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REVIEW 4 major objections 5 minor 58 references

Tensor-Network Finite Elements for Analytic Operator Equations

T0 review · 4 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read The paper claims that analytic operator equations, including nonlinear PDEs, reduce to a single linear residual equation once finite-element coefficients are lifted into a Fock space, and that tensor-network minimization of that residual re

desk verdict A plausible formal bridge between FE and tensor networks, but the advertised linearization of nonlinear PDEs is not proven and the numerics don't test it. read the letter →

arxiv 2607.13129 v1 pith:XYOQR3VV submitted 2026-07-14 math.NA cond-mat.othercs.NAhep-latphysics.comp-ph

Abhijatmedhi Chotrattanapituk , Michael J. Landry , Chu-Liang Fu , Mingda Li This is my paper · ORCID
classification math.NAcond-mat.othercs.NAhep-latphysics.comp-ph MSC 65N3065M60
keywords analyticoperatorequationstensornetworksfiniteelementmethodFock-spaceliftmatrixproductstatesnonlineardiffusionvariationaloptimizationmultilinearinteractiontensors
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 tries to establish a single variational language in which differential, integral, nonlinear, memory, and delay equations all become the same kind of object: a linear equation on an enlarged space of finite-element correlation coefficients. The trick is to lift the usual finite-element coefficients into a Fock space, where products and derivatives of the unknown function act as linear operators, and then to minimize the weak-form residual over a tensor-network manifold. If the construction works, highly nonlinear PDEs can be solved with tensor-network tools at a cost controlled by a bond dimension rather than by the exponential size of the full coefficient space. The paper demonstrates the proof of principle on one-dimensional linear and nonlinear diffusion, reproducing conventional solutions to within about two percent with a matrix-product-state ansatz of bond dimension one.

What carries the argument

The load-bearing object is the Fock-space lift: each element's local basis functions span a Hilbert space, the Fock space is the direct sum of its tensor powers, and each basis vector labels a sequence of element basis functions participating in a correlation. The FE-discretized operator becomes a tensor G acting linearly on coefficient vectors c in this Fock space, so nonlinearity is absorbed into the space itself. The second ingredient is the tensor-network variational manifold (e.g., matrix product state), which parametrizes the exponentially large coefficient space at polynomial cost and makes each single-site update a convex least-squares subproblem.

What would settle it

Solve a nonlinear PDE with a known analytic solution and a steepening front (e.g., Burgers' equation) on a fine mesh using the unconstrained Fock-space residual minimized over an MPS manifold of increasing bond dimension: if the physical projection does not converge to the known solution as bond dimension grows, or if different choices of the unspecified extra conditions give different physical projections, the central claim fails. A more direct check: verify that the null space of G projected to the physical sector is one-dimensional at every time step.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that finite-element discretization of an analytic operator equation produces a hierarchy of multilinear interaction tensors, and that lifting the element coefficients into a Fock space converts the nonlinear residual into a linear matrix equation G⊙c=0 on an augmented coefficient space. The physical solution is recovered by projecting the Fock-space solution back, and the residual is minimized as a weighted least-squares problem over tensor-network ansatz states. The authors claim this gives a common algebraic structure to PDEs, integro-differential equations, and memory-delay equations, and that matrix-product-state calculations for 1D diff

Load-bearing premise

The load-bearing premise is that the enlarged Fock-space residual problem can be constrained so that its solution's physical projection is the true finite-element solution; the paper asserts this but does not specify the extra constraints needed to make the projection unique.

Editorial extensions

If this is right

  • Nonlinear PDEs can be reformulated as linear residual equations on Fock space, so the same tensor-network solver applies across equation classes.
  • The cost of solving an equation scales with the bond dimension of the coefficient tensor-network, not with the exponential size of the full coefficient space, whenever the solution has limited inter-element correlation.
  • Implicit time stepping for initial-value problems reduces to a sequence of stationary variational problems, with element-level derivative and multiplication operators precomputed once and reused.
  • The framework preserves continuity and Neumann boundary conditions to numerical precision when the boundary residual terms are added to the objective.
  • Convergence in spatial resolution behaves like standard finite-difference/FEM methods: increasing basis order or element count only helps up to the fixed temporal resolution.

Reading between the lines

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

  • If the Fock-space lift is truly linear and the physical sector can be isolated, then quantum or quantum-inspired linear-system solvers could be applied to nonlinear PDEs—an implication the paper gestures at but does not develop.
  • The bond-dimension-one demonstration suggests the method works best when the solution's inter-element correlations are weak; a sharp test would be a problem with strong front propagation or shock formation where bond dimension one should break down.
  • The extra Fock-space degrees of freedom are asserted to require additional constraints (regularization or full equations) that are never specified; whether a unique physical projection exists is the point most in need of proof or counterexample.
  • The paper's precomputed element-level operators (derivative and multiplication) could be reused as building blocks for other nonlinear PDEs, such as reaction-diffusion or convection-dominated systems, as a drop-in variational scheme.
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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 / 5 minor

Summary. The paper proposes a tensor-network finite-element framework for analytic operator equations. It lifts the finite-element coefficient tensor into a Fock space, represents discretized operator equations as tensor contractions, and reformulates the problem as a weighted-residual minimization over tensor-network manifolds. The method is demonstrated on 1D linear and nonlinear diffusion initial-value problems using an MPS ansatz and DMRG sweeps. The central advertised feature is the conversion of nonlinear PDEs into linear matrix equations.

Significance. The topic is timely, and the paper provides clear tensor-circuit diagrams and element-local operator constructions that may be useful. The 1D numerical results show that an MPS-based minimization of the original nonlinear residual can match a Runge-Kutta reference to within about 2%, which is encouraging as a proof of principle. However, the central theoretical claim—that nonlinear OEs can be converted to linear matrix equations on Fock space—is not established. The required extra constraints are explicitly acknowledged but never specified, and the numerical experiments do not solve the lifted linear system. The paper is therefore stronger as a demonstration of tensor-network least-squares solvers for nonlinear finite-element residuals than as a general linearization framework. No machine-checked proofs or reproducible code are included, which further limits verifiability.

major comments (4)
  1. [Sec. IV.A, Eq. (20)] The claimed equivalence between the nonlinear FE system (9) and the linear Fock-space system (20) is unproven. As the text admits, the enlarged system 'has many more coefficients than Eq. (9). Hence, extra conditions... are required,' but these conditions are never given. Without them, Eq. (20) is underdetermined: its kernel contains vectors with a vanishing single-particle (physical) sector and nonzero higher sectors that cancel the residual, so a solution of (20) need not project to any solution of (9). The constraints needed to enforce the physical sector (e.g., c_i = c_1^{⊗i}) are nonlinear, so the advertised 'conversion to linear matrix equations' is not valid as stated.
  2. [Sec. VI, Eq. (44) and Fig. 6] The numerical demonstrations do not test the Fock-space linearization. The DMRG minimization is applied to the residual of the original nonlinear diffusion equation (44) with c(t) as the physical FE coefficient; no computation involves the lifted tensor G of Eq. (20) or the particle-number sectors. Thus, the experiments validate a tensor-network least-squares solver for nonlinear FE residuals, but provide no evidence for the claimed linear-matrix-equation formulation.
  3. [Sec. VI.A, Eqs. (38)–(42)] The local operators D_n and R_n are built from the inverse mass matrix Q_n^{-1}. The approximation error introduced by replacing the exact weak-form tensors with these contractions is not quantified, despite the paper's claim of 'controlled error.' The convergence study in Fig. 8 reports errors against an unspecified reference; without an error bound for the derivative/multiplication approximations, the error budget is incomplete.
  4. [Sec. VI, boundary-condition treatment] The paper states that a circuit is added to 'measure the residual' of continuity and Neumann conditions, and later claims these conditions are 'respected to within numerical precision.' It is unclear whether these residuals enter the objective as penalty terms or are post-hoc diagnostics. If penalties, the weights are not specified; if diagnostics, there is no mechanism enforcing them. This ambiguity undermines the claim that boundary conditions are preserved.
minor comments (5)
  1. [Sec. VI.B and Fig. 7] The text reports N=10 for the MPS experiments, while the Fig. 7 caption states N=11. Please reconcile.
  2. [Introduction] Line 1: 'quations' should be 'equations'.
  3. [Sec. II, Eq. (14)] The notation for basis vectors contains an extra opening brace; please clean up the formatting.
  4. [Sec. IV.B] The mapping to J pre-allocated slots with empty basis functions is not formally defined. It would help to specify injectivity or note any degeneracies in the representation.
  5. [Sec. VI.C, Fig. 8] The reference solution used to compute the mean absolute error is not defined. Please state whether it is the Runge-Kutta result, an analytic solution, or a refined FE solution.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation is self-contained, parameter-free with respect to reference data, and does not rely on load-bearing self-citations.

full rationale

The paper's core objects—the discretized operator tensors G, the local derivative D_n, and the multiplication R_n—are computed directly from the analytic OE kernels and the chosen FE basis via Eqs. (5)-(7), (17)-(19), (34)-(42). Nothing is fitted to the RK reference solutions; the TN calculation minimizes the weighted residual norm J(c) of Eqs. (22)-(24), and the reported <2% agreement with the Runge-Kutta solution is an independent numerical comparison, not a construction-level identity. There are no load-bearing self-citations: the reference list contains only standard external literature. The one passage that could be mistaken for circularity is the Fock-space lifting in Sec. IV.A, where products of FE coefficients are replaced by independent Fock-space coefficients and Eq. (20) is presented as a linear matrix equation. The paper itself acknowledges this lifting introduces extra degrees of freedom: 'Hence, extra conditions, which can take the form of regularization or full-blown equations, are required' (Sec. IV.A). Those constraints are never specified, and the numerical demonstration minimizes the original nonlinear residual of Eq. (44) rather than the lifted linear equation (20), so the advertised equivalence is an unproven correctness gap rather than a result that reduces to its inputs by definition. Missing proof and under-specification are real concerns, but they are not circularity under the stated criteria.

Assumptions & free parameters 5 free parameters · 4 assumptions · 2 invented entities

The framework's core is a formal rewrite of FE discretization into a tensor-network least-squares problem. The minimal mathematical overhead is a convergent Taylor expansion of the operator and a stable element basis. The significant uncharged assumptions are the recoverability of the physical solution from the larger Fock space (explicitly flagged as needing 'extra conditions' in Sec. IV.A) and the locality of correlations needed for the TN ansatz. The demonstrated runs add practical parameters (N, B, S, Δt, penalty weights) that are not documented to the level of a reproducible recipe.

free parameters (5)
  • Basis order B = 4
    Chosen for the demonstrations; no systematic optimality condition tied to the method's claims. While it is a standard FE resolution parameter rather than an ad hoc fit, the central accuracy results depend on its value.
  • Number of finite elements N = 11
    Chosen for the demonstration; the convergence study in Sec. VI.C shows error depends on N but the headline results use N=11.
  • Bond dimension S = 1
    Used for all shown results; the claim of 'controlled error' depends on this choice, and the paper does not report the same runs at higher S or the minimal S needed for 2% accuracy.
  • Time step Δt = not given explicitly
    Figure 8 varies Δt but the values are not reported in the text; the convergence behavior of the demonstration depends on them.
  • Penalty weights for continuity and Neumann residuals = not reported
    In Sec. VI.B, an 'additional circuit' measures continuity and boundary residuals in J(c); the relative weight of these residuals vs. the PDE residual is never specified, and it affects both accuracy and the claim that boundary conditions are respected.
assumptions (4)
  • domain assumption The operator L admits a convergent Fréchet–Taylor expansion of the form Eq. (4) over the relevant solution space.
    Sec. III.B introduces the expansion and cites [53–57]; this is a genuine restriction of the claimed generality, since many practical operators (e.g., non-smooth nonlinearities, monotone operators with fractional powers) are not analytic in this sense. The paper states 'analytic operator equations' in the title, so this is a stated domain assumption.
  • domain assumption The local FE spaces satisfy the element-local approximation Eq. (34) and (39) with an inverse mass matrix inversion that is well defined and stable.
    Introduced in Sec. VI.A: D_n and R_n are obtained by projecting derivatives and products onto the same element basis and applying the inverse Gram matrix. This requires the basis to be stable and the projection to be a good approximation; with Lagrange basis and small B,N this holds, but for general elements it is an assumption.
  • ad hoc to paper The physical FE solution can be recovered from the Fock-space solution c by projecting onto the single-particle subspace, and all spurious many-particle null-space components can be handled by unspecified extra constraints.
    Sec. IV.A states the enlarged system has many more coefficients and 'extra conditions... are required', but no explicit construction or uniqueness proof is given. The numerical examples implicitly assume the least-squares minimizer lies in the physical sector, which is not proven.
  • domain assumption Correlation locality: the solution's correlation structure is compatible with the chosen TN geometry (MPS for 1D).
    Sec. IV.B states the locality assumption explicitly ('we expected the correlation to only appear between neighboring elements') and notes strongly correlated solutions may require large bond dimensions. The convergence of the examples depends on this.
invented entities (2)
  • Fock space Ψ = ⊕_i Φ^⊗i of FE coefficients
    purpose: Lifts the nonlinear problem into a linear algebraic structure by treating products of the solution as higher-particle sectors
    It is a mathematical construction within the paper; no external falsifiable prediction follows from it alone. Its validity is judged by whether the projection back to the physical sector works, which is not demonstrated at the level of uniqueness.
  • Element-local derivative tensor D_n and multiplication tensor R_n
    purpose: Precomputed local operators that let spatial derivatives and multiplications act as linear maps inside each element, enabling reuse across time steps
    These are standard FE mass/stiffness-like matrices reorganized as tensors; they are not empirical entities and carry no independent evidence beyond the basis definitions that define them.

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Cite this review

Pith. "Pith review of Tensor-Network Finite Elements for Analytic Operator Equations." pith.science (2026). https://pith.science/paper/XYOQR3VV

@misc{pith2026260713129,
  author       = {Pith},
  title        = {Pith review of: Tensor-Network Finite Elements for Analytic Operator Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XYOQR3VV}},
  note         = {Machine review of arXiv:2607.13129}
}
read the original abstract

Operator equations (OEs) underpin quantitative modeling across science and engineering. Finite-element (FE) methods discretize continuous OEs into finite-dimensional algebraic systems, whereas tensor networks (TNs) provide flexible variational representations of correlated discrete systems. Here, we develop a framework that connects FE with TN for analytic OEs. The power of this method comes from its ability to convert highly non-linear partial differential equations into linear matrix equations. In particular, we show that FE discretization induces a hierarchy of multilinear interaction tensors, through which differential, integral, nonlinear, memory, and delay equations can be expressed within a common algebraic structure. The resulting systems are reformulated as weighted-residual optimization problems over TN degrees of freedom. Matrix-product-state calculations for one-dimensional linear and nonlinear diffusion reproduce conventional solutions with controlled error while preserving continuity and Neumann boundary conditions. The framework provides a common variational language for analytic OEs and establishes a direct connection between FE numerical formalism and TN variational algorithms, offering a general foundation for TN-based and quantum-inspired approaches to solving OEs.

Figures

Figures reproduced from arXiv: 2607.13129 by the authors.

Figure 1
Figure 1. FIG. 1. Contraction of i-linear interaction kernel [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Tensor network representation of the left hand side of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Tensor network ansatz of coefficient tensor with dif [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Tensor circuit representation of [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Tensor circuit representation of residual of non-linear [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Numerical simulation results for a non-linear diffusion IVPs ( [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Mean absolute error trend of TN variational method [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

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Reference graph

Works this paper leans on

58 extracted references

  1. [1]

    M. A. Krasnosel’skii, G. M. Vainikko, P. P. Zabreiko, Ya. B. Rutitskii, and V. Ya. Stetsenko.Approximate So- lution of Operator Equations. Wolters-Noordhoff Pub- lishing, Groningen, 1972

    1972

  2. [2]

    Springer, New York, 3 edition, 2009

    Kendall Atkinson and Weimin Han.Theoretical Numer- ical Analysis: A Functional Analysis Framework, vol- ume 39 ofTexts in Applied Mathematics. Springer, New York, 3 edition, 2009

    2009

  3. [3]

    Universitext

    Haim Brezis.Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer, New York, 2011

    2011

  4. [4]

    LeVeque.Finite Difference Methods for Or- dinary and Partial Differential Equations: Steady-State and Time-Dependent Problems

    Randall J. LeVeque.Finite Difference Methods for Or- dinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. Classics in Applied Mathematics. Society for Industrial and Applied Mathe- matics (SIAM), Philadelphia, PA, 2007

    2007

  5. [5]

    Strikwerda.Finite Difference Schemes and Par- tial Differential Equations

    John C. Strikwerda.Finite Difference Schemes and Par- tial Differential Equations. Society for Industrial and Ap- plied Mathematics (SIAM), Philadelphia, PA, 2 edition, 2004

    2004

  6. [6]

    J. W. Thomas.Numerical Partial Differential Equations: Finite Difference Methods, volume 22 ofTexts in Applied Mathematics. Springer, New York, 1995

    1995

  7. [7]

    Ciarlet.The Finite Element Method for El- liptic Problems, volume 40 ofClassics in Applied Mathe- matics

    Philippe G. Ciarlet.The Finite Element Method for El- liptic Problems, volume 40 ofClassics in Applied Mathe- matics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 North-Holland edition

    2002

  8. [8]

    Brenner and L

    Susanne C. Brenner and L. Ridgway Scott.The Math- ematical Theory of Finite Element Methods, volume 15 ofTexts in Applied Mathematics. Springer, New York, 3 edition, 2008

    2008

Show all 58 references
  1. [9]

    Thomas J. R. Hughes.The Finite Element Method: Lin- ear Static and Dynamic Finite Element Analysis. Dover Publications, Mineola, NY, 2000. Reprint of the 1987 Prentice-Hall edition

    2000

  2. [10]

    O. C. Zienkiewicz, R. L. Taylor, and J. Z. Zhu.The Finite Element Method: Its Basis and Fundamentals. Butterworth-Heinemann (Elsevier), Oxford, 7 edition, 2013

    2013

  3. [11]

    A practical introduction to tensor net- works: Matrix product states and projected entangled pair states.Annals of Physics, 349:117–158, 2014

    Rom´ an Or´ us. A practical introduction to tensor net- works: Matrix product states and projected entangled pair states.Annals of Physics, 349:117–158, 2014

    2014

  4. [12]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner. Finitely correlated states on quantum spin chains.Communica- tions in Mathematical Physics, 144(3):443–490, 1992

    1992

  5. [13]

    Klumper, A

    A. Klumper, A. Schadschneider, and J. Zittartz. Equiva- lence and solution of anisotropic spin-1 models and gen- eralized t-j fermion models in one dimension.Journal of Physics A: Mathematical and General, 24(16):L955, 1991

    1991

  6. [14]

    Kl¨ umper, A

    A. Kl¨ umper, A. Schadschneider, and J. Zittartz. Matrix product ground states for one-dimensional spin-1 quan- tum antiferromagnets.Europhysics Letters, 24(4):293, 1993

    1993

  7. [15]

    Verstraete and J

    F. Verstraete and J. I. Cirac. Renormalization algorithms for quantum-many body systems in two and higher di- mensions.arXiv, 2004

    2004

  8. [16]

    G. Vidal. Entanglement renormalization.Physical Re- view Letters, 99(22):220405, 2007

    2007

  9. [17]

    Steven R. White. Density matrix formulation for quan- tum renormalization groups.Phys. Rev. Lett., 69:2863– 2866, Nov 1992

    1992

  10. [18]

    Steven R. White. Density-matrix algorithms for quan- tum renormalization groups.Physical Review B, 48(14):10345–10356, 1993

    1993

  11. [19]

    H. C. Jiang, Z. Y. Weng, and D. N. Sheng. Density ma- trix renormalization group numerical study of the kagome antiferromagnet.Phys. Rev. Lett., 101:117203, Sep 2008

    2008

  12. [20]

    Spin-skagome quantum antiferromagnets in a field with tensor networks.Phys

    Thibaut Picot, Marc Ziegler, Rom´ an Or´ us, and Didier Poilblanc. Spin-skagome quantum antiferromagnets in a field with tensor networks.Phys. Rev. B, 93:060407, Feb 2016

    2016

  13. [21]

    Improved energy extrapolation with in- finite projected entangled-pair states applied to the two- dimensional hubbard model.Phys

    Philippe Corboz. Improved energy extrapolation with in- finite projected entangled-pair states applied to the two- dimensional hubbard model.Phys. Rev. B, 93:045116, Jan 2016

    2016

  14. [22]

    Dubail and N

    J. Dubail and N. Read. Tensor network trial states for chiral topological phases in two dimensions and a no-go theorem in any dimension.Physical Review B, 92(20):205307, 2015

    2015

  15. [23]

    Entanglement renor- malization and topological order.Phys

    Miguel Aguado and Guifr´ e Vidal. Entanglement renor- malization and topological order.Phys. Rev. Lett., 100:070404, Feb 2008

    2008

  16. [24]

    Explicit tensor network representation for the ground states of string-net models.Physical Review B, 79(8):085119, 2009

    Oliver Buerschaper, Miguel Aguado, and Guifr´ e Vi- dal. Explicit tensor network representation for the ground states of string-net models.Physical Review B, 79(8):085119, 2009

    2009

  17. [25]

    Tensor-product representations for string-net condensed states.Physical Review B, 79(8):085118, 2009

    Zheng-Cheng Gu, Michael Levin, Brian Swingle, and Xiao-Gang Wen. Tensor-product representations for string-net condensed states.Physical Review B, 79(8):085118, 2009

    2009

  18. [26]

    Markov and Yaoyun Shi

    Igor L. Markov and Yaoyun Shi. Simulating quan- tum computation by contracting tensor networks.SIAM Journal on Computing, 38(3):963–981, 2008

    2008

  19. [27]

    Ferris and David Poulin

    Andrew J. Ferris and David Poulin. Tensor networks and quantum error correction.Phys. Rev. Lett., 113:030501, Jul 2014

    2014

  20. [28]

    Towards quan- tum machine learning with tensor networks.Quantum Science and Technology, 4(2):024001, jan 2019

    William Huggins, Piyush Patil, Bradley Mitchell, K Bir- gitta Whaley, and E Miles Stoudenmire. Towards quan- tum machine learning with tensor networks.Quantum Science and Technology, 4(2):024001, jan 2019

    2019

  21. [29]

    Supervised learning with tensor networks

    Edwin Stoudenmire and David J Schwab. Supervised learning with tensor networks. In D. Lee, M. Sugiyama, U. Luxburg, I. Guyon, and R. Garnett, editors,Advances in Neural Information Processing Systems, volume 29. Curran Associates, Inc., 2016

    2016

  22. [30]

    Unsupervised generative modeling using matrix product states.Phys

    Zhao-Yu Han, Jun Wang, Heng Fan, Lei Wang, and Pan Zhang. Unsupervised generative modeling using matrix product states.Phys. Rev. X, 8:031012, Jul 2018

    2018

  23. [31]

    Quantum entanglement in deep learning archi- tectures.Physical Review Letters, 122(6):065301, 2019

    Yoav Levine, Or Sharir, Nadav Cohen, and Amnon Shashua. Quantum entanglement in deep learning archi- tectures.Physical Review Letters, 122(6):065301, 2019

    2019

  24. [32]

    Machine learning by unitary tensor network of hierarchi- cal tree structure.New Journal of Physics, 21(7):073059, jul 2019

    Ding Liu, Shi-Ju Ran, Peter Wittek, Cheng Peng, Raul Bl´ azquez Garc ´ ıa, Gang Su, and Maciej Lewenstein. Machine learning by unitary tensor network of hierarchi- cal tree structure.New Journal of Physics, 21(7):073059, jul 2019

    2019

  25. [33]

    Gallego and Rom´ an Or´ us

    ´Angel J. Gallego and Rom´ an Or´ us. Language design as information renormalization.SN Computer Science, 3(2):140, Jan 2022. 11

    2022

  26. [34]

    An introduction to the density matrix renormalization group ansatz in quantum chem- istry

    Garnet Kin-Lic Chan, Jonathan J Dorando, Debashree Ghosh, Johannes Hachmann, Eric Neuscamman, Haitao Wang, and Takeshi Yanai. An introduction to the density matrix renormalization group ansatz in quantum chem- istry. InFrontiers in quantum systems in chemistry and physics, pag...

    2008

  27. [35]

    Tensor product methods and entanglement op- timization for ab initio quantum chemistry.Interna- tional Journal of Quantum Chemistry, 115(19):1342– 1391, 2015

    Szil´ ard Szalay, Max Pfeffer, Valentin Murg, Gergely Bar- cza, Frank Verstraete, Reinhold Schneider, and ¨Ors Leg- eza. Tensor product methods and entanglement op- timization for ab initio quantum chemistry.Interna- tional Journal of Quantum Chemistry, 115(19):1342– 1391, 2015

    2015

  28. [36]

    Krumnow, L

    C. Krumnow, L. Veis, ¨O. Legeza, and J. Eisert. Fermionic orbital optimization in tensor network states.Phys. Rev. Lett., 117:210402, Nov 2016

    2016

  29. [37]

    Paredes, F

    B. Paredes, F. Verstraete, and J. I. Cirac. Exploiting quantum parallelism to simulate quantum random many- body systems.Phys. Rev. Lett., 95:140501, Sep 2005

    2005

  30. [38]

    Chandran, J

    A. Chandran, J. Carrasquilla, I. H. Kim, D. A. Abanin, and G. Vidal. Spectral tensor networks for many-body localization.Phys. Rev. B, 92:024201, Jul 2015

    2015

  31. [39]

    Ignacio Cirac, and S

    Frank Pollmann, Vedika Khemani, J. Ignacio Cirac, and S. L. Sondhi. Efficient variational diagonalization of fully many-body localized hamiltonians.Phys. Rev. B, 94:041116, Jul 2016

    2016

  32. [40]

    Verstraete and J

    F. Verstraete and J. I. Cirac. Continuous matrix prod- uct states for quantum fields.Physical Review Letters, 104(19):190405, 2010

    2010

  33. [41]

    Osborne, Henri Verschelde, and Frank Verstraete

    Jutho Haegeman, Tobias J. Osborne, Henri Verschelde, and Frank Verstraete. Entanglement renormalization for quantum fields in real space.Physical Review Letters, 110(10):100402, 2013

    2013

  34. [42]

    Entanglement renormalization and holog- raphy.Physical Review D, 86(6):065007, 2012

    Brian Swingle. Entanglement renormalization and holog- raphy.Physical Review D, 86(6):065007, 2012

    2012

  35. [43]

    Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations.Foundations of Computational Mathematics, 16(6):1423–1472, Dec 2016

    Markus Bachmayr, Reinhold Schneider, and Andr´ e Uschmajew. Tensor networks and hierarchical tensors for the solution of high-dimensional partial differential equations.Foundations of Computational Mathematics, 16(6):1423–1472, Dec 2016

    2016

  36. [44]

    van den Berg, Hessam Babaee, Peyman Givi, Martin Kiffner, and Dieter Jaksch

    Nikita Gourianov, Michael Lubasch, Sergey Dolgov, Quincy Y. van den Berg, Hessam Babaee, Peyman Givi, Martin Kiffner, and Dieter Jaksch. A quantum-inspired approach to exploit turbulence structures.Nature Com- putational Science, 2(1):30–37, 1 2022

    2022

  37. [45]

    Erika Ye and Nuno F. G. Loureiro. Quantum-inspired method for solving the vlasov-poisson equations.Phys. Rev. E, 106:035208, Sep 2022

    2022

  38. [46]

    Multigrid renormalization.Journal of Computational Physics, 372:587–602, 2018

    Michael Lubasch, Pierre Moinier, and Dieter Jaksch. Multigrid renormalization.Journal of Computational Physics, 372:587–602, 2018

    2018

  39. [47]

    Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations.Quantum, 5:431, 4 2021

    Juan Jos´ e Garc ´ ıa-Ripoll. Quantum-inspired algorithms for multivariate analysis: from interpolation to partial differential equations.Quantum, 5:431, 4 2021

    2021

  40. [48]

    Quantum algorithm and circuit design solving the poisson equation.New Journal of Physics, 15(1):013021, 1 2013

    Yudong Cao, Anargyros Papageorgiou, Iasonas Petras, Joseph Traub, and Sabre Kais. Quantum algorithm and circuit design solving the poisson equation.New Journal of Physics, 15(1):013021, 1 2013

    2013

  41. [49]

    Pedro C. S. Costa, Stephen Jordan, and Aaron Ostran- der. Quantum algorithm for simulating the wave equa- tion.Phys. Rev. A, 99:012323, 1 2019

    2019

  42. [50]

    Loureiro

    Abtin Ameri, Erika Ye, Paola Cappellaro, Hari Krovi, and Nuno F. Loureiro. Quantum algorithm for the linear vlasov equation with collisions.Phys. Rev. A, 107:062412, 6 2023

    2023

  43. [51]

    Variational quantum algorithm for the poisson equation.Phys

    Hai-Ling Liu, Yu-Sen Wu, Lin-Chun Wan, Shi-Jie Pan, Su-Juan Qin, Fei Gao, and Qiao-Yan Wen. Variational quantum algorithm for the poisson equation.Phys. Rev. A, 104:022418, 8 2021

    2021

  44. [52]

    Paine, and Vincent E

    Oleksandr Kyriienko, Annie E. Paine, and Vincent E. Elfving. Solving nonlinear differential equations with dif- ferentiable quantum circuits.Phys. Rev. A, 103:052416, 5 2021

    2021

  45. [53]

    Chua, and Charles A

    Stephen Boyd, Leon O. Chua, and Charles A. Desoer. Analytical foundations of Volterra series.IMA Journal of Mathematical Control and Information, 1(3):243–282, 1984

    1984

  46. [54]

    Stephen Boyd and Leon O. Chua. Fading memory and the problem of approximating nonlinear operators with Volterra series.IEEE Transactions on Circuits and Sys- tems, 32(11):1150–1161, 1985

    1985

  47. [55]

    John Wiley & Sons, New York, 1980

    Martin Schetzen.The Volterra and Wiener Theories of Nonlinear Systems. John Wiley & Sons, New York, 1980

    1980

  48. [56]

    North-Holland, Amsterdam, 1986

    Jorge Mujica.Complex Analysis in Banach Spaces: Holo- morphic Functions and Domains of Holomorphy in Finite and Infinite Dimensions, volume 120 ofNorth-Holland Mathematics Studies. North-Holland, Amsterdam, 1986

    1986

  49. [57]

    Springer Monographs in Mathematics

    Se´ an Dineen.Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics. Springer, London, 1999

    1999

  50. [58]

    Dormand and P.J

    J.R. Dormand and P.J. Prince. A family of embedded runge-kutta formulae.Journal of Computational and Ap- plied Mathematics, 6(1):19–26, 1980

    1980

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