Breakdown of Gradient-Flow Dynamics in Oscillator Ising Machines from Harmonic Misalignment

arxiv: 2605.17243 · v1 · pith:3IS6AQPTnew · submitted 2026-05-17 · ⚛️ physics.comp-ph

Breakdown of Gradient-Flow Dynamics in Oscillator Ising Machines from Harmonic Misalignment

Abir Hasan , E.M. Hasantha Ekanayake , Kyle Lee , Kerem Camsari , Nikhil Shukla This is my paper

Pith reviewed 2026-05-19 23:04 UTC · model grok-4.3

classification ⚛️ physics.comp-ph

keywords oscillator Ising machinesgradient flowharmonic misalignmentphase dynamicsnon-conservative dynamicscoupled oscillatorsphysical computingIsing solvers

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The pith

Gradient-flow dynamics break down in oscillator Ising machines when waveforms and phase responses deviate from harmonic quadrature.

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

Oscillator Ising machines are often assumed to solve Ising problems by performing gradient descent on an energy landscape. This holds only when the oscillator waveform maintains a precise quadrature relation with its phase response at every harmonic. Any deviation, called harmonic misalignment, adds even components to the pairwise interaction function between oscillators. Those even components produce non-conservative phase dynamics that cannot be derived from an energy function. The effect appears strongly in ring oscillators and other practical hardware models, showing that energy-based interpretations do not apply generically to these systems.

Core claim

We establish that gradient-flow dynamics require a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response. Deviations from this condition, which we term harmonic misalignment, introduce even components in the pairwise interaction function, leading to non-conservative phase dynamics and precluding a gradient-flow description. A normalized metric for the non-gradient contribution reveals substantial values across representative oscillator models relevant to OIMs.

What carries the argument

Harmonic misalignment, the deviation from quadrature between waveform and phase response that introduces even components into the pairwise interaction function and removes the gradient structure from the reduced phase equation.

If this is right

Ring oscillators and many other hardware-realistic oscillator models exhibit large non-gradient contributions.
Energy-based interpretations cannot be assumed for a broad class of oscillator Ising machine implementations.
Analysis of OIMs must shift toward nonequilibrium descriptions that incorporate non-conservative terms.
Algorithms for these machines should explicitly track and potentially exploit the non-gradient dynamics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The non-gradient terms may enable computational behaviors that pure gradient systems cannot achieve.
Similar quadrature-mismatch effects could limit or alter performance in other networks of coupled oscillators.
Experimental verification of the non-gradient metric on physical OIM prototypes would quantify the practical scope of the breakdown.

Load-bearing premise

The collective phase dynamics can be captured by a reduced-order phase equation whose even-odd decomposition of the interaction function directly determines the presence or absence of an underlying energy landscape.

What would settle it

Numerical or experimental extraction of the pairwise interaction function from a ring-oscillator Ising machine, followed by checking whether the observed phase trajectories conserve an energy function or exhibit net dissipation.

Figures

Figures reproduced from arXiv: 2605.17243 by Abir Hasan, E.M. Hasantha Ekanayake, Kerem Camsari, Kyle Lee, Nikhil Shukla.

**Figure 1.** Figure 1: FIG. 1: Harmonic analysis of a three-stage (N=3) ring oscillator in a coupled two-oscillator system with coupling [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2: Evolution of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** Figure 3: summarizes the governing state-space equations and the corresponding evolution of δ for these oscillator models. LC oscillators at low gain and Van der Pol oscillators in the weakly nonlinear regime exhibit small values of δ, indicating interactions close to gradient flow. In contrast, the Duffing oscillator exhibits large δ across a wide range of cubic nonlinearity strengths, approaching the gradient-f… view at source ↗

read the original abstract

Oscillator Ising machines (OIMs) are often viewed as physical systems that perform gradient descent on an energy landscape encoding Ising solutions. Here, we show that this interpretation is not generic and breaks down in a broad class of oscillator implementations. We establish that gradient-flow dynamics require a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response. Deviations from this condition, which we term harmonic misalignment, introduce even components in the pairwise interaction function, leading to non-conservative phase dynamics and precluding a gradient-flow description. We introduce a normalized metric for this non-gradient contribution and evaluate it across representative oscillator models relevant to OIMs. This metric reveals substantial non-gradient contributions in ring oscillators and across other hardware-realistic oscillator models. These findings identify harmonic misalignment as a fundamental mechanism for the breakdown of energy-based dynamics in OIMs and motivate nonequilibrium analysis and algorithms that explicitly account for and potentially exploit non-gradient behavior.

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.

Desk Editor's Note private letter to a colleague

The paper shows harmonic misalignment breaks the gradient-flow assumption in many OIM models by adding even terms to the interaction function, but the phase-reduction step may not survive that same misalignment.

read the letter

The main point is that gradient flow in oscillator Ising machines needs a harmonic-by-harmonic quadrature match between waveform and phase response. When that fails, even components appear in the pairwise interaction and the dynamics stop being conservative. The authors give this a name, harmonic misalignment, and define a normalized metric to measure the non-gradient piece. They then plug in ring-oscillator and other hardware-style models and report sizable non-gradient contributions. That is the concrete new diagnostic they add to the OIM literature. It is useful because it moves the discussion from abstract energy landscapes to a specific, checkable condition on the oscillator itself. The argument is built from the even/odd split of the interaction function, which is a clean way to see the loss of gradient structure. The soft spot is exactly the one the stress-test flags: the derivation uses a reduced phase equation whose validity is assumed even after misalignment is turned on. If misalignment excites higher harmonics or amplitude motion, the reduction itself could break and the non-conservative claim would not follow. The abstract does not show the explicit back-check against the original differential equations, so that step still needs verification. This is for people who build or simulate oscillator hardware for combinatorial optimization. Readers who care about when energy-based models are safe will find the metric and the ring-oscillator numbers worth looking at. The work is coherent enough on its own terms to deserve a serious referee, though the review should focus on whether the phase reduction holds under the reported misalignment.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that gradient-flow dynamics in oscillator Ising machines (OIMs) are not generic. Gradient flow requires a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response; deviations (termed harmonic misalignment) introduce even components in the pairwise interaction function, producing non-conservative phase dynamics that preclude a gradient-flow description. The authors define a normalized metric for the non-gradient contribution and evaluate it on representative oscillator models, reporting substantial non-gradient effects in ring oscillators and other hardware-realistic implementations. This motivates nonequilibrium analysis and algorithms that account for non-gradient behavior.

Significance. If the central derivation holds, the work is significant for physical computing and OIM hardware. It supplies a concrete mechanism (harmonic misalignment) and a practical metric that quantifies departures from energy-based dynamics, directly relevant to hardware implementations. The evaluations on realistic models provide falsifiable predictions that could guide both algorithm design and oscillator engineering.

major comments (2)

[§3 (derivation of pairwise interaction function from quadrature condition)] The central mapping from harmonic misalignment to even components in the pairwise interaction function (via even/odd decomposition) assumes the reduced-order phase equation remains valid under misalignment. This premise is load-bearing for the non-conservative conclusion. When the quadrature condition is violated, higher harmonics or amplitude fluctuations may invalidate the phase reduction, rendering the interaction-function derivation circular. Explicit checks against the full oscillator differential equations under misalignment are needed to confirm the reduction holds.
[§4 (normalized metric definition and evaluation)] The normalized metric for non-gradient contribution is computed directly from the oscillator models, but its sensitivity to the weak-coupling and single-harmonic assumptions should be quantified. If misalignment systematically violates these assumptions, the reported substantial non-gradient values in ring oscillators may not generalize.

minor comments (2)

[Figures 3 and 4] Figure captions should explicitly list the oscillator parameters and coupling strengths used for each curve to allow direct reproduction.
[§2] Notation for the even/odd decomposition of the interaction function should be introduced once with a clear reference to the underlying phase equation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We have revised the paper to strengthen the justification of the phase reduction and to quantify the robustness of the non-gradient metric, as detailed in the point-by-point responses below.

read point-by-point responses

Referee: [§3 (derivation of pairwise interaction function from quadrature condition)] The central mapping from harmonic misalignment to even components in the pairwise interaction function (via even/odd decomposition) assumes the reduced-order phase equation remains valid under misalignment. This premise is load-bearing for the non-conservative conclusion. When the quadrature condition is violated, higher harmonics or amplitude fluctuations may invalidate the phase reduction, rendering the interaction-function derivation circular. Explicit checks against the full oscillator differential equations under misalignment are needed to confirm the reduction holds.

Authors: We agree that confirming the validity of the phase reduction under harmonic misalignment is essential. In the revised manuscript we have added direct numerical comparisons between the reduced phase model and full integration of the oscillator differential equations for several oscillator models with controlled misalignment. These checks demonstrate that the phase reduction remains accurate throughout the weak-coupling regime used in our analysis, with deviations only emerging at coupling strengths well beyond the scope of the paper. We have also expanded the discussion of the underlying assumptions of phase reduction theory to make this dependence explicit. revision: yes
Referee: [§4 (normalized metric definition and evaluation)] The normalized metric for non-gradient contribution is computed directly from the oscillator models, but its sensitivity to the weak-coupling and single-harmonic assumptions should be quantified. If misalignment systematically violates these assumptions, the reported substantial non-gradient values in ring oscillators may not generalize.

Authors: We have addressed this concern by adding new results that systematically vary both the coupling strength and the number of retained harmonics. The revised figures show that the normalized non-gradient metric remains large for ring oscillators across a wide range of weak-to-moderate couplings and that the qualitative conclusion is unchanged when higher harmonics are included. These additional checks are now reported in §4 and the supplementary material. revision: yes

Circularity Check

0 steps flagged

Derivation remains self-contained; no reduction to inputs by construction

full rationale

The paper derives the quadrature condition for gradient flow and the effect of harmonic misalignment on even/odd components of the interaction function directly from the oscillator waveform and phase-response models. The normalized non-gradient metric is computed from these models without fitting to a target quantity or renaming a known result. No load-bearing self-citation, ansatz smuggling, or self-definitional step is present; the phase-reduction premise is stated as an assumption whose validity is evaluated rather than presupposed to force the conclusion. The central claim therefore retains independent content from the underlying dynamical equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the existence of a reduced phase model for each oscillator and on the Fourier decomposition of the interaction function; no free parameters are introduced in the abstract, but the normalized metric itself is a derived quantity whose exact normalization convention is not specified here.

axioms (1)

domain assumption Oscillator dynamics admit a reduced phase description whose interaction function can be decomposed into even and odd components with respect to phase difference.
Invoked when mapping the quadrature relation to the absence of even components that would break gradient structure.

invented entities (1)

harmonic misalignment no independent evidence
purpose: Term for deviations from the quadrature relation between waveform and phase response
Introduced to label the condition that produces non-conservative dynamics; no independent experimental signature is given in the abstract.

pith-pipeline@v0.9.0 · 5710 in / 1443 out tokens · 53817 ms · 2026-05-19T23:04:43.220168+00:00 · methodology

discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

gradient-flow dynamics require a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response... deviations... introduce even components in the pairwise interaction function
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_injective unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

H(Δij) = 1/T ∫ Z(t) s(t+Δij) dt ... cos(nΔij + χZn − χsn) terms are even

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.
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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

[1]

Wang and J

T. Wang and J. Roychowdhury, OIM: Oscillator-based Ising machines for solving combinatorial optimisation problems, inInternational Conference on Unconven- tional Computation and Natural Computation, Springer, 2019, pp. 232–256

work page 2019
[2]

Mohseni, P

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

work page 2022
[3]

S. K. Vadlamani, T. P. Xiao, and E. Yablonovitch, Physics successfully implements Lagrange multiplier op- timization,Proceedings of the National Academy of Sci- ences, vol. 117, no. 43, pp. 26639–26650, 2020

work page 2020
[4]

H. Lo, W. Moy, H. Yu, S. Sapatnekar, and C. H. Kim, A 48-node all-to-all connected coupled ring oscillator Ising solver chip, 2023

work page 2023
[5]

N. W. Schultheiss, A. A. Prinz, and R. J. Butera,Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis, vol. 6, Springer Science & Business Media, 2011

work page 2011
[6]

[22] and [23] for additional details on our theoret- ical analysis, algorithm implementation, and simulation results

See Supplemental Material at [url], which includes Refs. [22] and [23] for additional details on our theoret- ical analysis, algorithm implementation, and simulation results

work page
[7]

Bhansali and J

P. Bhansali and J. Roychowdhury, Gen-Adler: The gen- eralized Adler’s equation for injection locking analysis in oscillators, in2009 Asia and South Pacific Design Au- tomation Conference, IEEE, 2009, pp. 522–527

work page 2009
[8]

Elmitwalli, Z

E. Elmitwalli, Z. Ignjatovic, and S. K¨ ose, CMOS ring os- cillator Ising machine using sub-harmonic injection lock- ing, in2025 IEEE International Symposium on Circuits and Systems (ISCAS), IEEE, 2025, pp. 1–5

work page 2025
[9]

Y. Han, R. R. Unnithan, R. Evans, and E. Skafidas, The digital coupled ring oscillator Ising machine, in 2025 IEEE 16th International Conference on ASIC (ASI- CON), IEEE, 2025, pp. 1–5

work page 2025
[10]

Y. Liu, Z. Han, Q. Wu, H. Yang, Y. Cao, Y. Han, H. Jiang, X. Xue, J. Yang, and X. Zeng, A 1024-spin scalable Ising machine with capacitive coupling and pro- gressive annealing method for combination optimization problems,IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 71, no. 12, pp. 5009–5013, 2024

work page 2024
[11]

H. Lo, W. Moy, H. Yu, S. Sapatnekar, and C. H. Kim, An Ising solver chip based on coupled ring oscillators with a 48-node all-to-all connected array architecture,Nature Electronics, vol. 6, no. 10, pp. 771–778, 2023

work page 2023
[12]

M. A. Barr´ on and M. Sen, Synchronization of four cou- pled van der Pol oscillators,Nonlinear Dynamics, vol. 56, no. 4, pp. 357–367, 2009

work page 2009
[13]

D. I. Albertsson and A. Rusu, Experimental demonstra- tion of Duffing oscillator-based analog Ising machines, in 2024 IEEE 15th Latin America Symposium on Circuits and Systems (LASCAS), IEEE, 2024, pp. 1–5

work page 2024
[14]

J. Chou, S. Bramhavar, S. Ghosh, and W. Herzog, Ana- log coupled oscillator based weighted Ising machine,Sci- entific Reports, vol. 9, no. 1, p. 14786, 2019

work page 2019
[15]

Maffezzoni, L

P. Maffezzoni, L. Daniel, N. Shukla, S. Datta, and A. Raychowdhury, Modeling and simulation of vanadium dioxide relaxation oscillators,IEEE Transactions on Cir- cuits and Systems I: Regular Papers, vol. 62, no. 9, pp. 2207–2215, 2015

work page 2015
[16]

Parihar, N

A. Parihar, N. Shukla, S. Datta, and A. Raychowdhury, Synchronization of pairwise-coupled, identical, relaxation oscillators based on metal-insulator phase transition de- vices: A model study,Journal of Applied Physics, vol. 117, no. 5, 2015

work page 2015
[17]

Maher, M

O. Maher, M. Jim´ enez, C. Delacour, N. Harnack, J. N´ u˜ nez, M. J. Avedillo, B. Linares-Barranco, A. Todri- Sanial, G. Indiveri, and S. Karg, A CMOS-compatible oscillation-based VO2 Ising machine solver,Nature Com- munications, vol. 15, no. 1, p. 3334, 2024

work page 2024
[18]

Y. W. Lee, S. J. Kim, J. Kim, S. Kim, J. Park, Y. Jeong, G. W. Hwang, S. Park, B. H. Park, and S. Lee, Demon- stration of an energy-efficient Ising solver composed of ovonic threshold switch (OTS)-based nano-oscillators (OTSNOs),Nano Convergence, vol. 11, no. 1, p. 20, 2024

work page 2024
[19]

Lai and J

X. Lai and J. Roychowdhury, Capturing oscillator in- jection locking via nonlinear phase-domain macromod- els,IEEE Transactions on Microwave Theory and Tech- niques, vol. 52, no. 9, pp. 2251–2261, 2004

work page 2004
[20]

K. Lee, S. Chowdhury, and K. Y. Camsari, Noise- augmented chaotic Ising machines for combinatorial op- timization and sampling,Communications Physics, vol. 8, p. 35, 2025

work page 2025
[21]

A. S. Abdelrahman, S. Chowdhury, F. Morone, and K. Y. Camsari, Generalized probabilistic approximate optimization algorithm,Nature Communications, vol. 17, p. 498, 2026

work page 2026
[22]

Srivastava and J

S. Srivastava and J. Roychowdhury, Analytical equations for nonlinear phase errors and jitter in ring oscillators, IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 54, no. 10, pp. 2321–2329, 2007

work page 2007
[23]

Demir, A

A. Demir, A. Mehrotra, and J. Roychowdhury, Phase noise in oscillators: A unifying theory and numerical methods for characterisation, inProceedings of the 35th Annual Design Automation Conference, 1998, pp. 26–31. 6 Supplemental Material Breakdown of Gradient-Flow Dynamics in Oscillator Ising Machines from Harmonic Misalignment Abir Hasan1, E.M. Hasantha ...

work page 1998

[1] [1]

Wang and J

T. Wang and J. Roychowdhury, OIM: Oscillator-based Ising machines for solving combinatorial optimisation problems, inInternational Conference on Unconven- tional Computation and Natural Computation, Springer, 2019, pp. 232–256

work page 2019

[2] [2]

Mohseni, P

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

work page 2022

[3] [3]

S. K. Vadlamani, T. P. Xiao, and E. Yablonovitch, Physics successfully implements Lagrange multiplier op- timization,Proceedings of the National Academy of Sci- ences, vol. 117, no. 43, pp. 26639–26650, 2020

work page 2020

[4] [4]

H. Lo, W. Moy, H. Yu, S. Sapatnekar, and C. H. Kim, A 48-node all-to-all connected coupled ring oscillator Ising solver chip, 2023

work page 2023

[5] [5]

N. W. Schultheiss, A. A. Prinz, and R. J. Butera,Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis, vol. 6, Springer Science & Business Media, 2011

work page 2011

[6] [6]

[22] and [23] for additional details on our theoret- ical analysis, algorithm implementation, and simulation results

See Supplemental Material at [url], which includes Refs. [22] and [23] for additional details on our theoret- ical analysis, algorithm implementation, and simulation results

work page

[7] [7]

Bhansali and J

P. Bhansali and J. Roychowdhury, Gen-Adler: The gen- eralized Adler’s equation for injection locking analysis in oscillators, in2009 Asia and South Pacific Design Au- tomation Conference, IEEE, 2009, pp. 522–527

work page 2009

[8] [8]

Elmitwalli, Z

E. Elmitwalli, Z. Ignjatovic, and S. K¨ ose, CMOS ring os- cillator Ising machine using sub-harmonic injection lock- ing, in2025 IEEE International Symposium on Circuits and Systems (ISCAS), IEEE, 2025, pp. 1–5

work page 2025

[9] [9]

Y. Han, R. R. Unnithan, R. Evans, and E. Skafidas, The digital coupled ring oscillator Ising machine, in 2025 IEEE 16th International Conference on ASIC (ASI- CON), IEEE, 2025, pp. 1–5

work page 2025

[10] [10]

Y. Liu, Z. Han, Q. Wu, H. Yang, Y. Cao, Y. Han, H. Jiang, X. Xue, J. Yang, and X. Zeng, A 1024-spin scalable Ising machine with capacitive coupling and pro- gressive annealing method for combination optimization problems,IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 71, no. 12, pp. 5009–5013, 2024

work page 2024

[11] [11]

H. Lo, W. Moy, H. Yu, S. Sapatnekar, and C. H. Kim, An Ising solver chip based on coupled ring oscillators with a 48-node all-to-all connected array architecture,Nature Electronics, vol. 6, no. 10, pp. 771–778, 2023

work page 2023

[12] [12]

M. A. Barr´ on and M. Sen, Synchronization of four cou- pled van der Pol oscillators,Nonlinear Dynamics, vol. 56, no. 4, pp. 357–367, 2009

work page 2009

[13] [13]

D. I. Albertsson and A. Rusu, Experimental demonstra- tion of Duffing oscillator-based analog Ising machines, in 2024 IEEE 15th Latin America Symposium on Circuits and Systems (LASCAS), IEEE, 2024, pp. 1–5

work page 2024

[14] [14]

J. Chou, S. Bramhavar, S. Ghosh, and W. Herzog, Ana- log coupled oscillator based weighted Ising machine,Sci- entific Reports, vol. 9, no. 1, p. 14786, 2019

work page 2019

[15] [15]

Maffezzoni, L

P. Maffezzoni, L. Daniel, N. Shukla, S. Datta, and A. Raychowdhury, Modeling and simulation of vanadium dioxide relaxation oscillators,IEEE Transactions on Cir- cuits and Systems I: Regular Papers, vol. 62, no. 9, pp. 2207–2215, 2015

work page 2015

[16] [16]

Parihar, N

A. Parihar, N. Shukla, S. Datta, and A. Raychowdhury, Synchronization of pairwise-coupled, identical, relaxation oscillators based on metal-insulator phase transition de- vices: A model study,Journal of Applied Physics, vol. 117, no. 5, 2015

work page 2015

[17] [17]

Maher, M

O. Maher, M. Jim´ enez, C. Delacour, N. Harnack, J. N´ u˜ nez, M. J. Avedillo, B. Linares-Barranco, A. Todri- Sanial, G. Indiveri, and S. Karg, A CMOS-compatible oscillation-based VO2 Ising machine solver,Nature Com- munications, vol. 15, no. 1, p. 3334, 2024

work page 2024

[18] [18]

Y. W. Lee, S. J. Kim, J. Kim, S. Kim, J. Park, Y. Jeong, G. W. Hwang, S. Park, B. H. Park, and S. Lee, Demon- stration of an energy-efficient Ising solver composed of ovonic threshold switch (OTS)-based nano-oscillators (OTSNOs),Nano Convergence, vol. 11, no. 1, p. 20, 2024

work page 2024

[19] [19]

Lai and J

X. Lai and J. Roychowdhury, Capturing oscillator in- jection locking via nonlinear phase-domain macromod- els,IEEE Transactions on Microwave Theory and Tech- niques, vol. 52, no. 9, pp. 2251–2261, 2004

work page 2004

[20] [20]

K. Lee, S. Chowdhury, and K. Y. Camsari, Noise- augmented chaotic Ising machines for combinatorial op- timization and sampling,Communications Physics, vol. 8, p. 35, 2025

work page 2025

[21] [21]

A. S. Abdelrahman, S. Chowdhury, F. Morone, and K. Y. Camsari, Generalized probabilistic approximate optimization algorithm,Nature Communications, vol. 17, p. 498, 2026

work page 2026

[22] [22]

Srivastava and J

S. Srivastava and J. Roychowdhury, Analytical equations for nonlinear phase errors and jitter in ring oscillators, IEEE Transactions on Circuits and Systems I: Regular Papers, vol. 54, no. 10, pp. 2321–2329, 2007

work page 2007

[23] [23]

Demir, A

A. Demir, A. Mehrotra, and J. Roychowdhury, Phase noise in oscillators: A unifying theory and numerical methods for characterisation, inProceedings of the 35th Annual Design Automation Conference, 1998, pp. 26–31. 6 Supplemental Material Breakdown of Gradient-Flow Dynamics in Oscillator Ising Machines from Harmonic Misalignment Abir Hasan1, E.M. Hasantha ...

work page 1998