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

Abir Hasan; E.M. Hasantha Ekanayake; Kerem Camsari; Kyle Lee; Nikhil Shukla

arxiv: 2605.17243 · v2 · 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 2026-05-22 09:32 UTC · model grok-4.3

classification ⚛️ physics.comp-ph

keywords oscillator Ising machinesgradient flowharmonic misalignmentphase dynamicsnon-conservative dynamicsIsing optimizationhardware oscillators

0 comments

The pith

Gradient-flow dynamics in oscillator Ising machines require a specific quadrature relation between waveform and phase response that many hardware models violate.

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

The paper establishes that the widespread interpretation of oscillator Ising machines as performing gradient descent on an Ising energy landscape holds only under a strict condition: the oscillator waveform and its phase response must satisfy a harmonic-by-harmonic quadrature relation. When this relation is absent, a phenomenon called harmonic misalignment appears. It injects even components into the pairwise interaction function, which in turn produces non-conservative phase dynamics that cannot be described as gradient flow. A sympathetic reader would care because this breakdown means that common energy-based analyses and algorithms may not apply to many practical oscillator implementations, requiring instead nonequilibrium treatments that explicitly handle the extra non-gradient terms.

Core claim

Oscillator Ising machines are not generically gradient-flow systems. Gradient-flow dynamics require a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response. Deviations from this condition, termed harmonic misalignment, introduce even components in the pairwise interaction function, leading to non-conservative phase dynamics and precluding a gradient-flow description.

What carries the argument

Harmonic misalignment, defined as the absence of the required quadrature relation between waveform and phase response, which injects even components into the pairwise interaction function and renders the resulting phase dynamics non-conservative.

If this is right

A normalized metric quantifies the non-gradient contribution and shows substantial values for ring oscillators and other hardware-realistic models.
Energy-based analyses of OIMs must be restricted to models that satisfy the quadrature condition.
Practical hardware implementations require nonequilibrium analysis and algorithms that account for non-gradient behavior.

Where Pith is reading between the lines

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

Hardware designers could deliberately introduce or suppress harmonic misalignment to steer the system toward desired non-gradient behaviors such as faster escape from local minima.
The same misalignment mechanism may appear in other coupled-oscillator platforms used for combinatorial optimization or neuromorphic computing.
Algorithmic extensions could treat the even components as an additional driving term and derive correction protocols that restore effective gradient flow.

Load-bearing premise

The modeling choice that the pairwise interaction function derived from the waveform and phase response fully determines the phase dynamics, with no additional higher-order or non-pairwise effects present in the representative oscillator models.

What would settle it

Compute the phase dynamics for a ring oscillator or similar hardware model, extract the even and odd components of the effective interaction function, and test whether the observed trajectories conserve a scalar energy function or exhibit measurable non-conservative drift.

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 gradient flow in OIMs requires a specific quadrature condition per harmonic, and deviations create measurable non-conservative terms in common models.

read the letter

The key takeaway is that the usual energy-landscape picture for oscillator Ising machines does not hold generically. Gradient flow needs a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response; when that fails, even components appear in the pairwise interaction and the phase dynamics stop being conservative. The authors call this harmonic misalignment and give a normalized metric to size the non-gradient contribution. They then compute the metric on ring oscillators and a few other hardware-style models, finding it is large enough to matter. That is the concrete advance: a clear necessary condition plus a practical way to check it. The derivation from the quadrature requirement to the even/odd split in the interaction function looks straightforward and matches the abstract. The evaluation across representative models is useful for anyone who actually builds these circuits. The main soft spot is the modeling step that treats the pairwise interaction as fully determining the phase dynamics. The stress-test note is on point here. If the real oscillator models contain higher-harmonic or multi-oscillator corrections that are not odd under the same symmetry, those terms could offset the non-conservative piece even when misalignment is present. The paper measures the metric on the pairwise term alone and does not appear to supply an explicit bound or numerical check showing the omitted contributions stay small across the regimes of interest. That leaves the claim that gradient flow is precluded for the full system a little open. Readers who work on physical Ising solvers or analog optimization hardware will find this directly relevant; it pushes them toward nonequilibrium analysis rather than assuming an energy function. The work is coherent on its own terms and engages the literature honestly, so it deserves a serious referee even if the higher-order question needs tightening in revision.

Referee Report

2 major / 2 minor

Summary. The paper claims that gradient-flow dynamics in oscillator Ising machines require a harmonic-by-harmonic quadrature relation between the oscillator waveform and its phase response. Deviations from this condition, termed harmonic misalignment, introduce even components in the pairwise interaction function, leading to non-conservative phase dynamics and precluding a gradient-flow description. The authors introduce a normalized metric for the non-gradient contribution and evaluate it across representative oscillator models, finding substantial non-gradient effects in ring oscillators and other hardware-realistic implementations.

Significance. If the central claim holds, the work is significant for challenging the energy-landscape view of OIMs and motivating nonequilibrium analysis. Credit is given for the clear logical chain from the quadrature condition to even components in the interaction function and for introducing and evaluating the normalized non-gradient metric on relevant models.

major comments (2)

[Phase-reduced dynamics derivation] The central claim that harmonic misalignment precludes gradient-flow dynamics rests on the modeling assumption that the pairwise interaction function fully determines the phase dynamics. The stress-test concern lands: higher-order or non-pairwise couplings in the representative models (e.g., ring oscillators) could restore a gradient structure or cancel non-conservative terms. An explicit bound or numerical check showing these terms remain negligible across the parameter regimes is needed to secure the preclusion for the full system.
[Metric evaluation and model results] In the section evaluating the normalized metric, the metric is applied to the pairwise function alone. Without demonstrating that omitted higher-harmonic or multi-oscillator corrections do not alter the non-gradient character, the conclusion that gradient-flow dynamics break down in the full OIM models is not yet fully secured.

minor comments (2)

[Abstract] The abstract introduces the term 'harmonic misalignment' without a brief inline definition; adding one would improve accessibility for readers new to the topic.
[Figures] Figure captions would benefit from explicitly stating the normalization procedure and what the plotted metric values represent.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have prompted us to strengthen the presentation of our results. We address each major comment in turn below.

read point-by-point responses

Referee: [Phase-reduced dynamics derivation] The central claim that harmonic misalignment precludes gradient-flow dynamics rests on the modeling assumption that the pairwise interaction function fully determines the phase dynamics. The stress-test concern lands: higher-order or non-pairwise couplings in the representative models (e.g., ring oscillators) could restore a gradient structure or cancel non-conservative terms. An explicit bound or numerical check showing these terms remain negligible across the parameter regimes is needed to secure the preclusion for the full system.

Authors: We agree that the validity of the phase-reduced description for the full system merits explicit verification. Our analysis is grounded in the standard first-order phase reduction for weakly coupled oscillators, where pairwise interactions dominate at leading order in the coupling strength ε. Higher-order corrections enter at O(ε²) and are therefore negligible in the weak-coupling regime assumed for OIMs. To address the referee’s concern directly, we have added numerical simulations of the full ring-oscillator equations in the revised manuscript. These simulations compare the observed phase evolution against the phase-reduced prediction and confirm that the non-conservative terms induced by harmonic misalignment are not canceled by higher-order couplings within the relevant parameter ranges. The new results appear in an expanded Section IV together with a brief discussion of the validity of the reduction. revision: yes
Referee: [Metric evaluation and model results] In the section evaluating the normalized metric, the metric is applied to the pairwise function alone. Without demonstrating that omitted higher-harmonic or multi-oscillator corrections do not alter the non-gradient character, the conclusion that gradient-flow dynamics break down in the full OIM models is not yet fully secured.

Authors: We acknowledge that the normalized non-gradient metric is computed from the pairwise interaction function obtained via phase reduction. In the revised manuscript we have augmented the evaluation section with an explicit check that higher-harmonic corrections to the interaction function remain small relative to the leading term for the oscillator models considered. In addition, the full-system numerical comparisons introduced in response to the first comment also demonstrate that the non-gradient character persists when the complete nonlinear dynamics are simulated. These additions are now included in Section V and the associated figure captions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is mathematically self-contained

full rationale

The paper derives the necessity of harmonic-by-harmonic quadrature for gradient-flow dynamics directly from the phase-reduced model and the even/odd decomposition of the pairwise interaction function. This step is an explicit mathematical consequence of the stated assumptions on waveform and phase response, without reducing to a fitted parameter, self-citation chain, or redefinition of inputs. The normalized non-gradient metric is then evaluated on representative models as a separate diagnostic step. No load-bearing premise relies on prior work by the same authors in a way that would make the central claim tautological. The analysis remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the domain assumption that phase dynamics are governed by the derived pairwise interaction function and on the modeling choice that the selected oscillator models are representative of the broad class where misalignment occurs.

axioms (1)

domain assumption Phase dynamics of coupled oscillators in OIMs are captured by an interaction function obtained from the waveform and phase-response relation.
This is the standard modeling premise for OIMs that the paper uses to derive the breakdown condition.

invented entities (1)

harmonic misalignment no independent evidence
purpose: To name and quantify deviations from the quadrature relation that produce even components and non-conservative dynamics.
Newly introduced term whose only evidence is the derivation within the paper.

pith-pipeline@v0.9.0 · 5710 in / 1286 out tokens · 37204 ms · 2026-05-22T09:32:33.256128+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

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

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J uniqueness) echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

H(Δij) = Σ αn βn /2 cos(nΔij + χZ_n − χs_n); cos(Δχn)=0 ∀n with αnβn≠0 for oddness (gradient flow)
IndisputableMonolith/Constants.lean phi_golden_ratio echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

T=6 ln(φ)τ with φ=(1+√5)/2 for three-stage ring oscillator period

What do these tags mean?

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

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

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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

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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

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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

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See Supplemental Material at [url], which includes Refs

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and [22] for additional details on our theoretical anal- ysis, algorithm implementation, and simulation results

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Bhansali and J

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

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Elmitwalli, Z

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

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

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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

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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

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H. Chen, J. Jin, Z. Wang, and B. Zhang, A van der Pol-Duffing oscillator with indefinite degree,Qualitative Theory of Dynamical Systems, vol. 21, no. 4, p. 98, 2022

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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

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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

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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
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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

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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

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[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

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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

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

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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

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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, 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

[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]

See Supplemental Material at [url], which includes Refs

work page

[7] [7]

and [22] for additional details on our theoretical anal- ysis, algorithm implementation, and simulation results

work page

[8] [8]

Bhansali and J

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

work page 2009

[9] [9]

Elmitwalli, Z

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

work page 2025

[10] [10]

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

work page 2025

[11] [11]

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

[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]

H. Chen, J. Jin, Z. Wang, and B. Zhang, A van der Pol-Duffing oscillator with indefinite degree,Qualitative Theory of Dynamical Systems, vol. 21, no. 4, p. 98, 2022

work page 2022

[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 opti- mization 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