sequence that is not shown to hold for the Newton iterations, and it concerns a single time rather than the full sequential trajectory. The numerical experiments also do not probe the theorem's hypotheses (e.g., iteration number or sensor density). These gaps weaken the central claim but are potentially addressable.

major comments (4)
  1. [Section 3.2, Assumption 1(2) and Theorem 1] The theorem is conditional on Assumption 1(2), which posits a sequence θ(k) with vanishing sensor residual (29). The paper asserts that this sequence is obtained from the Newton-like iterations (25)/(27), but it never proves that those iterations converge to a zero of C(θ) - y. The cited local convergence result (Kelley, Theorem 2.4.2) requires Lipschitz continuity of C(θ), full row rank of Jr(θ*) and a sufficiently close initial guess; none of these is verified for the SMS ansatz. For example, with the tanh network (46), C(θ) is nonlinear in θ and Jr can be rank-deficient because of scaling redundancies among (ai, wi, bi, ci). Thus Theorem 1 does not, as it stands, apply to the actual DA-SMS algorithm. The authors should either prove the required properties for their ansatz or state Theorem 1 explicitly as a conditional statement about the Newton iterations, and then verify convergence of the iterations numerically (e.g., error vs iteration count at a fixed time).
  2. [Section 3.2, Assumption 1(1)] The Lipschitz constant L_û in Assumption 1(1) is used in Lemma 1 and Theorem 1 as a single number that bounds û(·,θ(k)) for every k in the sequence. The assumption, as written, only requires that each function û(·,θ) be Lipschitz; it does not require that the constants be uniformly bounded along the actual trajectory produced by the SMS ODEs and the Newton iterations. For the tanh ansatz (46), the Lipschitz constant grows with the magnitudes of the weights wi, and the paper gives no control on the parameters over the DA window. Without a uniform bound, the condition Δ < ε/(2(Lu+L_û)) may be impossible to satisfy, making the theorem vacuous. The authors need to add an explicit uniform Lipschitz assumption or prove boundedness of the parameters (or a compact constraint set) for their numerical experiments.
  3. [Section 3.1 and Theorem 1] Theorem 1 provides a uniform-in-space error bound for a single time ti, given a sequence θ(k) at that time. It does not address the sequential structure of Algorithm 1: it gives no bound on the error accumulated during the SMS forecast between observation times, nor on the error after multiple assimilation cycles. The abstract and introduction claim that 'the data assimilated SMS converges uniformly towards the true state of the system,' which suggests a statement about the whole trajectory over the DA window. The proved result is narrower. The authors should either extend the analysis to the full sequential algorithm (for instance, using stability or contraction of the SMS map between observations) or clearly delimit the claim to a single correction step.
  4. [Section 4.2 and Section 5(i)] The numerical experiments do not exercise the regime of Theorem 1. The theorem requires an infinite sequence of Newton iterations (k → ∞) at fixed time and enough sensors so that Δ < ε/(2(Lu+L_û)); in contrast, the KS and AD experiments use a single Newton iteration and relatively coarse sensor spacing (KS: r=10, Δt=2; AD: r=46), and no experiment reports the residual (26) as a function of iteration count. The paper honestly acknowledges in Section 5(i) that the theoretical bounds are pessimistic, but as a result the numerical evidence cannot validate the theorem; it only supports the weaker heuristic claim that one iteration works for the tested cases. A convergence study (error vs k at fixed ti, and error vs r) would materially strengthen the paper.
minor comments (6)
  1. [Algorithm 1] In the while loop, the line 'Jr = ∇θC(θ(k)_{i+1})' should read 'Jr = ∇θC(θ(k)_i)'; as written it evaluates the Jacobian at the next iterate, which is inconsistent with Eq. (27) and the surrounding text.
  2. [Section 3.3] Equation (37) is stated without the Tikhonov regularization parameter γ, while the derivation in Appendix A uses Mγ = M + γI. If γ is set to zero there, say so explicitly; if not, replace M by Mγ consistently.
  3. [Remark 1] The statement that 'equation (29) needs to be modified to read lim_k |...| ≤ |η|' is imprecise because the limit may not exist; it should be a limsup, and the expectation in (33) should be over the noise η with a clear definition.
  4. [Figure 4] The panel titles contain '~.' and 'RE' without definition; define the relative error (e.g., RE(t) = ||u - û||/||u||) and the tilde notation for the regularization parameter in the caption.
  5. [Sections 2 and Appendix A] Use 'Appendix A' and 'Appendix B' instead of 'A' and 'B' in the main text (e.g., 'As we show in A' and 'as described in B') for clarity.
  6. [Section 4.1] The sentence 'For DA-SMS, at every Δt = 0.5 time units...' uses a hyphen in 'DA-SMS' but elsewhere 'DA-SMS' is used; unify the terminology.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem 1 is a conditional interpolation bound, and the DA corrections use observational data rather than fitted predictions.

full rationale

The central convergence claim, Theorem 1, is not equivalent to its inputs. Assumption 1(2) postulates the existence of parameters whose sensor residuals tend to zero; the paper explicitly separates this from the algorithm ('Although in this paper we obtain this sequence through Newton-like iterations (25), alternative methods can also be used'), so the assumption is not defined in terms of the DA-SMS output. The theorem's content is Lemma 1's triangle-inequality transfer from sensor residuals to the uniform norm, which depends only on Lipschitz constants and sensor coverage. The Newton correction (25) minimizes the observation residual by construction, but that is the intended role of data assimilation and is not presented as an independent prediction; 'prediction' in the paper refers to the SMS forecast between observation times, against which DA-SMS is compared via independent DNS benchmarks. Heavy self-citation appears in the review of SMS foundations [1,2,3,7,11,27], but the convergence analysis does not rest on those citations. The genuinely weak point is that the paper does not verify the hypotheses of Kelley's Theorem 2.4.2 for the SMS ansatz, so Assumption 1(2) is not proved for the actual Newton iterates; this is an unverified assumption or missing proof, not a circular reduction. The paper's own Section 5 acknowledges the theory is pessimistic and that tighter bounds are needed. Therefore the derivation is self-contained and no circularity is present.

Assumptions & free parameters 5 free parameters · 3 assumptions · 0 invented entities

The central convergence theorem rests on standard Newton convergence theory and two structural assumptions about the SMS family; the latter are plausible but unverified, and the uniform Lipschitz bound is not stated as an explicit assumption.

free parameters (5)
  • Tikhonov regularization γ for SMS equations = 1e-3 (KS), 5e-2 (AD)
    Chosen by hand to counter stiffness; affects SMS trajectory and DA correction.
  • Tikhonov regularization γ̃ for Newton iterations = 1e-3 (KS), 5e-2 (AD)
    Chosen equal to SMS γ; controls noise amplification in pseudo-inverse.
  • Observation frequency Δt = 0.5 (NLS, AD), 2.0 (KS)
    Chosen ad hoc; Fig. 4(b) shows error varies modestly with Δt.
  • Number of sensors r = 3 (NLS), 10 (KS), 46 (AD)
    Chosen by hand; Fig. 4(c) shows error decreases with r.
  • DA window length T_DA = 35 (NLS), 30 (KS), 25 (AD)
    Chosen to end before forecast window; no criterion given.
assumptions (3)
  • standard math Local convergence of underdetermined Newton iterations (Kelley Thm 2.4.2) under Lipschitz continuity of C and full row rank of J_r.
    Used to justify the correction step (25) converging to a sensor-matching θ*; conditions are not verified.
  • domain assumption u and û are Lipschitz continuous in x with constants Lu and Lû, and Lû is uniform over the parameter sequence.
    Assumption 1(1); uniform bound on Lû is not proven, and weights may grow.
  • domain assumption There exists a parameter sequence satisfying (29) at sensor locations.
    Assumption 1(2); effectively assumes Newton can match observations, which is the key unproven step.

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Cite this review

Pith. "Pith review of Sequential data assimilation for PDEs using shape-morphing solutions." pith.science (2026). https://pith.science/paper/4ESVWTAF

@misc{pith2026241116593,
  author       = {Pith},
  title        = {Pith review of: Sequential data assimilation for PDEs using shape-morphing solutions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4ESVWTAF}},
  note         = {Machine review of arXiv:2411.16593}
}
read the original abstract

Shape-morphing solutions (also known as evolutional deep neural networks, reduced-order nonlinear solutions, and neural Galerkin schemes) are a new class of methods for approximating the solution of time-dependent partial differential equations (PDEs). Here, we introduce a sequential data assimilation method for incorporating observational data in a shape-morphing solution (SMS). Our method takes the form of a predictor-corrector scheme, where the observations are used to correct the SMS parameters using Newton-like iterations. Between observation points, the SMS equations (a set of ordinary differential equations) are used to evolve the solution forward in time. We prove that, under certain conditions, the data assimilated SMS (DA-SMS) converges uniformly towards the true state of the system. We demonstrate the efficacy of DA-SMS on three examples: the nonlinear Schrodinger equation, the Kuramoto-Sivashinsky equation, and a two-dimensional advection-diffusion equation. Our numerical results suggest that DA-SMS converges with relatively sparse observations and a single iteration of the Newton-like method.

Figures

Figures reproduced from arXiv: 2411.16593 by the authors.

Figure 1
Figure 1. Schematic illustration of the DA-SMS algorithm. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Here we compare the modulus of the envelope [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Comparisons of the state of the system over the entire spatial do [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Comparison of the relative error (RE) for DA-SMS with different pa [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]
Figure 5
Figure 5. Figure 5: Schematic illustration of the boundary conditions for the advection [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Space-time plots of the solutions to ADE ( [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

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Reference graph

Works this paper leans on

50 extracted references · 28 canonical work pages

  1. [1]

    Anderson, M

    W. Anderson, M. Farazmand, Evolution of nonlinear reduced-order so- lutions for PDEs with conserved quantities, SIAM Journal on Scientific Computing 44 (1) (2022) A176–A197. doi:10.1137/21M1415972

    work page doi:10.1137/21m1415972 2022

  2. [2]

    Anderson, M

    W. Anderson, M. Farazmand, Fast and scalable computation of shape- morphing nonlinear solutions with application to evolutional neural net- works, Journal of Computational Physics 498 (2024) 112649.doi:10.1016/ j.jcp.2023.112649

    arXiv 2024

  3. [3]

    Y. Du, T. A. Zaki, Evolutional deep neural network, Phys. Rev. E 104 (2021) 045303. doi:10.1103/PhysRevE.104.045303

    doi:10.1103/physreve.104.045303 2021

  4. [4]

    Bruna, B

    J. Bruna, B. Peherstorfer, V.-E. E, Neural Galerkin schemes with active learning for high-dimensional evolution equations, Journal of Computa- tional Physics 496 (2024) 112588. doi:10.1016/j.jcp.2023.112588

    arXiv 2024

  5. [5]

    K. Law, A. Stuart, K. Zygalakis, Data Assimilation: A Mathemati- cal Introduction, Texts in Applied Mathematics, Springer, 2015. doi: 10.1007/978-3-319-20325-6

    doi:10.1007/978-3-319-20325-6 2015

  6. [6]

    Finzi, A

    M. Finzi, A. Potapczynski, M. Choptuik, A. G. Wilson, A stable and scal- able method for solving initial value PDEs with neural networks (2023). arXiv:2304.14994

    arXiv 2023

  7. [7]

    Anderson, M

    W. Anderson, M. Farazmand, Fisher information and shape-morphing modes for solving the Fokker–Planck equation in higher dimensions, Ap- plied Mathematics and Computation 467 (2024) 128489. doi:10.1016/j. amc.2023.128489

    arXiv 2024

  8. [8]

    Berman, B

    J. Berman, B. Peherstorfer, Randomized sparse neural Galerkin schemes for solving evolution equations with deep networks, in: A. Oh, T. Nau- mann, A. Globerson, K. Saenko, M. Hardt, S. Levine (Eds.), Advances in Neural Information Processing Systems 36 (NeurIPS 2023), Vol. 36, Curran Associates, Inc., 2023, pp. 4097–4114

    work page 2023

Show all 50 references
  1. [9]

    Sukumar, A

    N. Sukumar, A. Srivastava, Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks, Computer Methods in Applied Mechanics and Engineering 389 (2022) 114333. doi: 10.1016/j.cma.2021.114333

    2022

  2. [10]

    M. Kast, J. S. Hesthaven, Positional embeddings for solving PDEs with evolutional deep neural networks, Journal of Computational Physics 508 (2024) 112986. doi:10.1016/j.jcp.2024.112986

    2024

  3. [11]

    Z. T. Hilliard, M. Farazmand, Enforcing conserved quantities in Galerkin truncation and finite volume discretization, Nonlinear Dynamics 112 (16) (2024) 14051–14069. doi:10.1007/s11071-024-09884-2 . 29

    2024 doi

  4. [12]

    Schwerdtner, P

    P. Schwerdtner, P. Schulze, J. Berman, B. Peherstorfer, Nonlinear em- beddings for conserving Hamiltonians and other quantities with neural Galerkin schemes, SIAM Journal on Scientific Computing 46 (5) (2024) C583–C607. doi:10.1137/23M1607799

    2024 doi

  5. [13]

    N. Gaby, X. Ye, H. Zhou, Neural control of parametric solutions for high- dimensional evolution PDEs, SIAM Journal on Scientific Computing 46 (2) (2024) C155–C185. doi:10.1137/23M1549870

    2024 doi

  6. [14]

    H. Kim, T. A. Zaki, Multi evolutional deep neural networks (Multi-EDNN), arXiv (arXiv.2407.12293) (2024). doi:arXiv.2407.12293

    2024 arXiv

  7. [15]

    R. E. Kalman, A new approach to linear filtering and prediction prob- lems, Journal of Basic Engineering 82 (1) (1960) 35–45. doi:10.1115/1. 3662552

    1960 doi

  8. [16]

    R. E. Kalman, R. S. Bucy, New results in linear filtering and prediction theory, Journal of Basic Engineering 83 (1) (1961) 95–108. doi:10.1115/ 1.3658902

    1961

  9. [17]

    B. A. McElhoe, An assessment of the navigation and course corrections for a manned flyby of Mars or Venus, IEEE Transactions on Aerospace and Electronic Systems AES-2 (4) (1966) 613–623. doi:10.1109/TAES.1966. 4501892

    1966 doi

  10. [18]

    S. F. Schmidt, Application of state-space methods to navigation problems, Vol. 3 of Advances in Control Systems, Elsevier, 1966, pp. 293–340. doi: B978-1-4831-6716-9.50011-4

    1966

  11. [19]

    Andersson, J

    E. Andersson, J. Haseler, P. Und´ en, P. Courtier, G. Kelly, D. Vasilje- vic, C. Brankovic, C. Gaffard, A. Hollingsworth, C. Jakob, P. Janssen, E. Klinker, A. Lanzinger, M. Miller, F. Rabier, A. Simmons, B. Strauss, P. Viterbo, C. Cardinali, J.-N. Th´ epaut, The ECMWF implemen...

    1998 doi

  12. [20]

    Courtier, E

    P. Courtier, E. Andersson, W. Heckley, D. Vasiljevic, M. Hamrud, A. Hollingsworth, F. Rabier, M. Fisher, J. Pailleux, The ECMWF im- plementation of three-dimensional variational assimilation (3d-var). I: For- mulation, Quarterly Journal of the Royal Meteorological Society 124 ...

    1998 doi

  13. [21]

    Rabier, A

    F. Rabier, A. McNally, E. Andersson, P. Courtier, P. Und´ en, J. Eyre, A. Hollingsworth, F. Bouttier, The ECMWF implementation of three- dimensional variational assimilation (3D-Var). II: Structure functions, Quarterly Journal of the Royal Meteorological Society 124 (550) (199...

    1998 doi

  14. [22]

    A. Apte, M. Hairer, A. M. Stuart, J. Voss, Sampling the posterior: An approach to non-Gaussian data assimilation, Physica D: Nonlinear Phe- nomena 230 (1) (2007) 50–64. doi:10.1016/j.physd.2006.06.009

    2007 doi

  15. [23]

    Reich, C

    S. Reich, C. Cotter, Probabilistic forecasting and Bayesian data assimila- tion, Cambridge University Press, 2015

    2015

  16. [24]

    X. Zou, I. M. Navon, F. X. Ledimet, An optimal nudging data assimila- tion scheme using parameter estimation, Quarterly Journal of the Royal Meteorological Society 118 (508) (1992) 1163–1186. doi:10.1002/qj. 49711850808

    1992 doi

  17. [25]

    Azouani, E

    A. Azouani, E. Olson, E. S. Titi, Continuous data assimilation using general interpolant observables, Journal of Nonlinear Science 24 (2) (2014) 277–304. doi:10.1007/s00332-013-9189-y

    2014 doi

  18. [26]

    Tr´ emolet, Accounting for an imperfect model in 4D-Var, Quarterly Journal of the Royal Meteorological Society 132 (621) (2006) 2483–2504

    Y. Tr´ emolet, Accounting for an imperfect model in 4D-Var, Quarterly Journal of the Royal Meteorological Society 132 (621) (2006) 2483–2504. doi:10.1256/qj.05.224

    2006 doi

  19. [27]

    Anderson, M

    W. Anderson, M. Farazmand, Shape-morphing reduced-order models for nonlinear Schr¨ odinger equations, Nonlinear Dynamics 108 (4) (2022) 2889–

    2022

  20. [28]

    Karniadakis, I

    G. Karniadakis, I. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, L. Yang, Physics-informed machine learning, Nat Rev Phys 3 (2021) 422–440. doi: 10.1038/s42254-021-00314-5

    2021 doi

  21. [29]

    Kovachki, Z

    N. Kovachki, Z. Li, B. Liu, K. Azizzadenesheli, K. Bhattacharya, A. Stuart, A. Anandkumar, Neural operator: Learning maps between function spaces with applications to PDEs, Journal of Machine Learning Research 24 (89) (2023) 1–97

    2023

  22. [30]

    L. Lu, P. Jin, G. Pang, Z. Zhang, G. E. Karniadakis, Learning nonlin- ear operators via DeepONet based on the universal approximation the- orem of operators, Nat Mach Intell 3 (2021) 218–229. doi:10.1038/ s42256-021-00302-5

    2021

  23. [31]

    X. Chen, T. Yamamoto, Newton-like methods for solving underdetermined nonlinear equations with nondifferentiable terms, Journal of Computa- tional and Applied Mathematics 55 (3) (1994) 311–324. doi:10.1016/ 0377-0427(94)90036-1

    1994

  24. [32]

    C. T. Kelley, Iterative Methods for Optimization, Society for Industrial and Applied Mathematics, 1999. doi:10.1137/1.9781611970920

    1999 doi

  25. [33]

    F. Pes, G. Rodriguez, The minimal-norm Gauss-Newton method and some of its regularized variants, ETNA - Electronic Transactions on Numerical Analysis 53 (2020) 459–480. doi:10.1553/etna_vol53s459. 31

    2020 doi

  26. [34]

    Rieder, On the regularization of nonlinear ill-posed problems via inexact Newton iterations, Inverse Problems 15 (1) (1999) 309

    A. Rieder, On the regularization of nonlinear ill-posed problems via inexact Newton iterations, Inverse Problems 15 (1) (1999) 309. doi:10.1088/ 0266-5611/15/1/028

    1999

  27. [35]

    I. J. Goodfellow, Y. Bengio, A. Courville, Deep Learning, MIT Press, Cam- bridge, MA, USA, 2016

    2016

  28. [36]

    Virmaux, K

    A. Virmaux, K. Scaman, Lipschitz regularity of deep neural networks: anal- ysis and efficient estimation, Advances in Neural Information Processing Systems 31 (2018). doi:arXiv.1805.10965

    2018 arXiv

  29. [37]

    Cousins, T

    W. Cousins, T. P. Sapsis, Reduced-order precursors of rare events in uni- directional nonlinear water waves, J. Fluid Mech 790 (2016) 368–388. doi:10.1017/jfm.2016.13

    2016 doi

  30. [38]

    Farazmand, T

    M. Farazmand, T. P. Sapsis, Reduced-order prediction of rogue waves in two-dimensional deep-water waves, J. Comput. Phys. 340 (2017) 418 – 434. doi:10.1016/j.jcp.2017.03.054

    2017 doi

  31. [39]

    V. E. Zakharov, Stability of periodic waves of finite amplitude on the sur- face of a deep fluid, Journal of Applied Mechanics and Technical Physics 9 (2) (1968) 190–194. doi:10.1007/BF00913182

    1968 doi

  32. [40]

    S. M. Cox, P. C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics 176 (2) (2002) 430–455. doi:10.1006/ jcph.2002.6995

    2002

  33. [41]

    V. P. Ruban, Gaussian variational ansatz in the problem of anomalous sea waves: Comparison with direct numerical simulation, Journal of Exper- imental and Theoretical Physics 120 (5) (2015) 925–932. doi:10.1134/ S1063776115050179

    2015

  34. [42]

    Chabchoub, N

    A. Chabchoub, N. P. Hoffmann, N. Akhmediev, Rogue wave observation in a water wave tank, Phys. Rev. Lett. 106 (20) (2011) 204502. doi: 10.1103/PhysRevLett.106.204502

    2011 doi

  35. [43]

    Kuramoto, Diffusion-induced chaos in reaction systems, Progress of The- oretical Physics Supplement 64 (1978) 346–367

    Y. Kuramoto, Diffusion-induced chaos in reaction systems, Progress of The- oretical Physics Supplement 64 (1978) 346–367. doi:10.1143/PTPS.64. 346

    1978 doi

  36. [44]

    G. I. Sivashinsky, Nonlinear analysis of hydrodynamic instability in laminar flames—I. Derivation of basic equations, Acta Astronautica 4 (11) (1977) 1177–1206. doi:10.1016/0094-5765(77)90096-0

    1977 doi

  37. [45]

    Farazmand, G

    M. Farazmand, G. Haller, Computing Lagrangian coherent structures from their variational theory, Chaos 22 (2012) 013128.doi:10.1063/1.3690153

    2012 doi

  38. [46]

    S. C. Shadden, F. Lekien, J. E. Marsden, Definition and properties of Lagrangian coherent structures from finite-time Lyapunov exponents in two-dimensional aperiodic flows, Physica D 212 (2005) 271–304. doi: 10.1016/j.physd.2005.10.007. 32

    2005 doi

  39. [47]

    Raissi, P

    M. Raissi, P. Perdikaris, G. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems in- volving nonlinear partial differential equations, Journal of Computational Physics 378 (2019) 686 – 707. doi:10.1016/j.jcp.2018.10.045

    2019 doi

  40. [48]

    Attia, A

    A. Attia, A. Alexanderian, A. K. Saibaba, Goal-oriented optimal design of experiments for large-scale Bayesian linear inverse problems, Inverse Prob- lems 34 (9) (2018) 095009. doi:10.1088/1361-6420/aad210

    2018 doi

  41. [49]

    Farazmand, Sparse discrete empirical interpolation method: State es- timation from few sensors, SIAM Journal on Scientific Computing 46 (6) (2024) A3658–A3680

    M. Farazmand, Sparse discrete empirical interpolation method: State es- timation from few sensors, SIAM Journal on Scientific Computing 46 (6) (2024) A3658–A3680. doi:10.1137/24M1636344. 33

    2024 doi

  42. [2902]

    doi:10.1007/s11071-022-07448-w

    doi

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