pith. sign in

arxiv: 2504.15516 · v2 · submitted 2025-04-22 · 🧮 math.NA · cs.NA

Derivation of Runge--Kutta Order Conditions via Functional Tree Tensor Networks

Junyuan He , Zhonghao Sun , Jizu Huang This is my paper

Pith reviewed 2026-05-22 19:22 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Runge-Kuttaorder conditionstree tensor networksTaylor expansionordinary differential equationsnumerical analysisfunctional derivativestensor networks
0
0 comments X

The pith

Runge-Kutta order conditions derived directly from tensor network expansions without induction

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

This paper introduces a framework based on tree tensor networks to handle high-order derivatives of functions under constraints. It uses this to derive the Taylor series for solutions of ODEs and for numerical approximations from Runge-Kutta methods. By comparing these expansions using tensor operators, the authors obtain order conditions for the methods. The approach yields sharper conditions for a given right-hand side function, showing when a method of claimed order p actually converges faster.

Core claim

The central claim is that the decomposition of the total derivative of a functional tree tensor network into a sum of networks for partial derivatives allows the Taylor expansion of numerical solutions under Runge-Kutta constraints to be expressed analogously to the exact solution. Order conditions then follow from equating coefficients in these expansions, bypassing mathematical induction. For a specific function f, this produces conditions stricter than the classical Butcher ones, pinpointing cases of superconvergence.

What carries the argument

The decomposition of the total derivative of a TTN into a summation of TTNs corresponding to its partial derivatives, applied to both exact and RK-constrained solutions.

Load-bearing premise

The tensor properties of partial derivatives combined with the separable structure of Runge-Kutta methods allow the Taylor expansions of numerical and exact solutions to be derived in an analogous way using tensor operators.

What would settle it

Compute the actual convergence order of a standard fourth-order Runge-Kutta method on a specific nonlinear function where the paper's sharper conditions predict order five, and check if the observed rate matches the higher prediction.

read the original abstract

Tree tensor networks (TTNs) provide a compact and structured representation of high-dimensional data, making them valuable in various areas of computational mathematics and physics. In this paper, we present a rigorous mathematical framework for expressing high-order derivatives of functional TTNs, both with or without constraints. Our framework decomposes the total derivative of a given TTN into a summation of TTNs, each corresponding to the partial derivatives of the original TTN. Using this decomposition, we derive the Taylor expansion of vector-valued functions subject to ordinary differential equation constraints or algebraic constraints imposed by Runge--Kutta (RK) methods. As a concrete application, we employ this framework to construct order conditions for RK methods. Due to the intrinsic tensor properties of partial derivatives and the separable tensor structure in RK methods, the Taylor expansion of numerical solutions can be obtained in a manner analogous to that of exact solutions using tensor operators. This enables the order conditions of RK methods to be established by directly comparing the Taylor expansions of the exact and numerical solutions, eliminating the need for mathematical induction. For a given function $\vector{f}$, we derive sharper order conditions that go beyond the classical ones, enabling the identification of situations where a standard RK scheme of order $p$ achieves unexpectedly higher convergence order for the particular function. These results establish new connections between tensor network theory and classical numerical methods, potentially opening new avenues for both analytical exploration and practical computation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a framework for expressing high-order derivatives of functional tree tensor networks (TTNs), decomposing the total derivative into a sum of TTNs corresponding to partial derivatives. This is applied to derive Taylor expansions for vector-valued functions under ODE constraints or algebraic constraints from Runge-Kutta methods. Order conditions for RK schemes are then obtained by direct comparison of the exact and numerical Taylor expansions using tensor operators, without induction. For a given f, the approach yields sharper order conditions than the classical Butcher conditions, identifying cases where a standard order-p RK method attains higher convergence order.

Significance. If the central TTN decomposition holds without implicit rank or separability restrictions on f, the work provides a new derivation route for RK order conditions and a mechanism for detecting super-convergence on specific f. The explicit connection between tensor-network representations and classical numerical analysis is a genuine novelty that could enable both analytical refinements and computational tools for order analysis.

major comments (2)
  1. [TTN derivative decomposition] The TTN derivative decomposition (abstract and the section on functional TTN derivatives) must be shown to reproduce every multi-index term appearing in the classical Taylor expansion for arbitrary smooth f. Please supply an explicit verification, e.g., by expanding a low-dimensional example to all rooted trees up to order 4 and confirming exact agreement with the Butcher series coefficients; otherwise the sharper order conditions risk being valid only under an unstated low-rank assumption on f.
  2. [Application to RK methods] § on application to RK methods: the claim that the separable tensor structure of RK stages permits an analogous tensor-operator expansion for the numerical solution must be accompanied by the explicit operator equality that equates the numerical and exact expansions term-by-term. Without this identity written out, it is unclear whether the comparison directly yields the classical conditions plus the additional sharper ones, or whether extra structural constraints are introduced.
minor comments (2)
  1. [Abstract] The abstract states that the framework applies to both ODE and algebraic constraints, yet the concrete RK application is described only for the ODE case; a short clarifying sentence on how algebraic constraints are handled would improve readability.
  2. [Notation] Notation for the tensor operators that represent the RK stages should be introduced once and used consistently; several passages refer to “tensor operators” without a preceding definition or reference to the relevant equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for providing constructive comments that will help strengthen our presentation. We respond to each major comment below.

read point-by-point responses

  1. Referee: [TTN derivative decomposition] The TTN derivative decomposition (abstract and the section on functional TTN derivatives) must be shown to reproduce every multi-index term appearing in the classical Taylor expansion for arbitrary smooth f. Please supply an explicit verification, e.g., by expanding a low-dimensional example to all rooted trees up to order 4 and confirming exact agreement with the Butcher series coefficients; otherwise the sharper order conditions risk being valid only under an unstated low-rank assumption on f.

    Authors: We agree that providing an explicit verification is important to confirm the generality of our framework. The TTN derivative decomposition is derived without any rank or separability restrictions on f, and it is intended to hold for arbitrary smooth functions. In the revised version, we will include a detailed example in an appendix, considering a low-dimensional vector-valued function (e.g., in 2 dimensions). We will expand the derivatives using the TTN approach up to order 4, listing all rooted trees and their coefficients, and verify exact agreement with the classical Butcher series terms. This will demonstrate that no implicit assumptions are made and that the sharper order conditions apply generally. revision: yes

  2. Referee: [Application to RK methods] § on application to RK methods: the claim that the separable tensor structure of RK stages permits an analogous tensor-operator expansion for the numerical solution must be accompanied by the explicit operator equality that equates the numerical and exact expansions term-by-term. Without this identity written out, it is unclear whether the comparison directly yields the classical conditions plus the additional sharper ones, or whether extra structural constraints are introduced.

    Authors: We appreciate this suggestion for improving clarity. In the revised manuscript, we will explicitly state the operator equality in the section on the application to Runge-Kutta methods. This will show how the tensor-operator expansion for the numerical solution, leveraging the separable structure of the RK stages, matches the exact expansion term by term. The identity will be written out to illustrate that the direct comparison produces the classical order conditions along with the sharper ones for specific f, without introducing additional constraints beyond those inherent to the method. revision: yes

Circularity Check

0 steps flagged

No circularity: independent tensor expansions compared directly

full rationale

The derivation constructs Taylor expansions of exact and numerical solutions separately via the TTN derivative decomposition into partial-derivative TTNs, then equates coefficients by direct comparison. This uses the stated intrinsic tensor properties of partial derivatives together with the separable structure of RK stages; no parameter is fitted to the target order conditions, no term is defined in terms of the result it is supposed to produce, and no self-citation chain is invoked to justify the central operator equality. The abstract explicitly frames the order conditions as arising from this side-by-side comparison rather than from any inductive or self-referential step, and the sharper conditions for particular f follow from the same explicit matching without reducing the general case to a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Abstract-only review prevents exhaustive enumeration; the framework appears to rest on the standard Taylor theorem for vector functions and on the newly stated decomposition property of constrained TTN derivatives.

axioms (2)
  • standard math Taylor expansion theorem for vector-valued functions subject to ODE or algebraic constraints
    Invoked to equate coefficients between exact and numerical solutions.
  • domain assumption Decomposition of the total derivative of a functional TTN into a sum of TTNs corresponding to partial derivatives
    Central modeling step stated in the abstract as the basis for all subsequent expansions.

pith-pipeline@v0.9.0 · 5787 in / 1448 out tokens · 71215 ms · 2026-05-22T19:22:10.427371+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Low Stage High Order Explicit Runge--Kutta Methods via Q- and D-Conditions: General Theory and Efficient Recursive Construction

    math.NA 2026-05 conditional novelty 7.0

    A Q/D-space reformulation of Butcher simplifying assumptions yields sufficient order conditions and a recursive linear-system construction for explicit Runge-Kutta methods of even order p with s(p)=(p²-2p+8)/4 stages.

Reference graph

Works this paper leans on

85 extracted references · 85 canonical work pages · cited by 1 Pith paper · 1 internal anchor

  1. [1]

    Acar and B

    E. Acar and B. Yener , Unsupervised multiway data analysis: A literature survey, IEEE Trans. Knowl. Data Eng., 21 (2008), pp. 6–20

    work page 2008

  2. [2]

    G. P. Agrawal, Nonlinear fiber optics , in Nonlinear Science at the Dawn of the 21st Century, Springer, 2000, pp. 195–211

    work page 2000

  3. [3]

    N. N. Akhmediev, A. Ankiewicz, et al. , Nonlinear pulses and beams , Springer, 88 (1997), pp. 23–43

    work page 1997

  4. [4]

    U. M. Ascher, S. J. Ruuth, and R. J. Spiteri , Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations , Appl. Numer. Math., 25 (1997), pp. 151–167

    work page 1997

  5. [5]

    Bachmayr, Low-rank tensor methods for partial differential equations , Acta Numer., 32 (2023), pp

    M. Bachmayr, Low-rank tensor methods for partial differential equations , Acta Numer., 32 (2023), pp. 1–121

    work page 2023

  6. [6]

    Bachmayr, R

    M. Bachmayr, R. Schneider, and A. Uschmajew , Tensor networks and hierarchical tensors for the solution of high-dimensional par tial differential equations, Found. Comput. Math., 16 (2016), pp. 1423–1472

    work page 2016

  7. [7]

    Berland, B

    H. Berland, B. Owren, and B. Skaflestad , B-series and order conditions for exponential integrators , SIAM J. Numer. Anal., 43 (2005), pp. 1715–1727

    work page 2005

  8. [8]

    Bruls and A

    O. Bruls and A. Cardona , On the use of Lie group time integrators in multibody dynamics, J. Comput. Nonlinear Dyn., 5 (2010). 36

    work page 2010

  9. [9]

    Burrage and P

    K. Burrage and P. M. Burrage , Order conditions of stochastic Runge–Kutta methods by B-series , SIAM J. Numer. Anal., 38 (2000), pp. 1626–1646

    work page 2000

  10. [10]

    J. C. Butcher , Coefficients for the study of Runge–Kutta integration processes , J. Aust. Math. Soc., 3 (1963), pp. 185–201

    work page 1963

  11. [11]

    J. C. Butcher , Numerical Methods for Ordinary Differential Equations , John Wiley & Sons, 2016

    work page 2016

  12. [12]

    J. C. Butcher , B-Series: Algebraic Analysis of Numerical Methods , vol. 55, Springer, 2021

    work page 2021

  13. [13]

    Butcher, Runge–Kutta methods for ordinary differential equations , Numer

    JC. Butcher, Runge–Kutta methods for ordinary differential equations , Numer. Anal. Optim. NAO-III Muscat Oman January 2014, (2015), pp. 37– 58

    work page 2014

  14. [14]

    Buvoli and B

    T. Buvoli and B. S. Southworth , A new class of Runge–Kutta meth- ods for nonlinearly partitioned systems , ArXiv Prepr. ArXiv240104859, (2024), https://arxiv.org/abs/2401.04859

    work page arXiv 2024

  15. [15]

    Ceruti, C

    G. Ceruti, C. Lubich, and H. W alach , Time integration of tree tensor networks, SIAM J. Numer. Anal., 59 (2021), pp. 289–313

    work page 2021

  16. [16]

    Chartier, E

    P. Chartier, E. Hairer, and G. Vilmart , Algebraic structures of b-series , Found. Comput. Math., 10 (2010), pp. 407–427

    work page 2010

  17. [17]

    F. F. Chen et al. , Introduction to Plasma Physics and Controlled Fusion , vol. 1, Springer, 1984

    work page 1984

  18. [18]

    Chill `a and J

    F. Chill `a and J. Schumacher , New perspectives in turbulent Rayleigh-B´ enard convection, Eur. Phys. J. E, 35 (2012), pp. 1–25

    work page 2012

  19. [19]

    Cichocki, N

    A. Cichocki, N. Lee, I. Oseledets, A.-H. Phan, Q. Zhao, D. P. Ma ndic, et al. , Tensor networks for dimensionality reduction and large-scale optimiza- tion: Part 1 low-rank tensor decompositions , Found. Trends Mach. Learn., 9 (2016), pp. 249–429

    work page 2016

  20. [20]

    Comon, G

    P. Comon, G. Golub, L.-H. Lim, and B. Mourrain , Symmetric tensors and symmetric tensor rank , SIAM Journal on Matrix Analysis and Applications, 30 (2008), pp. 1254–1279

    work page 2008

  21. [21]

    Dolgov, B

    S. Dolgov, B. Khoromskij, and D. Savostyanov , Superfast Fourier trans- form using QTT approximation , J. Fourier Anal. Appl., 18 (2012), pp. 915–953

    work page 2012

  22. [22]

    Dolgov, D

    S. Dolgov, D. Kressner, and C. Stro ¨ossner, Functional Tucker approx- imation using Chebyshev interpolation , SIAM J. Sci. Comput., 43 (2021), pp. A2190–A2210. 37

    work page 2021

  23. [23]

    Geometric Structures in Tensor Representations (Final Release)

    A. F alc ´o, W. Hackbusch, and A. Nouy , Geometric struc- tures in tensor representations , ArXiv Prepr. ArXiv150503027, (2015), https://arxiv.org/abs/1505.03027

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  24. [24]

    F alc´o, W

    A. F alc´o, W. Hackbusch, and A. Nouy , Tree-based tensor formats , SeMA J., 78 (2021), pp. 159–173

    work page 2021

  25. [25]

    V. L. Ginzburg, V. L. Ginzburg, and LD. Landau , On the Theory of Superconductivity, Springer, 2009

    work page 2009

  26. [26]

    Goldstone , Derivation of the Brueckner many-body theory , Proc

    J. Goldstone , Derivation of the Brueckner many-body theory , Proc. R. Soc. Lond. Ser. Math. Phys. Sci., 239 (1957), pp. 267–279

    work page 1957

  27. [27]

    Gorodetsky, S

    A. Gorodetsky, S. Karaman, and Y. Marzouk , A continuous analogue of the tensor-train decomposition , Comput. Methods Appl. Mech. Engrg., 347 (2019), pp. 59–84

    work page 2019

  28. [28]

    Gourianov , Exploiting the Structure of Turbulence with Tensor Networks , PhD thesis, University of Oxford, 2022

    N. Gourianov , Exploiting the Structure of Turbulence with Tensor Networks , PhD thesis, University of Oxford, 2022

    work page 2022

  29. [29]

    Gourianov, P

    N. Gourianov, P. Givi, D. Jaksch, and S. B. Pope , Tensor networks enable the calculation of turbulence probability distribut ions, ArXiv Prepr. ArXiv240709169, (2024), https://arxiv.org/abs/2407.09169

    work page arXiv 2024

  30. [30]

    Grasedyck, D

    L. Grasedyck, D. Kressner, and C. Tobler , A literature survey of low-rank tensor approximation techniques , GAMM-Mitteilungen, 36 (2013), pp. 53–78

    work page 2013

  31. [31]

    Hackbusch , Tensor Spaces and Numerical Tensor Calculus , vol

    W. Hackbusch , Tensor Spaces and Numerical Tensor Calculus , vol. 42, Springer, 2012

    work page 2012

  32. [32]

    Hairer , Order conditions for numerical methods for partitioned ord inary differential equations, Numer

    E. Hairer , Order conditions for numerical methods for partitioned ord inary differential equations, Numer. Math., 36 (1981), pp. 431–445

    work page 1981

  33. [33]

    Hairer, C

    E. Hairer, C. Lubich, and M. Roche , The Numerical Solution of Differential- Algebraic Systems by Runge–Kutta Methods , vol. 1409, Springer, 2006

    work page 2006

  34. [34]

    Hairer, C

    E. Hairer, C. Lubich, and G. W anner , Geometric Numerical Integration: Structure- preserving Algorithms for Ordinary Differential Equations, Springer Science & Business Media, 2006

    work page 2006

  35. [35]

    Hairer, S

    E. Hairer, S. P. Nørsett, and G. W anner , Solving Ordinary Differential Equations I: Nonstiff Problems: With 105 Figures , Springer Berlin, Heidelberg, 1987

    work page 1987

  36. [36]

    Hauschild and F

    J. Hauschild and F. Pollmann , Efficient numerical simulations with tensor networks: Tensor Network Python (TeNPy) , SciPost Phys. Lect. Notes, (2018), p. 005. 38

    work page 2018

  37. [37]

    Holtz, T

    S. Holtz, T. Rohwedder, and R. Schneider , The alternating linear scheme for tensor optimization in the tensor train format , SIAM J. Sci. Comput., 34 (2012), pp. A683–A713

    work page 2012

  38. [38]

    C. Hua, G. Rabusseau, and J. Tang , High-order pooling for graph neural networks with tensor decomposition , Adv. Neural Inf. Process. Syst., 35 (2022), pp. 6021–6033

    work page 2022

  39. [39]

    Humphries and AM

    AR. Humphries and AM. Stuart , Runge–Kutta methods for dissipative and gradient dynamical systems , SIAM J. Numer. Anal., 31 (1994), pp. 1452–1485

    work page 1994

  40. [40]

    Y. Ji, Q. W ang, X. Li, and J. Liu , A survey on tensor techniques and applications in machine learning , IEEE Access, 7 (2019), pp. 162950–162990

    work page 2019

  41. [41]

    C. A. Kennedy and M. H. Carpenter , Additive Runge–Kutta schemes for convection–diffusion–reaction equations, Appl. Numer. Math., 44 (2003), pp. 139– 181

    work page 2003

  42. [42]

    B. N. Khoromskij , Tensors-structured numerical methods in scientific com- puting: Survey on recent advances , Chemom. Intell. Lab. Syst., 110 (2012), pp. 1–19

    work page 2012

  43. [43]

    B. N. Khoromskij , Tensor numerical methods for multidimensional PDEs: Theoretical analysis and initial applications , ESAIM Proc. Surv., 48 (2015), pp. 1–28

    work page 2015

  44. [44]

    H. A. Kiers , Towards a standardized notation and terminology in multiway analysis, J. Chemom., 14 (2000), pp. 105–122

    work page 2000

  45. [45]

    T. G. Kolda and B. W. Bader , Tensor decompositions and applications , SIAM Rev., 51 (2009), pp. 455–500

    work page 2009

  46. [46]

    Kutta, Beitrag zur naherungsweisen integration totaler differential gleichun- gen, Z

    W. Kutta, Beitrag zur naherungsweisen integration totaler differential gleichun- gen, Z. Math. Phys., 46 (1901), pp. 435–453

    work page 1901

  47. [47]

    Lebedev, Y

    V. Lebedev, Y. Ganin, M. Rakhuba, I. Oseledets, and V. Lempitsk y, Speeding-up convolutional neural networks using fine-tuned CP-decomposition, in 3rd Int. Conf. Learn. Represent. ICLR 2015-Conf. Track Proc ., 2015

    work page 2015

  48. [48]

    Lee and A

    N. Lee and A. Cichocki , Fundamental tensor operations for large-scale data analysis using tensor network formats , Multidimens. Syst. Signal Process., 29 (2018), pp. 921–960

    work page 2018

  49. [49]

    Liu, S.-J

    D. Liu, S.-J. Ran, P. Wittek, C. Peng, R. B. Garc ´ıa, G. Su, and M. Lewenstein, Machine learning by unitary tensor network of hierarchical t ree structure, New J. Phys., 21 (2019), p. 073059

    work page 2019

  50. [50]

    J. Liu, S. Li, J. Zhang, and P. Zhang , Tensor networks for unsupervised 39 machine learning , Phys. Rev. E, 107 (2023), p. L012103

    work page 2023

  51. [51]

    Liu, Z.-F

    P. Liu, Z.-F. Gao, W. X. Zhao, Z.-Y. Xie, Z.-Y. Lu, and J.-R. We n, Enabling lightweight fine-tuning for pre-trained language model compression based on matrix product operators , in Proc. 59th Annu. Meet. Assoc. Comput. Linguist. 11th Int. Jt. Conf. Nat. Lang. Process. Vol. 1 Long Pap., 2021, p p. 5388–5398

    work page 2021

  52. [52]

    Lorenz , Deterministic nonperiodic flow , J

    E. Lorenz , Deterministic nonperiodic flow , J. Atmospheric Sci., 20 (1963)

    work page 1963

  53. [53]

    X. Ma, P. Zhang, S. Zhang, N. Duan, Y. Hou, M. Zhou, and D. Song , A tensorized transformer for language modeling , Adv. Neural Inf. Process. Syst., 32 (2019)

    work page 2019

  54. [54]

    I. L. Markov and Y. Shi , Simulating quantum computation by contracting tensor networks , SIAM J. Comput., 38 (2008), pp. 963–981

    work page 2008

  55. [55]

    R. D. Mattuck , A Guide to Feynman Diagrams in the Many-Body Problem , Courier Corporation, 1992

    work page 1992

  56. [56]

    Munthe–Kaas and B

    H. Munthe–Kaas and B. Owren , Computations in a free Lie algebra , Philos. Trans. R. Soc. Lond. Ser. Math. Phys. Eng. Sci., 357 (1999), pp. 957–981

    work page 1999

  57. [57]

    Munthe-Kaas , Lie-Butcher theory for Runge–Kutta methods , BIT Numer

    H. Munthe-Kaas , Lie-Butcher theory for Runge–Kutta methods , BIT Numer. Math., 35 (1995), pp. 572–587

    work page 1995

  58. [58]

    Munthe-Kaas , Runge–Kutta methods on Lie groups , BIT Numer

    H. Munthe-Kaas , Runge–Kutta methods on Lie groups , BIT Numer. Math., 38 (1998), pp. 92–111

    work page 1998

  59. [59]

    Munthe-Kaas , High order Runge–Kutta methods on manifolds , Appl

    H. Munthe-Kaas , High order Runge–Kutta methods on manifolds , Appl. Numer. Math., 29 (1999), pp. 115–127

    work page 1999

  60. [60]

    V. Murg, F. Verstraete, ¨O. Legeza, and R. M. Noack , Simulating strongly correlated quantum systems with tree tensor networks , Phys. Rev. B, 82 (2010), p. 205105

    work page 2010

  61. [61]

    Murua , On order conditions for partitioned symplectic methods , SIAM J

    A. Murua , On order conditions for partitioned symplectic methods , SIAM J. Numer. Anal., 34 (1997), pp. 2204–2211

    work page 1997

  62. [62]

    Nakatani and G

    N. Nakatani and G. K. Chan , Efficient tree tensor network states (TTNS) for quantum chemistry: Generalizations of the density matrix reno rmalization group algorithm, J. Chem. Phys., 138 (2013)

    work page 2013

  63. [63]

    Novikov, D

    A. Novikov, D. Podoprikhin, A. Osokin, and D. P. Vetrov , Tensorizing neural networks , Adv. Neural Inf. Process. Syst., 28 (2015)

    work page 2015

  64. [64]

    Or ´us, A practical introduction to tensor networks: Matrix product st ates and projected entangled pair states , Ann

    R. Or ´us, A practical introduction to tensor networks: Matrix product st ates and projected entangled pair states , Ann. Phys., 349 (2014), pp. 117–158. 40

    work page 2014

  65. [65]

    Or ´us, Tensor networks for complex quantum systems , Nat

    R. Or ´us, Tensor networks for complex quantum systems , Nat. Rev. Phys., 1 (2019), pp. 538–550

    work page 2019

  66. [66]

    I. V. Oseledets , Tensor-train decomposition, SIAM J. Sci. Comput., 33 (2011), pp. 2295–2317

    work page 2011

  67. [67]

    A.-H. Phan, K. Sobolev, K. Sozykin, D. Ermilov, J. Gusak, P. Tichavsk `y, V. Glukhov, I. Oseledets, and A. Cichocki , Stable low- rank tensor decomposition for compression of convolutional neu ral network , in Comput. Vision–ECCV 2020 16th Eur. Conf. Glasg. UK August 23–28 2020 Proc. Part XXIX 16, Springer, 2020, pp. 522–539

    work page 2020

  68. [68]

    D. S. G. Pollock , On Kronecker products, tensor products and matrix differential calculus, Int. J. Comput. Math., 90 (2013), pp. 2462–2476

    work page 2013

  69. [69]

    Ragnarsson , Structured Tensor Computations: Blocking, Symmetries and Kronecker Factorizations, PhD thesis, Cornell University, 2012

    S. Ragnarsson , Structured Tensor Computations: Blocking, Symmetries and Kronecker Factorizations, PhD thesis, Cornell University, 2012

    work page 2012

  70. [70]

    Riordan , An Introduction to Combinatorial Analysis , Princeton University Press, 2014

    J. Riordan , An Introduction to Combinatorial Analysis , Princeton University Press, 2014

    work page 2014

  71. [71]

    O. E. R ¨ossler, An equation for continuous chaos , Phys. Lett. A, 57 (1976), pp. 397–398

    work page 1976

  72. [72]

    Runge , ¨Uber die numerische Aufl¨ osung von Differentialgleichungen , Math

    C. Runge , ¨Uber die numerische Aufl¨ osung von Differentialgleichungen , Math. Ann., 46 (1895), pp. 167–178

    work page

  73. [73]

    F. A. Schr ¨oder, D. H. Turban, A. J. Musser, N. D. Hine, and A. W. Chin, Tensor network simulation of multi-environmental open quant um dynam- ics via machine learning and entanglement renormalisation , Nat. Commun., 10 (2019), p. 1062

    work page 2019

  74. [74]

    Schr ¨odinger, An undulatory theory of the mechanics of atoms and molecules, Phys

    E. Schr ¨odinger, An undulatory theory of the mechanics of atoms and molecules, Phys. Rev., 28 (1926), p. 1049

    work page 1926

  75. [75]

    Shi, L.-M

    Y.-Y. Shi, L.-M. Duan, and G. Vidal , Classical simulation of quantum many- body systems with a tree tensor network , Phys. Rev. A, 74 (2006), p. 022320

    work page 2006

  76. [76]

    Stoudenmire and D

    E. Stoudenmire and D. J. Schwab , Supervised learning with tensor networks , Adv. Neural Inf. Process. Syst., 29 (2016)

    work page 2016

  77. [77]

    J. Su, W. Byeon, J. Kossaifi, F. Huang, J. Kautz, and A. Anandku mar, Convolutional tensor-train LSTM for spatio-temporal learni ng, Adv. Neural Inf. Process. Syst., 33 (2020), pp. 13714–13726

    work page 2020

  78. [78]

    Temam , Navier–Stokes Equations: Theory and Numerical Analysis , vol

    R. Temam , Navier–Stokes Equations: Theory and Numerical Analysis , vol. 343, American Mathematical Society, 2024. 41

    work page 2024

  79. [79]

    Thaller , The Dirac Equation , Springer Science & Business Media, 2013

    B. Thaller , The Dirac Equation , Springer Science & Business Media, 2013

    work page 2013

  80. [80]

    B. K. Tran, B. S. Southworth, and T. Buvoli , Order conditions for non- linearly partitioned runge–kutta methods , ArXiv Prepr. ArXiv240112427, (2024), https://arxiv.org/abs/2401.12427

    work page arXiv 2024

Showing first 80 references.