Pith. sign in

REVIEW 5 minor 48 references

Exact Finite-Horizon Memory, Conditioning, and Dissipative Decay in Coarse Upwind Finite-Volume Prediction

T0 review · 0 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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

Pith's one-line read The paper proves an exact observation-rank law for coarse upwind advection: parent averages alone are not a predictive state, and the missing information is counted exactly, then shown to be partly unrecoverable or dissipatively erased.

desk verdict A correct, carefully scoped exact-rank and conditioning analysis for coarse upwind prediction; the topological lower bounds and perturbation model are narrower than the title might suggest, but the authors say so. read the letter →

arxiv 2608.08633 v1 pith:KKZB7SPP submitted 2026-08-09 math.NA cs.NA

Antonis Polemitis , Nicholas Christakis , Dimitris Drikakis This is my paper · ORCID
classification math.NAcs.NA MSC 65M0865F3593B0765M12
keywords finite-volumemethodscoarsegrainingpredictivememoryobservabilitynumericaldissipationstablerecoveryupwindschemeobservationrank
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

Coarse finite-volume averages are not, in general, an exact predictive state: two fine-grid states with the same parent averages can generate different future coarse histories under upwind advection. The paper's main result is an exact finite-horizon rank law for periodic scalar advection with $P$ parent cells, $r$ children per parent, and horizon $L$: the observation matrix has rank $P + (P-1)\min\{L,r-1\}$. It follows that any continuous centralized encoder needs $(P-1)\min\{L,r-1\}$ extra real coordinates beyond the parent averages, while any product-local encoder needs $\min\{L,r-1\}$ coordinates per parent, and an anchored flux-divergence queue attains the centralized bound. The paper then separates exact observability from stable and long-time relevance: the delayed subcell information enters through a triangular map whose smallest singular values scale like $\lambda^q$, and for $0<\lambda<1$ the nonconstant part of the solution is proved to contract toward a perturbation-controlled floor. This matters because reduced, multiscale, and learned models that claim closure on coarse observables must specify hidden information, horizon, conditioning, and dissipation; the scalar upwind hierarchy becomes a solvable benchmark for all of them.

What carries the argument

The central object is the finite-horizon observation matrix $O_L$, built from the restriction operator $R$ and the upwind shift $A_\lambda=(1-\lambda)I+\lambda S$; $O_L x$ is the sequence of parent averages from time $0$ to $L$. The rank law follows because the powers $A_\lambda^t$ span the same row space as $R,RS,\ldots,RS^L$, and each shift $S^{t+1}$ exposes one new right-collar child layer through the cyclic difference operator $D_t=r(RS^{t+1}-RS^t)$ whose rows sum to zero. The attaining memory is the anchored flux-divergence queue $JB\Phi^t$, the cyclic differences of the realized right-face fluxes with one coordinate deleted; at saturation its dimension $P-1$ per layer is exactly the rank increment. Conditioning is carried by the triangular collar-to-queue matrix $T_q$ with diagonal $\lambda,\ldots,\lambda^q$, and long-time decay by the per-step contraction rate $\rho_N=(1-4\lambda(1-\lambda)\sin^2(\pi/N))^{1/2}$.

What would settle it

Compute the singular values (or a rank-revealing factorization) of $O_L$ exactly, for example in rational arithmetic, for $P=8$, $r=6$, $\lambda=1/2$ across $L=0,\dots,9$; the theorem is false if the rank ever differs from $P+(P-1)\min\{L,5\}$.

Watch

Extended reading notes

Core claim

The central discovery is a finite-horizon observation-rank law for the periodic scalar upwind finite-volume hierarchy: for $L\ge 0$, $P\ge 2$, $r\ge 2$ and Courant number $0<\lambda\le 1$, the matrix $O_L=(R^\top,(RA_\lambda)^\top,\ldots,(RA_\lambda^L)^\top)^\top$ has rank $P+(P-1)\min\{L,r-1\}$. Each additional observation exposes one new child-cell layer and contributes $P-1$ independent directions, one fewer than the number of parents, because the spatially uniform flux gauge is invisible to the conservative update. The paper then constructs an anchored flux-divergence queue, the list of cyclic flux differences $J(B\Phi^0),\ldots,J(B\Phi^{q-1})$, that attains this centralized lower bound and, together with the parent averages, forms a minimal autonomous predictive state of dimension $Pr-r+1$ at saturation. Two companion results sharpen the practical meaning: the collar-to-queue map $T_q$ is triangular with diagonal $\lambda,\ldots,\lambda^q$, so its infinity-norm condition number is $((2-\lambda)/\lambda)^{q-1}$ and the deepest exposed layer is recoverable only with amplification of order $\lambda^{-q}$; and for $0<\lambda<1$ the nonconstant component of the computed solution contracts at rate $\rho_N$ under separately bounded mean and nonconstant perturbations, so information that is algebraically necessary for short-time closure can be numerically invisible or dissipatively erased later.

Load-bearing premise

The sharp lower-bound results assume a continuous encoder on an open support; with discontinuous encoders, or with non-open supports, the stated minimum extra-coordinate counts can be bypassed, and the decay theorem separately assumes a declared split of arithmetic error into mean and nonconstant parts.

Editorial extensions

If this is right

  • For any continuous centralized predictor on open support, the minimum number of auxiliary coordinates needed to reproduce $L+1$ steps of parent-average history is $(P-1)\min\{L,r-1\}$, and the anchored flux-divergence queue attains it.
  • At saturation ($L\ge r-1$), parent averages plus the queue form an autonomous linear predictive state of dimension $Pr-r+1$ that advances by a fixed matrix without revisiting the fine grid.
  • At small Courant number the deep subcell layers remain algebraically visible but numerically unrecoverable: their singular values fall like $\lambda^q$, so a declared SVD tolerance records an effective rank below the exact rank.
  • For $0<\lambda<1$ hidden subcell differences are first invisible, then become visible after a finite delay, and then contract at the rate $\rho_N$ toward a perturbation-controlled floor, so exact short-time closure and long-time practical relevance are distinct.

Reading between the lines

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

  • Repeating the same rank-and-conditioning program for higher-order scalar schemes would test whether a wider exposed collar per step changes the $P-1$ increment; the paper identifies this as a natural next step but does not prove it.
  • For nonperiodic boundaries the periodic flux gauge is replaced by boundary data, so the $r-1$ coordinate saving at saturation should shrink or vanish; recomputing the rank law with Dirichlet or absorbing interfaces would test this.
  • The perturbed-decay theorem treats $\eta$ and $\eta_m$ as declared per-step error bounds; measuring them for a concrete floating-point implementation would convert the contraction estimate into a certificate for that solver.
  • A learned closure trained on this hierarchy cannot reproduce all coarse histories at or beyond saturation with fewer than $Pr-r+1$ internal coordinates unless it is discontinuous; this gives a testable lower bound on the internal state dimension of such models.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 5 minor

Summary. The paper treats coarse finite-volume parent averages as a predictive-state question for the periodic scalar advection equation discretized by first-order upwind with forward Euler. Its main object is the finite-horizon observation matrix O_L = [R; RA_λ; ...; RA_λ^L], and Theorem 1 establishes the exact rank law rank O_L = P + (P−1) min{L, r−1} for P parents with r child cells per parent. From this it derives lower bounds of (P−1) min{L, r−1} additional real coordinates for continuous centralized encoders and min{L, r−1} coordinates per parent for product-local encoders, and constructs an anchored flux-divergence queue that attains the centralized bound and, at saturation, yields a minimal autonomous predictive state of dimension Pr−r+1. Theorem 2 gives the exact conditioning κ∞(T_q) = ((2−λ)/λ)^{q−1} for the triangular collar-to-queue map, and Theorem 3 proves contraction of the nonconstant fine-grid component under a declared bounded-perturbation arithmetic model, with separate control of mean drift. Numerical experiments illustrate the rank ladder, tolerance-dependent effective rank, queue conditioning, step-function flattening, and delayed coarse separation followed by dissipative decay.

Significance. When accepted as stated, the paper delivers a rare exact benchmark separating three properties that are often conflated in reduced and learned coarse models: the algebraic dimension of predictive memory, the conditioning with which delayed information can be recovered, and the time over which numerical dissipation erases it. The proofs of the rank theorem, the encoder lower bounds, the attaining queue construction, and the contraction estimate are internally consistent; I checked the row-rank construction, the triangular collar-to-queue inversion, and the Fourier eigenvalue estimate, and found no gaps. The encoder lower bounds are correctly scoped to continuous maps on open supports, and the bounded-perturbation model is declared rather than disguised as a derived IEEE roundoff bound. The public reproducibility package with executable certificates is an additional strength. The contribution is a well-posed solvable benchmark rather than a broad turbulence-closure theorem, and the paper is appropriately modest about that distinction.

minor comments (5)
  1. [§6, Theorem 3] The statement uses δ^n both for the perturbation vector and for its spatial mean: the displayed sentence 'Here δ^n = 1/N 1^T δ^n is the spatial mean of the arithmetic perturbation at step n' cannot be read literally. Please use an overbar for the mean and state the mean-drift assumption as |mean(δ^n)| ≤ η_m.
  2. [Abstract and §1.1] The abstract's sentence 'Centralized prediction therefore requires one fewer additional coordinate than the number of parent cells per exposed layer' states the lower bound without the qualifier 'for continuous encoders on open supports.' Section 8.1 gives the restriction correctly; adding the same qualifier to the abstract and contribution list would avoid an over-general reading by downstream users.
  3. [§7.2, Figure 1] The effective-rank curves for several λ overlap with the exact-rank curve and with each other, as the caption explains, but the degree of overlap makes the effective-rank loss hard to read in monochrome print. A short table of rankτ(O_L) values at L = 5, or a few marked points, would make the λ-dependence of the loss immediately visible.
  4. [§6, Remark 1] The remark correctly notes that the condition ζ > E_floor can fail when E_floor grows with N, but the mechanism would be clearer if it stated directly that E_floor = η/(1−ρ_N) scales like η N^2/[2π^2 λ(1−λ)] for fixed λ and η.
  5. [§7.5, Figure 4 caption] The caption states that dashed curves are the theoretical bounded-perturbation envelope, but it does not give the value of η used. Since η = 200 ε_mach √N is a declared diagnostic rather than a derived roundoff bound, include it in the caption or legend for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rank law, encoder bounds, queue attainment, conditioning estimates, and dissipative decay claims are derived from the definitions of the upwind scheme and observation operator, with no fitted input renamed as a prediction.

full rationale

The derivation chain is self-contained. Theorem 1's rank formula follows from explicit row-span arguments: (A_λ)^t is a polynomial in the shift S, the change from {RA_λ^t} to {RS^t} is triangular with diagonal entries 1,λ,...,λ^L, each new D_t row exposes one new child layer through the periodic difference operator, and the constructed square submatrix has determinant ±r^{-P}, giving the matching lower bound. The centralized and product-local encoder lower bounds are direct applications of Lemma 1/invariance of domain under the stated continuity and open-support hypotheses, which Section 8.1 explicitly declares rather than hides: 'The encoder lower bounds are topological dimension statements for continuous real-valued encoders on open sets. They are not bit-complexity bounds, statistical learning bounds, or claims about arbitrary discontinuous encoders.' The anchored flux-divergence queue attains the bound by the parent update identity y^{t+1}=y^t-(1/r)BΦ^t and the fixed recurrence (A_λ-(1-λ)I)^r=λ^r S^r; no ingredient is assumed that is being proved. Theorem 2's T_q inverse and norms are verified by binomial inversion, and Theorem 3's contraction constant ρ_N is computed from the Fourier eigenvalues of A_λ, with the perturbation δ^n an explicit hypothesis, not a fitted parameter. The hand-chosen η=200 ε_mach √N in Experiment 4 is explicitly labeled 'a conservative diagnostic perturbation floor used to visualize the bounded-perturbation estimate. It is not derived as a sharp implementation-specific IEEE 754 floating-point rounding constant,' so the experiments illustrate the conditional estimate rather than fitting it to force agreement. No load-bearing self-citation occurs: the cited works by the present authors (e.g., [11] and [28]) concern entropy generation and super-resolution forecasting and are not used to justify the rank, conditioning, or decay theorems. The paper's own limitations section identifies the scope (linear periodic scalar upwind, continuous encoders on open sets, declared perturbation model), and those restrictions, while important for application, are stated hypotheses rather than circular inputs. Accordingly no circular step is found.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central rank and conditioning theorems rest on standard linear algebra (row spans, circulant Fourier analysis, invariance of domain) and on the declared model assumptions (periodic uniform upwind FV, 0<λ≤1, continuous encoders on open supports). No new physical entities are introduced. The only hand-chosen constants affect diagnostics, not theorems.

free parameters (2)
  • C_svd (effective-rank tolerance multiplier) = 100
    Declared constant in the SVD tolerance τ = C_svd ε_mach max{(L+1)P, Pr} ||O_L||_2 (Section 5). It is chosen by hand and affects the reported effective ranks in Experiment 1 and Table 1, but the exact rank theorem is independent of it.
  • η (perturbation floor in step-flattening experiment) = 200 ε_mach sqrt(N)
    Chosen as a conservative diagnostic perturbation floor for the plots in Figure 4 and Table 2. Theorem 3 holds for any η; the value only marks the visualized envelope.
assumptions (6)
  • standard math Invariance of domain in finite-dimensional spaces
    Used in Lemma 1 and in the product-local lower bound in Theorem 1 to convert continuous injectivity into a dimension inequality.
  • standard math Circulant Fourier diagonalization of the periodic shift
    Used in Proposition 2 for the periodic difference operator and in Theorem 3 for the sharp contraction rate ρ_N.
  • domain assumption SVD-based numerical rank with a declared tolerance
    Section 5 defines effective rank using τ = C_svd ε_mach max{(L+1)P, Pr} ||O_L||_2; this is a diagnostic convention, not a theorem, and the exact rank statement does not depend on it.
  • domain assumption Periodic uniform grid, scalar linear advection, first-order upwind flux, forward Euler, 0<λ≤1
    Section 3 defines the hierarchy; all results are explicitly scoped to this model.
  • domain assumption Bounded-perturbation model for arithmetic
    Theorem 3 assumes the computed update equals the exact upwind update plus per-step perturbations with separately bounded mean and nonconstant components; this is a modeling assumption about floating-point behavior, acknowledged by the authors.
  • domain assumption Continuous encoders on open supports for minimal-dimension lower bounds
    The topological lower bounds in Theorem 1 and Corollary 5 require continuity and open support, as stated in Sections 1 and 4.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Exact Finite-Horizon Memory, Conditioning, and Dissipative Decay in Coarse Upwind Finite-Volume Prediction." pith.science (2026). https://pith.science/paper/KKZB7SPP

@misc{pith2026260808633,
  author       = {Pith},
  title        = {Pith review of: Exact Finite-Horizon Memory, Conditioning, and Dissipative Decay in Coarse Upwind Finite-Volume Prediction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKZB7SPP}},
  note         = {Machine review of arXiv:2608.08633}
}
read the original abstract

Coarse finite-volume averages do not generally form a predictive state: realized interface fluxes close a conservative update, but distinct fine-grid states with identical parent averages can generate different future coarse histories. We analyze this failure for periodic scalar advection discretized by a first-order upwind finite-volume method with forward Euler time integration. We derive a finite-horizon observation-rank law: each additional observation exposes one new child-cell layer and contributes one fewer independent direction than the number of parent cells, until the unresolved layers are exhausted. Centralized prediction therefore requires one fewer additional coordinate than the number of parent cells per exposed layer, whereas product-local prediction requires one coordinate per layer in each parent. An anchored flux-divergence queue attains the centralized bound, identifies the periodic flux gauge, and, at saturation, forms a minimal autonomous predictive state with the parent averages. We then distinguish exact observability from stable recoverability. The collar-to-queue map becomes rapidly ill-conditioned as the Courant number decreases, so algebraically visible delayed information may fall below a prescribed numerical tolerance. For Courant numbers strictly between zero and one, we prove contraction of the nonconstant component under bounded arithmetic perturbations, with separate control of mean drift, an explicit perturbation neighborhood, and a grid-dependent decay time. Numerical experiments illustrate the rank ladder, effective-rank loss, queue conditioning, step-function flattening, and delayed coarse separation followed by dissipative decay. The results provide a solvable benchmark for assessing state sufficiency in coarse, reduced, multiscale, and learned scientific models.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

48 extracted references · 20 canonical work pages

  1. [1]

    Cambridge Texts in Applied Mathematics

    LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002). https://doi.org/10.1017/ CBO9780511791253

    work page 2002

  2. [2]

    In: Handbook of Numerical Anal- ysis vol

    Eymard, R., Gallou¨ et, T., Herbin, R.: Finite volume methods. In: Handbook of Numerical Anal- ysis vol. 7, pp. 713–1018. Elsevier, Amsterdam (2000). https://doi.org/10.1016/S1570-8659(00) 07005-8

    work page doi:10.1016/s1570-8659(00 2000

  3. [3]

    Fluid Mechanics and Its Applications, vol

    Moukalled, F., Mangani, L., Darwish, M.: The Finite Volume Method in Computational Fluid Dynamics: An Advanced Introduction with OpenFOAM and Matlab. Fluid Mechanics and Its Applications, vol. 113. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-16874-6

    doi:10.1007/978-3-319-16874-6 2016

  4. [4]

    Mathematics of Computation34(149), 1–21 (1980) https://doi.org/10.1090/ S0025-5718-1980-0551288-3

    Crandall, M.G., Majda, A.: Monotone difference approximations for scalar conserva- tion laws. Mathematics of Computation34(149), 1–21 (1980) https://doi.org/10.1090/ S0025-5718-1980-0551288-3

    work page 1980

  5. [5]

    SIAM Journal on Numerical Analysis24(5), 967–984 (1987) https://doi.org/10.1137/0724063

    Berger, M.J.: On conservation at grid interfaces. SIAM Journal on Numerical Analysis24(5), 967–984 (1987) https://doi.org/10.1137/0724063

    work page doi:10.1137/0724063 1987

  6. [6]

    Journal of Computational Physics82(1), 64–84 (1989) https://doi.org/10.1016/0021-9991(89)90035-1 24

    Berger, M.J., Colella, P.: Local adaptive mesh refinement for shock hydrodynamics. Journal of Computational Physics82(1), 64–84 (1989) https://doi.org/10.1016/0021-9991(89)90035-1 24

    doi:10.1016/0021-9991(89)90035-1 1989

  7. [7]

    Journal of Computational Physics160(1), 241–282 (2000) https://doi.org/10.1006/jcph.2000.6459

    Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. Journal of Computational Physics160(1), 241–282 (2000) https://doi.org/10.1006/jcph.2000.6459

    arXiv 2000

  8. [8]

    SIAM Review43(1), 89–112 (2001) https://doi.org/10.1137/S003614450036757X

    Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Review43(1), 89–112 (2001) https://doi.org/10.1137/S003614450036757X

    doi:10.1137/s003614450036757x 2001

Show all 48 references
  1. [9]

    Mathematics of Computation41(164), 321–336 (1983) https: //doi.org/10.1090/S0025-5718-1983-0717689-8

    Osher, S., Sanders, R.: Numerical approximations to nonlinear conservation laws with locally varying time and space grids. Mathematics of Computation41(164), 321–336 (1983) https: //doi.org/10.1090/S0025-5718-1983-0717689-8

    1983 doi

  2. [10]

    Journal of Scientific Computing33(3), 239–278 (2007) https://doi.org/10.1007/ s10915-007-9151-y

    Constantinescu, E.M., Sandu, A.: Multirate timestepping methods for hyperbolic conserva- tion laws. Journal of Scientific Computing33(3), 239–278 (2007) https://doi.org/10.1007/ s10915-007-9151-y

    2007

  3. [11]

    Journal of Computational Physics 227(10), 4853–4872 (2008) https://doi.org/10.1016/j.jcp.2008.01.035

    Thornber, B., Drikakis, D., Williams, R.J.R., Youngs, D.: On entropy generation and dissipation of kinetic energy in high-resolution shock-capturing schemes. Journal of Computational Physics 227(10), 4853–4872 (2008) https://doi.org/10.1016/j.jcp.2008.01.035

    2008 doi

  4. [12]

    Progress of Theoretical Physics 33(3), 423–455 (1965) https://doi.org/10.1143/PTP.33.423

    Mori, H.: Transport, collective motion, and brownian motion. Progress of Theoretical Physics 33(3), 423–455 (1965) https://doi.org/10.1143/PTP.33.423

    1965 doi

  5. [13]

    Physical Review124(4), 983–992 (1961) https://doi.org/10.1103/PhysRev.124.983

    Zwanzig, R.: Memory effects in irreversible thermodynamics. Physical Review124(4), 983–992 (1961) https://doi.org/10.1103/PhysRev.124.983

    1961 doi

  6. [14]

    Physica D: Nonlinear Phenomena166(3–4), 239–257 (2002) https://doi.org/10.1016/S0167-2789(02)00446-3

    Chorin, A.J., Hald, O.H., Kupferman, R.: Optimal prediction with memory. Physica D: Nonlinear Phenomena166(3–4), 239–257 (2002) https://doi.org/10.1016/S0167-2789(02)00446-3

    2002 doi

  7. [15]

    Physical Review Fluids2(1), 014604 (2017) https://doi.org/10.1103/ PhysRevFluids.2.014604 arXiv:1611.03311

    Parish, E.J., Duraisamy, K.: Non-markovian closure models for large eddy simulations using the Mori–Zwanzig formalism. Physical Review Fluids2(1), 014604 (2017) https://doi.org/10.1103/ PhysRevFluids.2.014604 arXiv:1611.03311

    2017 arXiv

  8. [16]

    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences473(2205), 20170385 (2017) https://doi.org/10.1098/rspa.2017.0385

    Gouasmi, A., Parish, E.J., Duraisamy, K.: A priori estimation of memory effects in reduced- order models of nonlinear systems using the Mori–Zwanzig formalism. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences473(2205), 20170385 (2017) https:/...

    2017

  9. [17]

    SIAM Journal on Applied Dynamical Systems20(4), 2558–2601 (2021) https://doi.org/10.1137/21M1401759

    Lin, Y.T., Tian, Y., Livescu, D., Anghel, M.: Data-driven learning for the Mori–Zwanzig formalism: A generalization of the koopman learning framework. SIAM Journal on Applied Dynamical Systems20(4), 2558–2601 (2021) https://doi.org/10.1137/21M1401759

    2021 doi

  10. [18]

    Journal of Computational Physics556, 114810 (2026) https://doi.org/10.1016/j.jcp.2026.114810 arXiv:2507.17051 [math.NA]

    Agdestein, S.D., Verstappen, R., Sanderse, B.: Exact expressions for the unresolved stress in a finite-volume based large-eddy simulation. Journal of Computational Physics556, 114810 (2026) https://doi.org/10.1016/j.jcp.2026.114810 arXiv:2507.17051 [math.NA]

    2026

  11. [19]

    Annual Review of Fluid Mechanics51, 357–377 (2019) https://doi.org/10.1146/annurev-fluid-010518-040547

    Duraisamy, K., Iaccarino, G., Xiao, H.: Turbulence modeling in the age of data. Annual Review of Fluid Mechanics51, 357–377 (2019) https://doi.org/10.1146/annurev-fluid-010518-040547

    2019 doi

  12. [20]

    Nature Reviews Physics3, 422–440 (2021) https://doi.org/10.1038/ s42254-021-00314-5

    Karniadakis, G.E., Kevrekidis, I.G., Lu, L., Perdikaris, P., Wang, S., Yang, L.: Physics- informed machine learning. Nature Reviews Physics3, 422–440 (2021) https://doi.org/10.1038/ s42254-021-00314-5

    2021

  13. [21]

    Nature Computational Science2(6), 358–366 (2022) https://doi.org/10.1038/s43588-022-00264-7

    Vinuesa, R., Brunton, S.L.: Enhancing computational fluid dynamics with machine learning. Nature Computational Science2(6), 358–366 (2022) https://doi.org/10.1038/s43588-022-00264-7

    2022 doi

  14. [22]

    Proceedings of the National Academy of Sciences116(31), 15344–15349 (2019) https://doi.org/10.1073/pnas.1814058116 arXiv:1808.04930 25

    Bar-Sinai, Y., Hoyer, S., Hickey, J., Brenner, M.P.: Learning data-driven discretizations for partial differential equations. Proceedings of the National Academy of Sciences116(31), 15344–15349 (2019) https://doi.org/10.1073/pnas.1814058116 arXiv:1808.04930 25

    2019 arXiv

  15. [23]

    Proceedings of the National Academy of Sciences 118(21), 2101784118 (2021) https://doi.org/10.1073/pnas.2101784118

    Kochkov, D., Smith, J.A., Alieva, A., Wang, Q., Brenner, M.P., Hoyer, S.: Machine learning- accelerated computational fluid dynamics. Proceedings of the National Academy of Sciences 118(21), 2101784118 (2021) https://doi.org/10.1073/pnas.2101784118

    2021 doi

  16. [24]

    Physical Review Fluids6, 064605 (2021) https://doi.org/10.1103/PhysRevFluids.6.064605

    Zhuang, J., Kochkov, D., Bar-Sinai, Y., Brenner, M.P., Hoyer, S.: Learned discretizations for passive scalar advection in a two-dimensional turbulent flow. Physical Review Fluids6, 064605 (2021) https://doi.org/10.1103/PhysRevFluids.6.064605

    2021 doi

  17. [25]

    Computers & Fluids307, 106978 (2026) https://doi.org/10.1016/j.compfluid.2026

    Rom´ emont, G., Renac, F., Nunez, J., Gueyffier, D., Chinesta, F.: A data-driven learned discretiza- tion approach in finite volume schemes for hyperbolic conservation laws and varying boundary conditions. Computers & Fluids307, 106978 (2026) https://doi.org/10.1016/j.compflui...

    2026 doi

  18. [26]

    Physics of Fluids33(7), 073603 (2021) https://doi.org/10.1063/5.0054312 arXiv:2011.02364 [physics.flu-dyn]

    Gao, H., Sun, L., Wang, J.-X.: Super-resolution and denoising of fluid flow using physics-informed convolutional neural networks without high-resolution labels. Physics of Fluids33(7), 073603 (2021) https://doi.org/10.1063/5.0054312 arXiv:2011.02364 [physics.flu-dyn]

    2021 arXiv

  19. [27]

    Computer Methods in Applied Mechanics and Engineering427, 117023 (2024) https://doi.org/10.1016/j.cma.2024.117023 arXiv:2311.07896 [physics.flu-dyn]

    Gao, H., Han, X., Fan, X., Sun, L., Liu, L.-P., Duan, L., Wang, J.-X.: Bayesian conditional diffusion models for versatile spatiotemporal turbulence generation. Computer Methods in Applied Mechanics and Engineering427, 117023 (2024) https://doi.org/10.1016/j.cma.2024.117023 ar...

    2024

  20. [28]

    Physics of Fluids37(1), 016124 (2025) https://doi.org/ 10.1063/5.0250509

    Sofos, F., Drikakis, D., Kokkinakis, I.W., Spottswood, S.M.: Spatiotemporal super-resolution forecasting of high-speed turbulent flows. Physics of Fluids37(1), 016124 (2025) https://doi.org/ 10.1063/5.0250509

    2025 doi

  21. [29]

    Journal of Scientific Computing107(1), 3 (2026) https://doi.org/10.1007/s10915-026-03210-1 arXiv:2411.01746 [math.NA]

    Liu, L., Li, T., Gelb, A., Lee, Y.: Entropy stable conservative flux form neural networks. Journal of Scientific Computing107(1), 3 (2026) https://doi.org/10.1007/s10915-026-03210-1 arXiv:2411.01746 [math.NA]

    2026 arXiv

  22. [30]

    SIAM Journal on Scientific Computing47(6), 1196–1222 (2025) https://doi.org/10.1137/24M1697566 arXiv:2403.14848 [math.NA]

    Charles, P., Ray, D.: Learning WENO for entropy stable schemes to solve conservation laws. SIAM Journal on Scientific Computing47(6), 1196–1222 (2025) https://doi.org/10.1137/24M1697566 arXiv:2403.14848 [math.NA]

    2025 arXiv

  23. [31]

    the basic exper- iment

    Smagorinsky, J.: General circulation experiments with the primitive equations: I. the basic exper- iment. Monthly Weather Review91(3), 99–164 (1963) https://doi.org/10.1175/1520-0493(1963) 091⟨0099:GCEWTP⟩2.3.CO;2

    1963 doi

  24. [32]

    Physics of Fluids A: Fluid Dynamics3(7), 1760–1765 (1991) https://doi.org/10.1063/1.857955

    Germano, M., Piomelli, U., Moin, P., Cabot, W.H.: A dynamic subgrid-scale eddy viscosity model. Physics of Fluids A: Fluid Dynamics3(7), 1760–1765 (1991) https://doi.org/10.1063/1.857955

    1991 doi

  25. [33]

    Communications in Mathematical Sciences1(4), 715–762 (2003) https://doi.org/10.4310/CMS.2003.v1.n4.a5

    Kevrekidis, I.G., Gear, C.W., Hyman, J.M., Kevrekidis, P.G., Runborg, O., Theodoropoulos, C.: Equation-free, coarse-grained multiscale computation: Enabling microscopic simulators to perform system-level analysis. Communications in Mathematical Sciences1(4), 715–762 (2003) htt...

    2003 doi

  26. [34]

    Communications in Mathematical Sciences1(1), 87–132 (2003) https://doi.org/10.4310/CMS.2003.v1.n1.a8

    E, W., Engquist, B.: The heterogeneous multiscale methods. Communications in Mathematical Sciences1(1), 87–132 (2003) https://doi.org/10.4310/CMS.2003.v1.n1.a8

    2003 doi

  27. [35]

    Sirovich, L.: Turbulence and the dynamics of coherent structures. i. coherent structures. Quarterly of Applied Mathematics45(3), 561–571 (1987) https://doi.org/10.1090/qam/910462

    1987 doi

  28. [36]

    Journal of Fluid Mechanics656, 5–28 (2010) https://doi.org/10.1017/S0022112010001217

    Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. Journal of Fluid Mechanics656, 5–28 (2010) https://doi.org/10.1017/S0022112010001217

    2010 doi

  29. [37]

    Proceedings of the National Academy of Sciences 113(15), 3932–3937 (2016) https://doi.org/10.1073/pnas.1517384113

    Brunton, S.L., Proctor, J.L., Kutz, J.N.: Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proceedings of the National Academy of Sciences 113(15), 3932–3937 (2016) https://doi.org/10.1073/pnas.1517384113

    2016 doi

  30. [38]

    Journal of Scientific Computing81, 594–622 (2019) https://doi.org/10.1007/s10915-019-01031-z

    Barsukow, W., Hohm, J., Klingenberg, C., Roe, P.L.: The active flux scheme on cartesian 26 grids and its low mach number limit. Journal of Scientific Computing81, 594–622 (2019) https://doi.org/10.1007/s10915-019-01031-z

    2019 doi

  31. [39]

    Journal of Scientific Computing94, 54 (2023) https://doi.org/10.1007/ s10915-023-02106-8 arXiv:2205.07572 [math.NA]

    Calhoun, D., Chudzik, E., Helzel, C.: The cartesian grid active flux method with adaptive mesh refinement. Journal of Scientific Computing94, 54 (2023) https://doi.org/10.1007/ s10915-023-02106-8 arXiv:2205.07572 [math.NA]

    2023 arXiv

  32. [40]

    Society for Industrial and Applied Mathematics, Philadelphia (1997)

    Trefethen, L.N., Bau, D.: Numerical Linear Algebra. Society for Industrial and Applied Mathematics, Philadelphia (1997)

    1997

  33. [41]

    Society for Industrial and Applied Mathematics, Philadelphia (2002)

    Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2002). https://doi.org/10.1137/1.9780898718027

    2002 doi

  34. [42]

    SIAM Journal on Numerical Analysis25(3), 586–617 (1988) https://doi.org/10.1137/0725037

    Cohn, S.E., Dee, D.P.: Observability of discretized partial differential equations. SIAM Journal on Numerical Analysis25(3), 586–617 (1988) https://doi.org/10.1137/0725037

    1988 doi

  35. [43]

    Regelungstechnik14(12), 545–548 (1966) https://doi.org/10.1524/auto.1966.14.112

    Ho, B.L., Kalman, R.E.: Effective construction of linear state-variable models from input/output functions. Regelungstechnik14(12), 545–548 (1966) https://doi.org/10.1524/auto.1966.14.112. 545

    1966 doi

  36. [44]

    IEEE Transactions on Automatic Control26(1), 17–32 (1981) https: //doi.org/10.1109/TAC.1981.1102568

    Moore, B.C.: Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Transactions on Automatic Control26(1), 17–32 (1981) https: //doi.org/10.1109/TAC.1981.1102568

    1981

  37. [45]

    Journal of Computational Physics14(2), 159–179 (1974) https://doi.org/10.1016/0021-9991(74)90011-4

    Warming, R.F., Hyett, B.J.: The modified equation approach to the stability and accuracy analysis of finite-difference methods. Journal of Computational Physics14(2), 159–179 (1974) https://doi.org/10.1016/0021-9991(74)90011-4

    1974 doi

  38. [46]

    SIAM Review25(1), 35–61 (1983) https://doi.org/10.1137/1025002

    Harten, A., Lax, P.D., Leer, B.: On upstream differencing and godunov-type schemes for hyperbolic conservation laws. SIAM Review25(1), 35–61 (1983) https://doi.org/10.1137/1025002

    1983 doi

  39. [47]

    Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conserva- tion laws. i. Mathematics of Computation49(179), 91–103 (1987) https://doi.org/10.1090/ S0025-5718-1987-0890255-3

    1987

  40. [48]

    International Journal for Numerical Methods in Fluids47(10-11), 1173–1179 (2005) https://doi.org/10.1002/ fld.862 https://onlinelibrary.wiley.com/doi/pdf/10.1002/fld.862 27

    Margolin, L.G., Rider, W.J.: The design and construction of implicit les models. International Journal for Numerical Methods in Fluids47(10-11), 1173–1179 (2005) https://doi.org/10.1002/ fld.862 https://onlinelibrary.wiley.com/doi/pdf/10.1002/fld.862 27

    2005 doi

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.