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Thermodynamics-Inspired Computing with Oscillatory Neural Networks for Inverse Matrix Computation

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Pith's one-line read Under a small-phase approximation, a coupled Kuramoto oscillator network computes a matrix inverse as the stationary covariance of its phase fluctuations, $A^{-1} = K_n K \Sigma_s$.

desk verdict Known thermodynamic inverse result re-derived through a small-angle Kuramoto linearization; the contribution is the ONN mapping and an honest parameter study, but the central claim is conditional on a small-phase regime that the paper does not fully specify. read the letter →

arxiv 2507.22544 v1 pith:VRKFD636 submitted 2025-07-30 cs.LG cs.ET

George Tsormpatzoglou , Filip Sabo , Aida Todri-Sanial This is my paper
classification cs.LGcs.ET MSC 68T0765F0560H1034C15
keywords oscillatoryneuralnetworksKuramotomodelinversematrixcomputationthermodynamiccomputingLangevindynamicsOrnstein-Uhlenbeckprocessphasecovariancesmall-angleapproximation
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

This paper proposes that a network of coupled oscillators—the kind already studied as Ising machines for combinatorial optimization—can solve linear algebra problems, specifically computing the inverse of a symmetric positive definite matrix. The authors show that when phase differences remain small, the nonlinear Kuramoto dynamics linearize into a quadratic energy landscape whose stationary phase covariance is proportional to the inverse of the encoded matrix. They validate the identity $A^{-1} = K_n K \Sigma_s$ both by sampling the Boltzmann distribution of the oscillator energy and by simulating the stochastic Kuramoto Langevin equation. If the identity holds in hardware, reading out phase covariances from a physical oscillator array would give a noise-harnessing, analog route to matrix inversion.

What carries the argument

The load-bearing object is the linearized Kuramoto energy gradient under the small-phase approximation: $\partial E/\partial \phi_i \approx K \sum_j J_{ij}(\phi_i - \phi_j) + K_{si} \phi_i = K(A\phi)_i$, where $A$ is the encoded matrix. This linearization connects an oscillator network to the quadratic potential $U(x) = \tfrac{1}{2} x^T A x - b^T x$, whose thermal equilibrium is Gaussian with covariance $\beta^{-1} A^{-1}$. The stochastic dynamics are an Ornstein-Uhlenbeck process, and the stationary covariance equation $A\Sigma_s + \Sigma_s A^T = B$ fixes the noise scale $L = \sqrt{2/K_n}\,I$; multiplying the pieces together yields $A^{-1} = K_n K \Sigma_s$.

What would settle it

Run the paper's own stochastic Kuramoto SDE for a fixed $3\times3$ matrix at the reported optimal coupling $K \approx 10^3$ and again at $K$ one order of magnitude smaller, computing $K_n K \Sigma_s$ from the measured phases in both cases. If the relative error against $A^{-1}$ stays small at both couplings, the small-phase condition is not the operative constraint; if the error rises sharply at the smaller $K$ while staying near one percent at the larger one, the identity's validity is bounded exactly by the small-phase approximation.

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Extended reading notes

Core claim

The central claim is the identity $A^{-1} = K_n K \Sigma_s$ (Eq. 20): the matrix to be inverted is encoded as $A$, with off-diagonal entries set by the oscillator coupling strengths $J_{ij}$ and diagonal entries by the harmonic injection strengths $K_{si}$; $K$ and $K_n$ are known constants; and $\Sigma_s$ is the stationary covariance matrix of the oscillator phases. The derivation Taylor-expands $\sin(\phi) \approx \phi$, turning the gradient of the oscillator energy into the linear form $\nabla E \approx K A \phi$, which matches the quadratic potential whose equilibrium distribution has covariance $\beta^{-1} A^{-1}$. The stochastic Kuramoto equation is then an Ornstein-Uhlenbeck process, and its stationary covariance satisfies the Lyapunov equation $A \Sigma_s + \Sigma_s A^T = B$, which fixes the noise scale and closes the identity. Numerical simulations of both the equilibrium Boltzmann distribution and the full SDE dynamics support the identity for tuned $K$ and $K_n$.

Load-bearing premise

The derivation works only when every phase difference is small enough that the sine of an angle can be approximated by the angle itself; once noise or coupling pushes phases away from zero, the linear energy landscape and the covariance identity no longer describe the oscillator network.

Editorial extensions

If this is right

  • A physical oscillator network that reaches equilibrium could emit the inverse matrix directly as a covariance readout, bypassing the $O(d^3)$ arithmetic of digital matrix inversion.
  • The encoding is explicit: coupling strengths set off-diagonal entries, harmonic injection strengths set diagonal entries, so any symmetric positive definite matrix can be loaded into the network.
  • Accuracy is governed by two tunable parameters, $K$ and $K_n$, with the scaling rule $K = 10^3/\text{scale}$ keeping error nearly flat as matrix entries change magnitude.
  • Time-to-solution has two parts, synchronization plus sampling, and physical sampling is limited to one phase measurement per oscillation period, so fast-switching oscillators are a prerequisite for practical speed.

Reading between the lines

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

  • The paper leaves linear-system solving implicit: with a nonzero bias term $b$ in the same quadratic energy, the stationary mean phase would equal $A^{-1} b$, so the same hardware could solve $Ax = b$ and iteratively refine an inverse without redesign.
  • The 1000-random-matrix test, where roughly half of matrices scored below five percent relative error but large errors remained frequent, suggests practical use will need an automatic calibration or feedback loop that keeps phases in the linear regime; the paper does not propose one.
  • In hardware the readout circuitry, not the oscillators, may dominate cost: phase-to-digital conversion and per-period sampling could make the measurement subsystem larger than the network, so the thermodynamic advantage would only materialize if covariance can be accumulated in the analog domain.
  • A testable refinement suggested by the derivation: the identity relies on additive white noise with covariance $(2/K_n)I$; introducing colored or multiplicative noise should break the Lyapunov relation, giving an experimental signature of the regime where the claim operates.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes using coupled Kuramoto oscillators as a thermodynamic-style solver for matrix inversion. The central idea is that if the oscillator phases are small enough to linearize sin(phi) to phi, the ONN gradient reduces to a linear form K A phi, and the stationary covariance of the resulting Ornstein-Uhlenbeck process satisfies A^{-1} = K_n K Sigma_s (Eq. 20). The authors validate this with two numerical approaches: sampling the Boltzmann distribution of the ONN energy and simulating the stochastic Kuramoto SDE. They report error trends as a function of coupling strength K, noise parameter K_n, matrix scale, and matrix size, and they run a 1000-matrix random survey whose error distribution has a heavy tail.

Significance. If the central identity were valid in the claimed operating regime, the paper would provide a physically plausible route for analog oscillator hardware to solve a core linear algebra task, and the derivation connecting the Kuramoto Lyapunov function to the OU covariance is elegant and mostly correct. The paper also deserves credit for testing the method against random matrices and for explicitly discussing measurement and sampling bottlenecks. However, the validity of the identity is conditional on a small-phase, symmetric-positive-definite regime that the paper neither ensures analytically nor controls in experiments; the paper's own random-matrix sweep shows about half of trials above 5% error and a heavy tail of extreme outliers, which the authors themselves call 'unreliable.' The significance is therefore real but strictly conditional, and the manuscript currently overstates its generality.

major comments (4)
  1. [Section III.A, Eqs. (15)-(20)] The derivation assumes not only small phases but also that A is symmetric positive definite. As defined in the displayed matrix after Eq. (17), A is symmetric only if J_ij = J_ji, and SPD is neither guaranteed nor stated as an assumption for the inverse-matrix claim. Without SPD, the stationary solution to the Lyapunov equation (14) is not guaranteed to equal (K_n K)^{-1} A^{-1}, and the Gaussian form of Eq. (18) is unjustified. The paper should state these conditions explicitly and verify that the matrices used in the experiments satisfy them.
  2. [Section IV.B, Eq. (30)] The proposed scaling K = 10^3 / scale depends only on the element magnitude of A and does not control the phase variance along soft eigendirections. From Eqs. (19)-(20), the variance of the projection v^T phi in an eigendirection v of A is approximately (v^T A^{-1} v) / (K_n K), so keeping all phases in the linear regime requires K_n K lambda_min(A) >> 1. Eq. (30) ignores lambda_min, so ill-conditioned matrices violate the small-phase condition and land in the large-error tail of Fig. 6. This is not a numerical artifact but a failure of the parameter-selection rule in the exact regime the theorem requires.
  3. [Section IV.B, Fig. 6] The random-matrix experiment is the relevant test of the general claim, and the paper reports that roughly 50% of trials have less than 5% error and that large errors are frequent enough for the model to be considered unreliable. The abstract and conclusions state without qualification that ONNs compute the inverse matrix, but a method with approximately a 50% failure rate on random SPD-sized matrices is at best conditionally valid. The manuscript should reframe the claim as holding only when a condition-number-aware parameter rule keeps all phases in the linear regime, or it should provide a quantitative method to detect or avoid the unreliable regime.
  4. [Section III.B, Eq. (21)] The energy-function approach samples exp(-K_n E(phi)) with the nonlinear ONN energy of Eq. (3), while Eq. (20) is derived only from the linearized quadratic energy. For larger phase excursions, the Boltzmann covariance of the nonlinear energy will not satisfy A^{-1} = K_n K Sigma_s. The paper validates the energy-function method only on a 2-oscillator example and does not address this mismatch; either the nonlinearity should be included in the error analysis or the claim should be restricted to the linearized regime.
minor comments (6)
  1. [Eq. (23)] The relative error is written as an elementwise division of matrices, which is not well defined when entries of A^{-1} are zero; please define the error using a matrix norm or specify an elementwise convention.
  2. [Section III.A, Eq. (17)] The notation A for the matrix being inverted is overloaded with the linear algebra notation A in Eqs. (5)-(9); consider using a distinct symbol such as M for the encoded matrix to avoid confusion.
  3. [Section IV.B, Fig. 6] The caption states that values exceeding 120% are grouped into the final bin, but the text reports 'errors above 120%' for K_n = 20; please clarify whether the final bin includes all larger errors and how the reported percentages treat those outliers.
  4. [Section II.B, Eqs. (10)-(12)] The relationship between B, L, and the SDE noise is written in a compressed way; explicitly stating B = L L^T and the meaning of L in Eq. (28) would improve reproducibility.
  5. [Introduction] The paper cites thermodynamic linear algebra work [4, 17] but does not clearly delineate what new capability the Kuramoto-oscillator implementation adds relative to those existing thermodynamic solvers; a brief comparison would help position the contribution.
  6. [General] The manuscript does not state code or data availability; given the stochastic nature of the simulations and the sensitivity to K and K_n, providing the random-matrix generation procedure and solver settings would materially aid replication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inverse-matrix result follows from an explicitly encoded matrix A and the standard Ornstein-Uhlenbeck covariance identity, not from assuming the target.

full rationale

The derivation is self-contained and does not reduce to its inputs by construction. The paper explicitly constructs the matrix A from the coupling coefficients J_ij and injection terms K_si (Eqs. 15-17 and the displayed 3x3 form), so the energy gradient is engineered to be K A phi under the stated small-phase approximation. The target result A^{-1} = K_n K Sigma_s is then obtained by applying the standard quadratic Boltzmann / Ornstein-Uhlenbeck covariance relation, which is an independent mathematical identity rather than a restatement of the target. The small-phase linearization is openly declared rather than smuggled in through a citation, and the dynamics-based validation uses the full nonlinear Kuramoto SDE (Eq. 29), providing an independent numerical check of the linearized prediction. Self-citations in the paper are background references to prior ONN work and are not load-bearing for the inverse-matrix claim. The empirical scaling rule K = 10^3/scale in Eq. (30) is an accuracy heuristic, not a fitted parameter that is later renamed as a prediction. The acknowledged sensitivity to the small-phase assumption and the large-error tail in Fig. 6 is a correctness and robustness limitation, not a circularity in the derivation.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central result combines two standard blocks: the OU/Boltzmann inverse-covariance identity (Eqs. 8-14, from refs [4,17]) and a linearization of the Kuramoto energy (Eq. 16). The only paper-specific input is the mapping A_ij = -J_ij for off-diagonal and A_ii = sum J + Ks/K for diagonal. The free parameters are K, Kn, and Ns, all tuned by hand. No new physical entities are introduced.

free parameters (3)
  • K (overall coupling strength) = Tuned per matrix; heuristic K = 10^3/scale (Eq. 30)
    Chosen by hand to make phase differences small so the linear approximation holds; not derived from first principles. Figures 3-5 show the error strongly depends on K.
  • Kn (generalized noise) = Values from 1 to 1e12 tested; optimal around 1e4 for 3x3 random matrices
    Controls noise level and phase magnitude. In the exact Gaussian limit the result is independent of Kn, but finite-size and nonlinear effects make accuracy Kn-dependent, so the authors tune it.
  • Ns (number of SDE steps/samples) = 5e7 for 3x3, 1e8 for 10x10, 5e7 for 20x20, 5e5 for random tests
    Large step counts are required for convergence; chosen by the authors to balance accuracy and runtime. This numerical parameter affects reproducibility and the reported errors.
assumptions (6)
  • domain assumption Kuramoto model with symmetric coupling and harmonic injection provides a valid model of oscillatory neural networks.
    The paper relies on Eqs. 1-3 from refs [8,34,35] without deriving the model from a physical circuit; the reality of ONN hardware is assumed.
  • standard math The coupled oscillator dynamics is a gradient flow with a global Lyapunov function given by Eq. 3.
    This is a standard result for sinusoidally coupled Kuramoto with symmetric J, but the symmetry requirement is not explicitly stated in the paper.
  • standard math Additive white noise leads to a stationary Boltzmann distribution f(phi) proportional to exp(-Kn E(phi)).
    Known result for gradient SDEs with isotropic noise; used in Eqs. 21 and 29.
  • ad hoc to paper Small-phase linearization sin(phi) approximately phi is valid for all oscillators.
    Central approximation at Eq. 16; the paper tunes K and Kn to force it to hold, but provides no a priori error bound.
  • domain assumption The target matrix A is symmetric positive definite and the coupling matrix J is symmetric.
    Background Eq. 5 assumes an SPD quadratic energy; the mapping in Eqs. 17-19 requires A to be symmetric and positive definite for the covariance to exist, but these conditions are not stated in the methods.
  • domain assumption The system reaches equilibrium within the simulation time.
    Covariance is measured from stationary OU solutions; convergence is assumed, and large Ns values are used to approximate it.

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Pith. "Pith review of Thermodynamics-Inspired Computing with Oscillatory Neural Networks for Inverse Matrix Computation." pith.science (2026). https://pith.science/paper/VRKFD636

@misc{pith2026250722544,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics-Inspired Computing with Oscillatory Neural Networks for Inverse Matrix Computation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VRKFD636}},
  note         = {Machine review of arXiv:2507.22544}
}
read the original abstract

We describe a thermodynamic-inspired computing paradigm based on oscillatory neural networks (ONNs). While ONNs have been widely studied as Ising machines for tackling complex combinatorial optimization problems, this work investigates their feasibility in solving linear algebra problems, specifically the inverse matrix. Grounded in thermodynamic principles, we analytically demonstrate that the linear approximation of the coupled Kuramoto oscillator model leads to the inverse matrix solution. Numerical simulations validate the theoretical framework, and we examine the parameter regimes that computation has the highest accuracy.

Figures

Figures reproduced from arXiv: 2507.22544 by the authors.

Figure 1
Figure 1. Illustration of the analytical approach for computing [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Boltzmann probability landscape along with Langevin dynamics for a two-coupled oscillator system visualized in a) [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Relative error between computed and actual inverse [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Average relative error as a function of the overall [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: Average relative error with respect to a) the magnitude scale of matrix [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Average relative error distribution of 1000 random 3 [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

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Works this paper leans on

39 extracted references · 37 canonical work pages

  1. [1]

    Linear algebra as the mathematical founda- tion of artificial intelligence: concepts, applications, and future prospects,

    M. K. Rani, “Linear algebra as the mathematical founda- tion of artificial intelligence: concepts, applications, and future prospects,” International Journal of Engineering Science and Humanities , vol. 14, no. Special Issue 1, pp. 123–137, 2024

    work page 2024

  2. [2]

    An introduction to convolu- tional neural networks,

    K. O’Shea and R. Nash, “An introduction to convolu- tional neural networks,” 2015

    work page 2015

  3. [3]

    Linear regression,

    X. Su, X. Yan, and C.-L. Tsai, “Linear regression,” Wi- ley Interdisciplinary Reviews: Computational Statistics , vol. 4, no. 3, pp. 275–294, 2012

    work page 2012

  4. [4]

    Thermodynamic Linear Algebra

    M. Aifer, K. Donatella, M. H. Gordon, S. Duffield, T. Ahle, D. Simpson, G. E. Crooks, and P. J. Coles, “Thermodynamic linear algebra,” 2024. [Online]. Available: https://arxiv.org/abs/2308.05660

    work page Pith review arXiv 2024

  5. [5]

    First draft of a report on the edvac,

    J. von Neumann, “First draft of a report on the edvac,” IEEE Annals of the History of Computing , vol. 15, no. 4, pp. 27–75, 1993

    work page 1993

  6. [6]

    A survey of different approaches for overcoming the processor- memory bottleneck,

    D. Efnusheva, A. Cholakoska, and A. Tentov, “A survey of different approaches for overcoming the processor- memory bottleneck,” International Journal of Computer Science and Information Technology , vol. 9, no. 2, pp. 151–163, 2017

    work page 2017

  7. [7]

    The end of moore’s law: A new beginning for information technology,

    T. N. Theis and H.-S. P. Wong, “The end of moore’s law: A new beginning for information technology,” Comput- ing in science & engineering , vol. 19, no. 2, pp. 41–50, 2017

    work page 2017

  8. [8]

    Computing with oscillators from theoretical underpin- nings to applications and demonstrators,

    A. Todri-Sanial, C. Delacour, M. Abernot, and F. Sabo, “Computing with oscillators from theoretical underpin- nings to applications and demonstrators,” npj Unconven- tional Computing , vol. 1, no. 1, p. 14, Dec. 2024

    work page 2024

Show all 39 references
  1. [9]

    A vision to com- pute like nature: Thermodynamically,

    T. Hylton, T. M. Conte, and M. D. Hill, “A vision to com- pute like nature: Thermodynamically,” Communications of the ACM , vol. 64, no. 6, pp. 35–38, 2021

    2021

  2. [10]

    Opportunities for neuromorphic computing algorithms and applications,

    C. D. Schuman, S. R. Kulkarni, M. Parsa, J. P. Mitchell, P. Date, and B. Kay, “Opportunities for neuromorphic computing algorithms and applications,” Nature Compu- tational Science , vol. 2, no. 1, pp. 10–19, 2022

    2022

  3. [11]

    A full-stack view of proba- bilistic computing with p-bits: Devices, architectures, and algorithms,

    S. Chowdhury, A. Grimaldi, N. A. Aadit, S. Niazi, M. Mohseni, S. Kanai, H. Ohno, S. Fukami, L. Theog- arajan, G. Finocchio et al. , “A full-stack view of proba- bilistic computing with p-bits: Devices, architectures, and algorithms,” IEEE Journal on Exploratory Solid-State Com...

    2023

  4. [12]

    Adiabatic computing: A contemporary re- view,

    S. Samanta, “Adiabatic computing: A contemporary re- view,” in2009 4th international conference on computers and devices for communication (CODEC) . IEEE, 2009, pp. 1–4

    2009

  5. [13]

    Quantum computing review: A decade of research,

    S. K. Sood et al. , “Quantum computing review: A decade of research,” IEEE Transactions on Engineering Management, vol. 71, pp. 6662–6676, 2023

    2023

  6. [14]

    Thermodynamic computing: an intellectual and technological frontier,

    T. Hylton, “Thermodynamic computing: an intellectual and technological frontier,” inProceedings, vol. 47, no. 1. MDPI, 2020, p. 23

    2020

  7. [15]

    Thermodynamic com- puting,

    T. Conte, E. DeBenedictis, N. Ganesh, T. Hylton, J. P. Strachan, R. S. Williams, A. Alemi, L. Altenberg, G. Crooks, J. Crutchfield et al. , “Thermodynamic com- puting,” arXiv preprint arXiv:1911.01968 , 2019

    1911 arXiv

  8. [16]

    Thermodynamic neural network,

    T. Hylton, “Thermodynamic neural network,” Entropy, vol. 22, no. 3, p. 256, 2020

    2020

  9. [17]

    Thermodynamic computing system for AI applications,

    D. Melanson, M. A. Khater, M. Aifer, K. Donatella, M. H. Gordon, T. Ahle, G. Crooks, A. J. Martinez, F. Sbahi, and P. J. Coles, “Thermodynamic computing system for AI applications,” Nature Communications , vol. 16, no. 1, p. 3757, 2025

    2025

  10. [18]

    Applications of electrical noise,

    M.-S. Gupta, “Applications of electrical noise,” Proceed- ings of the IEEE , vol. 63, no. 7, pp. 996–1010, 2005

    2005

  11. [19]

    Thermodynamic ai and the fluctuation frontier,

    P. J. Coles, C. Szczepanski, D. Melanson, K. Donatella, A. J. Martinez, and F. Sbahi, “Thermodynamic ai and the fluctuation frontier,” 2023. [Online]. Available: https://arxiv.org/abs/2302.06584

    2023 arXiv

  12. [20]

    Pattern recognition via synchronization in phase-locked loop neural net- works,

    F. Hoppensteadt and E. Izhikevich, “Pattern recognition via synchronization in phase-locked loop neural net- works,” IEEE Transactions on Neural Networks , vol. 11, no. 3, pp. 734–738, 2000

    2000

  13. [21]

    Oim: Oscillator-based ising machines for solving combinatorial optimisation problems,

    T. Wang and J. Roychowdhury, “Oim: Oscillator-based ising machines for solving combinatorial optimisation problems,” in International Conference on Unconven- tional Computation and Natural Computation . Springer International Publishing Cham, 2019, pp. 232–256

    2019

  14. [22]

    How frequency injection locking can train oscillatory neural networks to compute in phase,

    A. Todri-Sanial, S. Carapezzi, C. Delacour, M. Abernot, T. Gil, E. Corti, S. F. Karg, J. N ¨u˜nez, M. Jim ´en`ez, M. J. Avedillo, and B. Linares-Barranco, “How frequency injection locking can train oscillatory neural networks to compute in phase,” IEEE transactions on neural n...

    1996

  15. [23]

    Digital implementation of oscillatory neural network for image recognition applications,

    M. Abernot, T. Gil, M. Jim ´enez, J. N ´u˜nez, M. J. Avel- lido, B. Linares-Barranco, T. Gonos, T. Hardelin, and A. Todri-Sanial, “Digital implementation of oscillatory neural network for image recognition applications,”Fron- tiers in Neuroscience , vol. 15, p. 713054, 2021. 9

    2021

  16. [24]

    Training energy-based single-layer hopfield and oscillatory networks with un- supervised and supervised algorithms for image classi- fication,

    M. Abernot and A. Todri-Sanial, “Training energy-based single-layer hopfield and oscillatory networks with un- supervised and supervised algorithms for image classi- fication,” Neural Computing and Applications , vol. 35, no. 25, pp. 18 505–18 518, 2023

    2023

  17. [25]

    Digital implementation of on-chip hebbian learn- ing for oscillatory neural network,

    E. Luhulima, M. Abernot, F. Corradi, and A. Todri- Sanial, “Digital implementation of on-chip hebbian learn- ing for oscillatory neural network,” in 2023 IEEE/ACM International Symposium on Low Power Electronics and Design (ISLPED) . IEEE, 2023, pp. 1–6

    2023

  18. [26]

    Classonn: Classification with oscillatory neural networks using the kuramoto model,

    F. Sabo and A. Todri-Sanial, “Classonn: Classification with oscillatory neural networks using the kuramoto model,” in 2024 Design, Automation & Test in Europe Conference & Exhibition (DATE) . IEEE, 2024, pp. 1–2

    2024

  19. [27]

    Ising ma- chines as hardware solvers of combinatorial optimization problems,

    N. Mohseni, P. L. McMahon, and T. Byrnes, “Ising ma- chines as hardware solvers of combinatorial optimization problems,” Nature Reviews Physics , vol. 4, no. 6, pp. 363–379, 2022

    2022

  20. [28]

    Efficient optimization with higher-order ising machines,

    C. Bybee, D. Kleyko, D. E. Nikonov, A. Khosrow- shahi, B. A. Olshausen, and F. T. Sommer, “Efficient optimization with higher-order ising machines,” Nature Communications, vol. 14, no. 1, p. 6033, 2023

    2023

  21. [29]

    Novel computing paradigms using oscillators,

    T. Wang, “Novel computing paradigms using oscillators,” Ph.D. dissertation, EECS Department, University of California, Berkeley, Jan 2020. [Online]. Available: http://www2.eecs.berkeley.edu/Pubs/TechRpts/2020/EECS- 2020-12.html

    2020

  22. [30]

    Energy-performance assessment of oscillatory neural networks based on vo 2 devices for future edge ai computing,

    C. Delacour, S. Carapezzi, M. Abernot, and A. Todri- Sanial, “Energy-performance assessment of oscillatory neural networks based on vo 2 devices for future edge ai computing,” IEEE Transactions on Neural Networks and Learning Systems , 2023

    2023

  23. [31]

    Oscillatory neural networks for edge ai computing,

    C. Delacour, S. Carapezzi, M. Abernot, G. Boschetto, N. Azemard, J. Salles, T. Gil, and A. Todri-Sanial, “Oscillatory neural networks for edge ai computing,” in 2021 IEEE computer society annual symposium on VLSI (ISVLSI). IEEE, 2021, pp. 326–331

    2021

  24. [32]

    Mapping hebbian learning rules to coupling resistances for oscillatory neural networks,

    C. Delacour and A. Todri-Sanial, “Mapping hebbian learning rules to coupling resistances for oscillatory neural networks,” Frontiers in neuroscience , vol. 15, p. 694549, 2021

    2021

  25. [33]

    A mixed-signal oscillatory neural network for scalable analog computa- tions in phase domain,

    C. Delacour, S. Carapezzi, G. Boschetto, M. Abernot, T. Gil, N. Azemard, and A. Todri-Sanial, “A mixed-signal oscillatory neural network for scalable analog computa- tions in phase domain,” Neuromorphic Computing and Engineering, vol. 3, no. 3, p. 034004, 2023

    2023

  26. [34]

    Kuramoto, Chemical turbulence

    Y . Kuramoto, Chemical turbulence . Springer, 1984

    1984

  27. [35]

    Ising machines: Theory and practice,

    N. Sagan and J. Roychowdhury, “Ising machines: Theory and practice,” Master’s thesis, EECS Department, University of California, Berkeley, May 2023. [Online]. Available: http://www2.eecs.berkeley.edu/Pubs/TechRpts/2023/EECS- 2023-138.html

    2023

  28. [36]

    Sur la th ´eorie du mouvement brown- ien,

    P. Langevin et al., “Sur la th ´eorie du mouvement brown- ien,” CR Acad. Sci. Paris , vol. 146, no. 530-533, p. 530, 1908

    1908

  29. [37]

    C. W. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences , ser. Springer Series in Synergetics. Springer, 1985

    1985

  30. [38]

    Nonequilibrium gaussian distribution for coupled linear langevin equations,

    H. Sakaguchi, “Nonequilibrium gaussian distribution for coupled linear langevin equations,” arXiv preprint arXiv:2503.21220, 2025

    2025 arXiv

  31. [39]

    Increasing the clock speed of a thermodynamic computer by adding noise,

    S. Whitelam, “Increasing the clock speed of a thermodynamic computer by adding noise,” 2025. [Online]. Available: https://arxiv.org/abs/2410.12211

    2025

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