Pith. sign in

REVIEW 1 major objections 3 minor 109 references

Unifying Physical Backpropagation

T0 review · 1 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper shows that physical hardware can compute the exact gradient of its own cost on-device: reciprocity for linear systems, a time-reversal mirror for nonlinear trajectories, an intertwining condition for non-reciprocal ones.

desk verdict A genuine unification of on-device physical backprop with a clean linear/nonlinear split; the nonlinear trajectory 'exactness' needs a noise-aware qualifier before publication. read the letter →

arxiv 2608.11585 v1 pith:ENGNWSS5 submitted 2026-08-12 cond-mat.dis-nn cs.ETcs.LGphysics.optics

Cyrill Bösch , Yigithan Gediz , Hakan Türeci This is my paper · ORCID
classification cond-mat.dis-nncs.ETcs.LGphysics.optics MSC 49K1568T0581Q1235Q41
keywords physicalbackpropagationadjointmethodon-devicegradientcomputationreciprocitytime-reversalmirrorequilibriumpropagationnon-Hermitiansystemsneuralnetworks
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

Physical computers—optical, mechanical, or electrical networks that compute through their own dynamics—are hard to train because their exact gradients are usually computed in a digital model that never quite matches the real device. This paper asks when the device itself can produce the gradient it needs, and answers with a unified theory built on the adjoint method, the standard technique of computing a cost gradient from two solves. The answer splits by regime: a linear system needs only reciprocity (its mass, damping, and stiffness matrices symmetric, in the Euclidean or Onsager sense), and then one extra finite-amplitude run of the same hardware—with sources and initial conditions swapped and time dependences reversed—yields the exact gradient, damping and gain included. A nonlinear trajectory system needs more: zero damping, a time-reversal mirror that regenerates the reversed trajectory from the final state, and a symmetric linearized internal force, with the gradient extracted from the infinitesimal response of a nudged reversed trajectory. The same framework recovers Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training, and photonic in-situ backpropagation as special cases, and a generalized intertwining condition extends exact on-device gradients to non-reciprocal systems.

What carries the argument

The central object is the forward-time adjoint field $a(t)=b(T-t)$: the Lagrange multiplier of the constrained optimization, re-indexed so it runs forward in time. It satisfies the adjoint equation, which involves transposed and linearized operators, and the gradient is built from overlap integrals between $a$ and parameter sensitivities of the forward operators. The load-bearing identity is that the adjoint propagator coincides with the forward propagator on the same hardware when the transposed operators are conjugate to the forward ones under a constant invertible matrix $S$—$S K^T S^{-1}=K$ (with $S=I$ for reciprocity, $S=V$ for Onsager reciprocity, and orthogonal $S$ for twisted reciprocity). For nonlinear trajectories the mechanism is the time-reversal mirror: initializing with the measured final state and flipped velocity regenerates the reversed trajectory $w(t)=u(T-t)$, and an infinitesimal nudge $\epsilon P\theta_u$ produces a response $\delta w/\epsilon$ that obeys the adjoint equation when $D=0$ and $F_u^T=F_u$ along the trajectory.

What would settle it

On a damped resistor–capacitor network with unequal capacitances, run the forward trajectory, then run the single finite-amplitude adjoint experiment proposed here and compare its gradient with central-difference gradients of the measured cost; the theory predicts exact agreement, and deliberately adding a non-reciprocal element such as a gyrator should make the adjoint-run gradient deviate exactly as the intertwining condition predicts. Repeating the comparison on a one-way-gain Hatano–Nelson chain with mirror-symmetric on-site potentials would settle the twisted-reciprocal extension.

Watch

Extended reading notes

Core claim

The paper's central claim is that the adjoint field needed for formally exact gradients can be produced by the same hardware that ran the forward computation, under sufficient conditions that depend on the regime. For linear systems, reciprocity—$M^T=M$, $D^T=D$, $K^T=K$ at every instant, or the Onsager version $V A^T V^{-1}=A$—makes the adjoint equation identical to the forward equation, so the adjoint is obtained in a single finite-amplitude experiment with the loss derivatives as sources and initial conditions and any explicit time dependence replayed in reverse; damping and gain are admissible. For nonlinear trajectory systems the conditions are the existence of a time-reversal mirror (which regenerates $w(t)=u(T-t)$ from the measured final state and flipped velocity), an undamped dynamics $D=0$, and reciprocal linearization $F_u^T=F_u$ along the trajectory; the adjoint is then the infinitesimal limit $\lim_{\epsilon\to 0}(w_\epsilon-w)/\epsilon$ of a nudged reversed trajectory, so multiple experiments and small nudges are required. In stationary problems the nonlinear construction reduces to the difference between free and nudged equilibria—Equilibrium Propagation—provided the fixed point is stable and its tangent operators self-adjoint. Finally, reciprocity is shown to be only the simplest case of an intertwining condition $S K^T S^{-1}=K$ for a constant invertible $S$, which yields exact physical gradients even in non-Hermitian, non-reciprocal systems, exemplified by a Hatano–Nelson chain with mirror-symmetric on-site potentials. The conditions are sufficient, not necessary, and the listed algorithms all follow as instances of the one adjoint construction.

Load-bearing premise

The whole construction depends on the hardware being reversible in a specific sense: linear devices must be reciprocal, and nonlinear trajectory devices must be undamped and able to measure and re-inject their final state without error.

Editorial extensions

If this is right

  • Linear reciprocal platforms with damping or gain can deliver exact trajectory gradients with one extra finite-amplitude run, so energy-dissipating hardware is trainable without a digital twin.
  • Nonlinear trajectory training on hardware is only exact in undamped, time-reversal-symmetric systems; damped nonlinear trajectories cannot produce exact gradients on the same device, but their steady states remain trainable by Equilibrium Propagation.
  • Existing exact algorithms—Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training, photonic in-situ backpropagation—are all special cases of one adjoint construction, giving them a common language for extension.
  • Symmetry-odd non-reciprocity (one-way gain on a mirror-symmetric structure) is trainable exactly, so non-Hermitian platforms like the Hatano–Nelson chain become usable for physical learning.
  • Encoding inputs in the parameters of a linear, damped, reciprocal device (structural nonlinearity) combines nonlinear computation with finite-amplitude exact gradients, a route the paper singles out as particularly attractive.

Reading between the lines

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

  • The sufficiency, not necessity, of the conditions invites stronger statements: one can likely relax the intertwiner to time-dependent or parameter-dependent $S$ with closed-form updates, buying broader trainability at the price of dynamic range and recomputation.
  • The paper's separation of 'knowing the model' from 'knowing the parameters' predicts a concrete robustness effect: in devices whose parameters enter linearly, gradient estimates from measured forward and adjoint fields should be immune to drift and aging without recalibration—a testable laboratory prediction.
  • The appendix's mass-proportional damping construction, with scalar exponential reweighting, suggests a practical bridge for weakly damped nonlinear systems: even when the strict $D=0$ theorem fails, an approximate same-device gradient may be recovered and its error bounded by the size of the damping term.
  • The twisted-reciprocity condition is a trainability version of known non-Hermitian symmetry classifications; viewed that way, any hardware symmetry (rotation, inversion, exchange) with the correct parity signature under transposition confers exact on-device gradients, which could be used to screen candidate platforms before building them.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 3 minor

Summary. The paper develops a unifying theory, based on the adjoint method, for when the adjoint field required for exact gradient computation can be generated on the same physical hardware that ran the forward computation. It derives sufficient conditions for linear and nonlinear second-order, first-order, Schrödinger-type, and stationary systems; recovers Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training, and photonic in situ backpropagation; and introduces an intertwining condition that extends exact same-hardware adjoints to a class of non-reciprocal systems illustrated by a Hatano–Nelson chain. The appendices provide self-contained derivations of the adjoint equations and gradient formulas for all four system classes, including the damping obstruction for nonlinear trajectories and the V-reciprocal generalization.

Significance. If the results hold as stated, this is a valuable unification: it places several existing exact physical-learning algorithms in a common adjoint framework, clarifies the differing roles of linearity versus nonlinearity, and provides constructive sufficient conditions for designing new on-device gradient schemes. The derivations are self-contained; Theorems 6.1, 7.1, 8.1, 9.1, and 10.1 are explicit; and the Hatano–Nelson example gives a concrete, falsifiable construction. The paper also correctly distinguishes exact from approximate prior algorithms. The main limitation is operational: the nonlinear trajectory construction is exact only in an idealized noiseless infinitesimal limit, and the paper does not quantify how finite measurement and actuation precision degrade the extracted gradient.

major comments (1)
  1. [Sec. 6.2, Eq. (31); also Sec. 8.2, Eq. (58) and Sec. 9.2, Eq. (76)] The claim that the nonlinear adjoint is obtained exactly from a(t) = lim_{ε→0}(w_ε(t)-w(t))/ε assumes that the full final state (u(T), -u̇(T)) can be measured and re-injected without error and that w_ε can be recorded without error. With measurement/re-injection error η, the extracted field carries error O(η/ε), while the truncation bias is O(ε); balancing the two gives ε* ~ √η and a gradient accuracy floor ~ √η. Thus 'formally exact' is accurate only in the jointly noiseless, infinitesimal limit. The same η/ε structure affects the PT-TRM Schrödinger construction (Eq. (58)) and the Equilibrium Propagation stationary limit (Eq. (76)). The authors should add a quantitative error-propagation statement, or explicitly qualify every 'formally exact same-hardware' claim as holding in the noiseless mathematical model, rather than leaving the idealization implicit.
minor comments (3)
  1. [Sec. 6.2, Eq. (30)] The initial conditions w_ε(0) = u(T) - ε M^{-1} Pψ_u̇ and ẇ_ε(0) = -u̇(T) - ε M^{-1} Pψ_u are written with what appear to be unprojected loss derivatives; consider making the notation ψ_u = ∂ψ/∂u and ψ_u̇ = ∂ψ/∂u̇ explicit so the connection to the adjoint initial conditions (6)-(7) is immediate.
  2. [Sec. 8.1, 'Holographic vs. intensity measurements'] The sentence 'The construction requires time reversal symmetry, so that complex conjugating c(T) re-generates the adjoint state evolution' is valid only when K is real; the preceding paragraph has only K^T = K. The authors do state later that time reversal symmetry makes K real, but the logic would be clearer if this assumption were introduced before the intensity-reconstruction protocol.
  3. [Table 1, nonlinear damped trajectory rows] The table entries for damped nonlinear trajectory systems say 'None' and mark the case obstructed; this is correct for the same-device setting, but the mass-proportional damping construction of Appendix C (which requires flipping damping to gain and hence a different hardware setting) is summarized only in the text. A parenthetical pointer to Appendix C in the table would prevent the impression that all damped nonlinear trajectory gradients are unobtainable in every experimental configuration.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the sufficient conditions are derived from the adjoint equations, and existing algorithms are recovered as corollaries rather than used as premises.

full rationale

The central derivations are self-contained. The adjoint equations (4)-(13) are derived from Lagrange multipliers in Appendix A; the linear recovery follows by substituting M^T=M, D^T=D, and K^T=K into the adjoint equation and comparing with the forward equation (26)-(27). The nonlinear result is proven by comparing the nudged perturbation equation (29) with the adjoint equation (4) and invoking ODE uniqueness, with the sufficient conditions D=0, M^T=M, and F_u^T=F_u stated explicitly in Sec. 6.2 and Appendix B. The stationary Equilibrium Propagation construction is derived by linearizing the nudged steady-state equation (74)-(76), and the intertwining Theorem 10.1 is an algebraic substitution d=Sa into the adjoint equation (80)-(82). The claimed recoveries of Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training, and photonic in situ backpropagation are cross-referenced after independent derivation rather than used as inputs to the proofs. The only self-citations, [52] and [100], are contextual or illustrate a standard Wirtinger viewpoint; neither is load-bearing for the main theorems. The skeptic's concerns about the epsilon-to-zero limit and measurement noise are operational correctness/feasibility limits, not circularity of the derivation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central claims are theorem-based; the ledger lists the modeling assumptions about the physical setup and the standard mathematical background. No parameters are fitted to data. The adjoint field, the intertwiner S, and the V-metric are mathematical constructs borrowed from optimization and linear algebra, not new physical entities.

assumptions (5)
  • domain assumption The physical system's forward dynamics is exactly described by the stated ODE (or PDE) with known operators M, D, F, f; model-reality gap is out of scope.
    Sec 1 and Sec 4: the 'knowing the model' requirement. If the model is wrong, the overlap integrals (8)-(13) are biased.
  • domain assumption Adjoint sources, initial conditions, and time-reversal of explicit time dependences can be physically injected or replayed on the same hardware.
    Sec 3: 'The only allowed modifications between the forward and adjoint experiments are changes of initial conditions, external forces, and reversal of explicit time dependences.'
  • domain assumption For nonlinear trajectory systems, a time-reversal mirror (TRM) exists on the same hardware, i.e., the reversed trajectory is generated by initializing with (u(T), -u'(T)) (or via PT symmetry), which requires D=0 in the real second-order case.
    Theorem 6.1 and Appendix B; the damping obstruction shows D=0 is required.
  • domain assumption The linearized dynamics along the forward trajectory is reciprocal: F_u^T = F_u (or V-reciprocal), for nonlinear trajectory and stationary constructions.
    Theorems 6.1 and 9.1; this is a premise of the sufficient conditions.
  • standard math Standard results from ODE theory, optimization, and calculus apply: Lagrange multiplier method, integration by parts, Picard-Lindelöf existence, Poincaré lemma, Wirtinger calculus.
    Used throughout Appendices A-H.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Unifying Physical Backpropagation." pith.science (2026). https://pith.science/paper/ENGNWSS5

@misc{pith2026260811585,
  author       = {Pith},
  title        = {Pith review of: Unifying Physical Backpropagation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ENGNWSS5}},
  note         = {Machine review of arXiv:2608.11585}
}
read the original abstract

Physical computing systems exploit device dynamics for computation, but their gradient-based optimization is challenging: backpropagation through a digital twin suffers from model-reality gap. On-device gradient computation could resolve this issue, and a handful of theoretical and experimental studies have proposed ways to achieve it. Yet a unifying theory identifying when a physical system can compute the gradient of its own performance has been missing. Here we develop such a unification, based on the adjoint method: we identify sufficient conditions under which the adjoint field required for formally exact gradients can be generated on the same hardware that performs the computation. Linear and nonlinear systems obey fundamentally different conditions: for linear systems damping or gain is admissible provided reciprocity is preserved. For nonlinear trajectory systems the sufficient conditions are reciprocity of the linearized system and the existence of a time-reversal mirror. Algorithmically, the nonlinear case requires infinitesimal nudging, whereas linear systems admit a finite-amplitude experiment. We recover Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training and in situ gradient methods in integrated-photonic and free-space-optical systems. We further show that reciprocity is only the simplest instance of a more general intertwining condition, which extends exact on-device gradient computation to a class of non-Hermitian, non-reciprocal systems. Further generalizations include time-dependent parameters, Onsager-reciprocal dynamics and nonlinear, PT-symmetric Schr\"odinger equations. Our work provides a unified theoretical basis for formally exact physical learning algorithms and a template for constructing them across a range of physical systems.

Figures

Figures reproduced from arXiv: 2608.11585 by the authors.

Figure 1
Figure 1. Conceptual overview of optimizing a physical system with the adjoint method. The physical [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. A schematic illustration of the experiments to obtain the adjoint field physically. For the linear [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. The Hatano–Nelson chain of Eq. (86) for N = 4: sites with on-site potentials εn, coupled by asymmetric nearest-neighbor hoppings, amplified to the right (Je+h , blue) and attenuated to the left (Je−h , red), so that KT ̸= K and the chain transmits with one-way gain. Transposition reverses the arrow of every hop; the spatial inversion n → N + 1 − n about the marked inversion center reverses it again. If the on-site p… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

109 extracted references · 68 canonical work pages

  1. [1]

    Toward a formal theory for computing machines made out of whatever physics offers.Nature Communications, 14(1):4911, 2023

    Herbert Jaeger, Beatriz Noheda, and Wilfred G Van Der Wiel. Toward a formal theory for computing machines made out of whatever physics offers.Nature Communications, 14(1):4911, 2023

    2023

  2. [2]

    Wave physics as an analog recurrent neural network.Science Advances, 5(12):eaay6946, 2019

    Tyler W Hughes, Ian AD Williamson, Momchil Minkov, and Shanhui Fan. Wave physics as an analog recurrent neural network.Science Advances, 5(12):eaay6946, 2019

    2019

  3. [3]

    Analogue computing with metamaterials.Nature Reviews Materials, 6(3):207–225, 2021

    Farzad Zangeneh-Nejad, Dimitrios L Sounas, Andrea Al` u, and Romain Fleury. Analogue computing with metamaterials.Nature Reviews Materials, 6(3):207–225, 2021. 26

    2021

  4. [4]

    In- sensor passive speech classification with phononic metamaterials.Advanced Functional Materials, 34(17):2311877, 2024

    Tena Dubˇ cek, Daniel Moreno-Garcia, Thomas Haag, Parisa Omidvar, Henrik R Thomsen, Theodor S Becker, Lars Gebraad, Christoph B¨ arlocher, Fredrik Andersson, Sebastian D Huber, et al. In- sensor passive speech classification with phononic metamaterials.Advanced Functional Materials, 34(17):2311877, 2024

    2024

  5. [5]

    Demonstration of decentralized physics-driven learning.Physical Review Applied, 18(1):014040, 2022

    Sam Dillavou, Menachem Stern, Andrea J Liu, and Douglas J Durian. Demonstration of decentralized physics-driven learning.Physical Review Applied, 18(1):014040, 2022

    2022

  6. [6]

    Memory devices and applications for in-memory computing.Nature Nanotechnology, 15(7):529–544, 2020

    Abu Sebastian, Manuel Le Gallo, Riduan Khaddam-Aljameh, and Evangelos Eleftheriou. Memory devices and applications for in-memory computing.Nature Nanotechnology, 15(7):529–544, 2020

    2020

  7. [7]

    Physics for neuromorphic computing.Nature Reviews Physics, 2(9):499–510, 2020

    Danijela Markovi´ c, Alice Mizrahi, Damien Querlioz, and Julie Grollier. Physics for neuromorphic computing.Nature Reviews Physics, 2(9):499–510, 2020

    2020

  8. [8]

    Photonics for artificial intelligence and neuro- morphic computing.Nature Photonics, 15(2):102–114, 2021

    Bhavin J Shastri, Alexander N Tait, Thomas Ferreira de Lima, Wolfram HP Pernice, Harish Bhaskaran, C David Wright, and Paul R Prucnal. Photonics for artificial intelligence and neuro- morphic computing.Nature Photonics, 15(2):102–114, 2021

    2021

Show all 109 references
  1. [9]

    Scalable optical learning operator.Nature Computational Science, 1(8):542–549, 2021

    U˘ gur Te˘ gin, Mustafa Yıldırım,˙Ilker O˘ guz, Christophe Moser, and Demetri Psaltis. Scalable optical learning operator.Nature Computational Science, 1(8):542–549, 2021

    2021

  2. [10]

    Inference in artificial intelligence with deep optics and photonics.Nature, 588(7836):39–47, 2020

    Gordon Wetzstein, Aydogan Ozcan, Sylvain Gigan, Shanhui Fan, Dirk Englund, Marin Soljaˇ ci´ c, Cornelia Denz, David AB Miller, and Demetri Psaltis. Inference in artificial intelligence with deep optics and photonics.Nature, 588(7836):39–47, 2020

    2020

  3. [11]

    All-optical machine learning using diffractive deep neural networks.Science, 361(6406):1004– 1008, 2018

    Xing Lin, Yair Rivenson, Nezih T Yardimci, Muhammed Veli, Yi Luo, Mona Jarrahi, and Aydogan Ozcan. All-optical machine learning using diffractive deep neural networks.Science, 361(6406):1004– 1008, 2018

    2018

  4. [12]

    The physics of optical computing.Nature Reviews Physics, 5(12):717–734, 2023

    Peter L McMahon. The physics of optical computing.Nature Reviews Physics, 5(12):717–734, 2023

    2023

  5. [13]

    Arbitrary control over multimode wave propagation for machine learning.Nature Physics, 22:164–171, 2026

    Tatsuhiro Onodera, Martin M Stein, Benjamin A Ash, Mandar M Sohoni, Melissa Bosch, Ryotatsu Yanagimoto, Marc Jankowski, Timothy P McKenna, Tianyu Wang, Gennady Shvets, Maxim R Shcherbakov, Logan G Wright, and Peter L McMahon. Arbitrary control over multimode wave propagation f...

    2026

  6. [14]

    Programmable on-chip nonlinear photonics.Nature, 649(8096):330–337, 2026

    Ryotatsu Yanagimoto, Benjamin A Ash, Mandar M Sohoni, Martin M Stein, Yiqi Zhao, Federico Pre- sutti, Marc Jankowski, Logan G Wright, Tatsuhiro Onodera, and Peter L McMahon. Programmable on-chip nonlinear photonics.Nature, 649(8096):330–337, 2026

    2026

  7. [15]

    Recent advances in physical reservoir computing: A review.Neural Networks, 115:100–123, 2019

    Gouhei Tanaka, Toshiyuki Yamane, Jean Benoit H´ eroux, Ryosho Nakane, Naoki Kanazawa, Seiji Takeda, Hidetoshi Numata, Daiju Nakano, and Akira Hirose. Recent advances in physical reservoir computing: A review.Neural Networks, 115:100–123, 2019

    2019

  8. [16]

    Training of physical neural networks.Nature, 645(8079):53–61, 2025

    Ali Momeni, Babak Rahmani, Benjamin Scellier, Logan G Wright, Peter L McMahon, Clara C Wanjura, Yuhang Li, Anas Skalli, Natalia G Berloff, Tatsuhiro Onodera, et al. Training of physical neural networks.Nature, 645(8079):53–61, 2025

    2025

  9. [17]

    Inverse-designed low-index-contrast structures on a silicon photonics platform for vector– matrix multiplication.Nature Photonics, 18(5):501–508, 2024

    Vahid Nikkhah, Ali Pirmoradi, Farshid Ashtiani, Brian Edwards, Firooz Aflatouni, and Nader En- gheta. Inverse-designed low-index-contrast structures on a silicon photonics platform for vector– matrix multiplication.Nature Photonics, 18(5):501–508, 2024

    2024

  10. [18]

    Deep physical neural networks trained with backpropagation.Nature, 601(7894):549–555, 2022

    Logan G Wright, Tatsuhiro Onodera, Martin M Stein, Tianyu Wang, Darren T Schachter, Zoey Hu, and Peter L McMahon. Deep physical neural networks trained with backpropagation.Nature, 601(7894):549–555, 2022. 27

    2022

  11. [19]

    Optical diffusion models for image generation.Advances in Neural Information Processing Systems, 37:59150–59173, 2024

    Ilker Oguz, Niyazi Ulas Dinc, Mustafa Yildirim, Junjie Ke, Innfarn Yoo, Qifei Wang, Feng Yang, Christophe Moser, and Demetri Psaltis. Optical diffusion models for image generation.Advances in Neural Information Processing Systems, 37:59150–59173, 2024

    2024

  12. [20]

    Trainable hardware for dynamical computing using error backpropagation through physical media

    Michiel Hermans, Micha¨ el Burm, Thomas Van Vaerenbergh, Joni Dambre, and Peter Bienstman. Trainable hardware for dynamical computing using error backpropagation through physical media. Nature Communications, 6(1):6729, 2015

    2015

  13. [21]

    In situ optical backpropagation training of diffractive optical neural networks

    Tiankuang Zhou, Lu Fang, Tao Yan, Jiamin Wu, Yipeng Li, Jingtao Fan, Huaqiang Wu, Xing Lin, and Qionghai Dai. In situ optical backpropagation training of diffractive optical neural networks. Photonics Research, 8(6):940–953, 2020

    2020

  14. [22]

    Threading light through dynamic complex media.Nature Photonics, 19(4):434–440, 2025

    Chaitanya K Mididoddi, Robert J Kilpatrick, Christina Sharp, Philipp Del Hougne, Simon AR Horsley, and David B Phillips. Threading light through dynamic complex media.Nature Photonics, 19(4):434–440, 2025

    2025

  15. [23]

    Experimentally realized in situ backpropagation for deep learning in photonic neural networks.Science, 380(6643):398–404, 2023

    Sunil Pai, Zhanghao Sun, Tyler W Hughes, Taewon Park, Ben Bartlett, Ian AD Williamson, Momchil Minkov, Maziyar Milanizadeh, Nathnael Abebe, Francesco Morichetti, et al. Experimentally realized in situ backpropagation for deep learning in photonic neural networks.Science, 380(6...

    2023

  16. [24]

    Training all-mechanical neural networks for task learning through in situ backpropagation.Nature Communications, 15(1):10528, 2024

    Shuaifeng Li and Xiaoming Mao. Training all-mechanical neural networks for task learning through in situ backpropagation.Nature Communications, 15(1):10528, 2024

    2024

  17. [25]

    How to train your resistive net- work: Generalized equilibrium propagation and analytical learning.arXiv preprint arXiv:2602.03546, 2026

    Jonathan Lin, Aman Desai, Frank Barrows, and Francesco Caravelli. How to train your resistive net- work: Generalized equilibrium propagation and analytical learning.arXiv preprint arXiv:2602.03546, 2026

    2026

  18. [26]

    Self-learning machines based on Hamiltonian echo backpropagation.Physical Review X, 13(3):031020, 2023

    V´ ıctor L´ opez-Pastor and Florian Marquardt. Self-learning machines based on Hamiltonian echo backpropagation.Physical Review X, 13(3):031020, 2023

    2023

  19. [27]

    Routledge, 2018

    Lev Semenovich Pontryagin.Mathematical theory of optimal processes. Routledge, 2018

    2018

  20. [28]

    Springer Science & Business Media, 2010

    Andreas Fichtner.Full seismic waveform modelling and inversion. Springer Science & Business Media, 2010

    2010

  21. [29]

    In-situ physical adjoint comput- ing in multiple-scattering electromagnetic environments for wave control.Nature Communications, 16(1):11466, 2025

    John Guillamon, Cheng-Zhen Wang, Zin Lin, and Tsampikos Kottos. In-situ physical adjoint comput- ing in multiple-scattering electromagnetic environments for wave control.Nature Communications, 16(1):11466, 2025

    2025

  22. [30]

    Equilibrium propagation: Bridging the gap between energy- based models and backpropagation.Frontiers in Computational Neuroscience, 11:24, 2017

    Benjamin Scellier and Yoshua Bengio. Equilibrium propagation: Bridging the gap between energy- based models and backpropagation.Frontiers in Computational Neuroscience, 11:24, 2017

    2017

  23. [31]

    Learning long range dependencies through time reversal symmetry breaking.arXiv preprint arXiv:2506.05259, 2025

    Guillaume Pourcel and Maxence Ernoult. Learning long range dependencies through time reversal symmetry breaking.arXiv preprint arXiv:2506.05259, 2025

    2025 arXiv

  24. [32]

    Fully forward mode training for optical neural networks.Nature, 632(8024):280–286, 2024

    Zhiwei Xue, Tiankuang Zhou, Zhihao Xu, Shaoliang Yu, Qionghai Dai, and Lu Fang. Fully forward mode training for optical neural networks.Nature, 632(8024):280–286, 2024

    2024

  25. [33]

    Training of photonic neural networks through in situ backpropagation and gradient measurement.Optica, 5(7):864–871, 2018

    Tyler W Hughes, Momchil Minkov, Yu Shi, and Shanhui Fan. Training of photonic neural networks through in situ backpropagation and gradient measurement.Optica, 5(7):864–871, 2018

    2018

  26. [34]

    Multilayer optical learning networks.Applied Optics, 26(23):5061–5076, 1987

    Kelvin Wagner and Demetri Psaltis. Multilayer optical learning networks.Applied Optics, 26(23):5061–5076, 1987. 28

    1987

  27. [35]

    Backpropagation through nonlinear units for the all-optical training of neural networks.Photonics Research, 9(3):B71– B80, 2021

    Xianxin Guo, Thomas D Barrett, Zhiming M Wang, and Alexander I Lvovsky. Backpropagation through nonlinear units for the all-optical training of neural networks.Photonics Research, 9(3):B71– B80, 2021

    2021

  28. [36]

    Training neural networks with end-to-end optical backpropagation.Advanced Photonics, 7(1):016004, 2025

    James Spall, Xianxin Guo, and Alexander I Lvovsky. Training neural networks with end-to-end optical backpropagation.Advanced Photonics, 7(1):016004, 2025

    2025

  29. [37]

    Fully nonlinear neuromorphic computing with linear wave scattering.Nature Physics, 20(9):1434–1440, 2024

    Clara C Wanjura and Florian Marquardt. Fully nonlinear neuromorphic computing with linear wave scattering.Nature Physics, 20(9):1434–1440, 2024

    2024

  30. [38]

    Topological mechanical neural networks as classifiers through in situ backpropagation learning.arXiv preprint arXiv:2503.07796, 2025

    Shuaifeng Li and Xiaoming Mao. Topological mechanical neural networks as classifiers through in situ backpropagation learning.arXiv preprint arXiv:2503.07796, 2025

    2025 arXiv

  31. [39]

    Localization transitions in non-Hermitian quantum mechan- ics.Physical Review Letters, 77(3):570–573, 1996

    Naomichi Hatano and David R Nelson. Localization transitions in non-Hermitian quantum mechan- ics.Physical Review Letters, 77(3):570–573, 1996

    1996

  32. [40]

    Supervised learning in physical networks: From machine learning to learning machines.Physical Review X, 11(2):021045, 2021

    Menachem Stern, Daniel Hexner, Jason W Rocks, and Andrea J Liu. Supervised learning in physical networks: From machine learning to learning machines.Physical Review X, 11(2):021045, 2021

    2021

  33. [41]

    Learning without neurons in physical systems.Annual Review of Condensed Matter Physics, 14(1):417–441, 2023

    Menachem Stern and Arvind Murugan. Learning without neurons in physical systems.Annual Review of Condensed Matter Physics, 14(1):417–441, 2023

    2023

  34. [42]

    Machine learning without a processor: Emergent learning in a nonlinear analog network

    Sam Dillavou, Benjamin D Beyer, Menachem Stern, Andrea J Liu, Marc Z Miskin, and Douglas J Durian. Machine learning without a processor: Emergent learning in a nonlinear analog network. Proceedings of the National Academy of Sciences, 121(28):e2319718121, 2024

    2024

  35. [43]

    Training self-learning circuits for power-efficient solutions.APL Machine Learning, 2(1):016114, 2024

    Menachem Stern, Sam Dillavou, Dinesh Jayaraman, Douglas J Durian, and Andrea J Liu. Training self-learning circuits for power-efficient solutions.APL Machine Learning, 2(1):016114, 2024

    2024

  36. [44]

    Coercivity and local convergence of physical learning in linear circuits.arXiv preprint arXiv:2606.15443, 2026

    Joshua A McGinnis, Xinbo Li, and Yoichiro Mori. Coercivity and local convergence of physical learning in linear circuits.arXiv preprint arXiv:2606.15443, 2026

    2026

  37. [45]

    Multimodal deep learning using on-chip diffractive optics with in situ training capability.Nature Communications, 15(1):6189, 2024

    Junwei Cheng, Chaoran Huang, Jialong Zhang, Bo Wu, Wenkai Zhang, Xinyu Liu, Jiahui Zhang, Yiyi Tang, Hailong Zhou, Qiming Zhang, Min Gu, Jianji Dong, and Xinliang Zhang. Multimodal deep learning using on-chip diffractive optics with in situ training capability.Nature Communica...

    2024

  38. [46]

    Multimodal oscillator networks learn to solve a classification problem.npj Metamaterials, 2(1):3, 2026

    Daan de Bos and Marc Serra-Garcia. Multimodal oscillator networks learn to solve a classification problem.npj Metamaterials, 2(1):3, 2026

    2026

  39. [47]

    Forward–forward training of an optical neural network

    Ilker Oguz, Junjie Ke, Qifei Weng, Feng Yang, Mustafa Yildirim, Niyazi Ulas Dinc, Jih-Liang Hsieh, Christophe Moser, and Demetri Psaltis. Forward–forward training of an optical neural network. Optics Letters, 48(20):5249–5252, 2023

    2023

  40. [48]

    Backpropagation-free training of deep physical neural networks.Science, 382(6676):1297–1303, 2023

    Ali Momeni, Babak Rahmani, Matthieu Mall´ ejac, Philipp Del Hougne, and Romain Fleury. Backpropagation-free training of deep physical neural networks.Science, 382(6676):1297–1303, 2023

    2023

  41. [49]

    Training deep physical neural networks with local physical information bottleneck.arXiv preprint arXiv:2602.09569, 2026

    Hao Wang, Ziao Wang, Xiangpeng Liang, Han Zhao, Jianqi Hu, Junjie Jiang, Xing Fu, Jianshi Tang, Huaqiang Wu, Sylvain Gigan, et al. Training deep physical neural networks with local physical information bottleneck.arXiv preprint arXiv:2602.09569, 2026

    2026

  42. [50]

    Using memristors for robust local learning of hardware restricted Boltzmann machines.Scientific Reports, 9(1):1851, 2019

    Maxence Ernoult, Julie Grollier, and Damien Querlioz. Using memristors for robust local learning of hardware restricted Boltzmann machines.Scientific Reports, 9(1):1851, 2019

    2019

  43. [51]

    Marcelo Guzman, Simone Ciarella, and Andrea J Liu. Unsupervised and probabilistic learning with contrastive local learning networks: The restricted Kirchhoff machine.Proceedings of the National Academy of Sciences, 123(21):e2525792123, 2026. 29

    2026

  44. [52]

    Local learning rules for out-of-equilibrium physical generative models.arXiv preprint arXiv:2506.19136, 2025

    Cyrill B¨ osch, Geoffrey Roeder, Marc Serra-Garcia, and Ryan P Adams. Local learning rules for out-of-equilibrium physical generative models.arXiv preprint arXiv:2506.19136, 2025

    2025 arXiv

  45. [53]

    Nonlinear computation with linear optics via source-position encoding.arXiv preprint arXiv:2504.20401, 2025

    Nick Richardson, Cyrill B¨ osch, and Ryan P Adams. Nonlinear computation with linear optics via source-position encoding.arXiv preprint arXiv:2504.20401, 2025

    2025 arXiv

  46. [54]

    Nonlinear optical encoding enabled by recurrent linear scattering.Nature Photonics, 18(10):1067–1075, 2024

    Fei Xia, Kyungduk Kim, Yaniv Eliezer, SeungYun Han, Liam Shaughnessy, Sylvain Gigan, and Hui Cao. Nonlinear optical encoding enabled by recurrent linear scattering.Nature Photonics, 18(10):1067–1075, 2024

    2024

  47. [55]

    Nonlinear processing with linear optics.Nature Photonics, 18(10):1076–1082, 2024

    Mustafa Yildirim, Niyazi Ulas Dinc, Ilker Oguz, Demetri Psaltis, and Christophe Moser. Nonlinear processing with linear optics.Nature Photonics, 18(10):1076–1082, 2024

    2024

  48. [56]

    Giles and Niles A

    Michael B. Giles and Niles A. Pierce. An introduction to the adjoint approach to design.Flow, Turbulence and Combustion, 65(3-4):393–415, December 2000

    2000

  49. [57]

    R.-E. Plessix. A review of the adjoint-state method for computing the gradient of a functional with geophysical applications.Geophysical Journal International, 167(2):495–503, November 2006

    2006

  50. [58]

    Dutton, and Shanhui Fan

    Georgios Veronis, Robert W. Dutton, and Shanhui Fan. Method for sensitivity analysis of photonic crystal devices.Optics Letters, 29(19):2288–2290, Oct 2004

    2004

  51. [59]

    Jan Werschnik and E. K. U. Gross. Quantum optimal control theory.Journal of Physics B: Atomic, Molecular and Optical Physics, 40(18):R175–R211, 2007

    2007

  52. [60]

    Optimal control of PDEs in a complex space setting: Application to the Schr¨ odinger equation.SIAM Journal on Control and Optimization, 57(2):1390–1412, 2019

    Mar´ ıa Soledad Aronna, Joseph Fr´ ed´ eric Bonnans, and Axel Kr¨ oner. Optimal control of PDEs in a complex space setting: Application to the Schr¨ odinger equation.SIAM Journal on Control and Optimization, 57(2):1390–1412, 2019

    2019

  53. [61]

    D. H. Brandwood. A complex gradient operator and its application in adaptive array theory.IEE Proceedings F - Communications, Radar and Signal Processing, 130(1):11–16, 1983

    1983

  54. [62]

    Complex-valued matrix differentiation: Techniques and key results.IEEE Transactions on Signal Processing, 55(6):2740–2746, 2007

    Are Hjørungnes and David Gesbert. Complex-valued matrix differentiation: Techniques and key results.IEEE Transactions on Signal Processing, 55(6):2740–2746, 2007

    2007

  55. [63]

    Unconstrained optimization of real functions in complex variables.SIAM Journal on Optimization, 22(3):879–898, 2012

    Laurent Sorber, Marc Van Barel, and Lieven De Lathauwer. Unconstrained optimization of real functions in complex variables.SIAM Journal on Optimization, 22(3):879–898, 2012

    2012

  56. [64]

    Springer Netherlands, 2009

    Michael Hinze, Ren´ e Pinnau, Michael Ulbrich, and Stefan Ulbrich.Optimization with PDE Con- straints. Springer Netherlands, 2009

    2009

  57. [65]

    Training overdamped dynamics.arXiv preprint arXiv:2602.19122, 2026

    Marc Berneman and Daniel Hexner. Training overdamped dynamics.arXiv preprint arXiv:2602.19122, 2026

    2026

  58. [66]

    Equilibrium propagation for dissipative dynamics.arXiv preprint arXiv:2506.20402, 2025

    Marc Berneman and Daniel Hexner. Equilibrium propagation for dissipative dynamics.arXiv preprint arXiv:2506.20402, 2025

    2025 arXiv

  59. [67]

    Holomorphic equilibrium propagation computes exact gradi- ents through finite size oscillations.Advances in Neural Information Processing Systems, 35:12950– 12963, 2022

    Axel Laborieux and Friedemann Zenke. Holomorphic equilibrium propagation computes exact gradi- ents through finite size oscillations.Advances in Neural Information Processing Systems, 35:12950– 12963, 2022

    2022

  60. [68]

    Training nonlinear optical neural networks with scattering backpropagation.arXiv preprint arXiv:2508.11750, 2025

    Nicola Dal Cin, Florian Marquardt, and Clara C Wanjura. Training nonlinear optical neural networks with scattering backpropagation.arXiv preprint arXiv:2508.11750, 2025

    2025 arXiv

  61. [69]

    Near-equilibrium propagation training in nonlinear wave systems.arXiv preprint arXiv:2510.16084, 2025

    Karol Sajnok and Micha l Matuszewski. Near-equilibrium propagation training in nonlinear wave systems.arXiv preprint arXiv:2510.16084, 2025. 30

    2025

  62. [70]

    Gen- eralization of equilibrium propagation to vector field dynamics.arXiv preprint arXiv:1808.04873, 2018

    Benjamin Scellier, Anirudh Goyal, Jonathan Binas, Thomas Mesnard, and Yoshua Bengio. Gen- eralization of equilibrium propagation to vector field dynamics.arXiv preprint arXiv:1808.04873, 2018

    2018 arXiv

  63. [71]

    Improving equilibrium propagation without weight symmetry through Jacobian homeostasis

    Axel Laborieux and Friedemann Zenke. Improving equilibrium propagation without weight symmetry through Jacobian homeostasis. InInternational Conference on Learning Representations, 2024

    2024

  64. [72]

    Equilibrium propagation for non-conservative systems.arXiv preprint arXiv:2602.03670, 2026

    Antonino Emanuele Scurria, Dimitri Vanden Abeele, Bortolo Matteo Mognetti, and Serge Massar. Equilibrium propagation for non-conservative systems.arXiv preprint arXiv:2602.03670, 2026

    2026 arXiv

  65. [73]

    Agnostic physics-driven deep learning.arXiv preprint arXiv:2205.15021, 2022

    Benjamin Scellier, Siddhartha Mishra, Yoshua Bengio, and Yann Ollivier. Agnostic physics-driven deep learning.arXiv preprint arXiv:2205.15021, 2022

    2022 arXiv

  66. [74]

    Reciprocal relations in irreversible processes

    Lars Onsager. Reciprocal relations in irreversible processes. I.Physical Review, 37(4):405–426, 1931

    1931

  67. [75]

    Fluctuations and irreversible processes.Physical Review, 91(6):1505–1512, 1953

    Lars Onsager and Stefan Machlup. Fluctuations and irreversible processes.Physical Review, 91(6):1505–1512, 1953

    1953

  68. [76]

    Topological mechanics of gyroscopic metamaterials.Proceedings of the National Academy of Sciences, 112(47):14495–14500, 2015

    Lisa M Nash, Dustin Kleckner, Alismari Read, Vincenzo Vitelli, Ari M Turner, and William T M Irvine. Topological mechanics of gyroscopic metamaterials.Proceedings of the National Academy of Sciences, 112(47):14495–14500, 2015

    2015

  69. [77]

    Observation of PT phase transition in a simple mechanical system.American Journal of Physics, 81(3):173–179, 2013

    Carl M Bender, Bjorn K Berntson, David Parker, and E Samuel. Observation of PT phase transition in a simple mechanical system.American Journal of Physics, 81(3):173–179, 2013

    2013

  70. [78]

    Time reversal of ultrasonic fields

    Mathias Fink. Time reversal of ultrasonic fields. I. Basic principles.IEEE Transactions on Ultra- sonics, Ferroelectrics, and Frequency Control, 39(5):555–566, 1992

    1992

  71. [79]

    Lagrangian-based equilib- rium propagation: generalisation to arbitrary boundary conditions & equivalence with Hamiltonian echo learning.arXiv preprint arXiv:2506.06248, 2025

    Guillaume Pourcel, Debabrota Basu, Maxence Ernoult, and Aditya Gilra. Lagrangian-based equilib- rium propagation: generalisation to arbitrary boundary conditions & equivalence with Hamiltonian echo learning.arXiv preprint arXiv:2506.06248, 2025

    2025 arXiv

  72. [80]

    Michael Spivak.Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus. W. A. Benjamin, 1965

    1965

  73. [81]

    Scaling equilibrium propagation to deep convnets by drastically reducing its gradient estimator bias.Frontiers in Neuroscience, 15:633674, 2021

    Axel Laborieux, Maxence Ernoult, Benjamin Scellier, Yoshua Bengio, Julie Grollier, and Damien Querlioz. Scaling equilibrium propagation to deep convnets by drastically reducing its gradient estimator bias.Frontiers in Neuroscience, 15:633674, 2021

    2021

  74. [82]

    Equilibrium propagation for learning in Lagrangian dynamical systems.Physical Review E, 112(3):035304, 2025

    Serge Massar. Equilibrium propagation for learning in Lagrangian dynamical systems.Physical Review E, 112(3):035304, 2025

    2025

  75. [83]

    Springer Science & Business Media, 1991

    Reinhold Remmert.Theory of complex functions, volume 122. Springer Science & Business Media, 1991

    1991

  76. [84]

    A short tutorial on Wirtinger calculus with applications in quantum information.arXiv preprint arXiv:2312.04858, 2023

    Kelvin Koor, Yixian Qiu, Leong Chuan Kwek, and Patrick Rebentrost. A short tutorial on Wirtinger calculus with applications in quantum information.arXiv preprint arXiv:2312.04858, 2023

    2023 arXiv

  77. [85]

    Integrated photonic neural network with on-chip backpropagation training.Nature, 651(8107):927–932, 2026

    Farshid Ashtiani, Mohamad Hossein Idjadi, and Kwangwoong Kim. Integrated photonic neural network with on-chip backpropagation training.Nature, 651(8107):927–932, 2026

    2026

  78. [86]

    Training end-to-end analog neural networks with equilibrium propagation.arXiv preprint arXiv:2006.01981, 2020

    Jack Kendall, Ross Pantone, Kalpana Manickavasagam, Yoshua Bengio, and Benjamin Scel- lier. Training end-to-end analog neural networks with equilibrium propagation.arXiv preprint arXiv:2006.01981, 2020

    2006 arXiv

  79. [87]

    Training an Ising machine with equilibrium propagation.Nature Communications, 15(1):3671, 2024

    J´ er´ emie Laydevant, Danijela Markovi´ c, and Julie Grollier. Training an Ising machine with equilibrium propagation.Nature Communications, 15(1):3671, 2024. 31

    2024

  80. [88]

    Equivalence of equilibrium propagation and recurrent back- propagation.Neural Computation, 31(2):312–329, 2019

    Benjamin Scellier and Yoshua Bengio. Equivalence of equilibrium propagation and recurrent back- propagation.Neural Computation, 31(2):312–329, 2019

    2019

  81. [89]

    Updates of equilibrium prop match gradients of backprop through time in an RNN with static input.Advances in Neural Information Processing Systems, 32, 2019

    Maxence Ernoult, Julie Grollier, Damien Querlioz, Yoshua Bengio, and Benjamin Scellier. Updates of equilibrium prop match gradients of backprop through time in an RNN with static input.Advances in Neural Information Processing Systems, 32, 2019

    2019

  82. [90]

    Frequency propagation: Multimechanism learning in nonlinear physical networks.Neural Computation, 36(4):596–620, 2024

    Vidyesh Rao Anisetti, Ananth Kandala, Benjamin Scellier, and JM Schwarz. Frequency propagation: Multimechanism learning in nonlinear physical networks.Neural Computation, 36(4):596–620, 2024

    2024

  83. [91]

    Training coupled phase oscillators as a neuromorphic platform using equilibrium propagation.Neuromorphic Computing and Engineering, 4(3):034014, 2024

    Qingshan Wang, Clara C Wanjura, and Florian Marquardt. Training coupled phase oscillators as a neuromorphic platform using equilibrium propagation.Neuromorphic Computing and Engineering, 4(3):034014, 2024

    2024

  84. [92]

    Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits.Nature Physics, 16(7):747– 750, 2020

    Tobias Helbig, Tobias Hofmann, Stefan Imhof, Mohamed Abdelghany, Tobias Kiessling, Laurens W Molenkamp, Ching Hua Lee, Alexander Szameit, Martin Greiter, and Ronny Thomale. Generalized bulk–boundary correspondence in non-Hermitian topolectrical circuits.Nature Physics, 16(7):7...

    2020

  85. [93]

    Topological funneling of light.Science, 368(6488):311–314, 2020

    Sebastian Weidemann, Mark Kremer, Tobias Helbig, Tobias Hofmann, Alexander Stegmaier, Mar- tin Greiter, Ronny Thomale, and Alexander Szameit. Topological funneling of light.Science, 368(6488):311–314, 2020

    2020

  86. [94]

    Sound isolation and giant linear nonreciprocity in a compact acoustic circulator.Science, 343(6170):516–519, 2014

    Romain Fleury, Dimitrios L Sounas, Caleb F Sieck, Michael R Haberman, and Andrea Al` u. Sound isolation and giant linear nonreciprocity in a compact acoustic circulator.Science, 343(6170):516–519, 2014

    2014

  87. [95]

    Odd elasticity.Nature Physics, 16(4):475–480, 2020

    Colin Scheibner, Anton Souslov, Debarghya Banerjee, Piotr Sur´ owka, William T M Irvine, and Vincenzo Vitelli. Odd elasticity.Nature Physics, 16(4):475–480, 2020

    2020

  88. [96]

    On Onsager’s principle of microscopic reversibility.Reviews of Modern Physics, 17(2-3):343–350, 1945

    Hendrik B G Casimir. On Onsager’s principle of microscopic reversibility.Reviews of Modern Physics, 17(2-3):343–350, 1945

    1945

  89. [97]

    Symmetry and topology in non-Hermitian physics.Physical Review X, 9(4):041015, 2019

    Kohei Kawabata, Ken Shiozaki, Masahito Ueda, and Masatoshi Sato. Symmetry and topology in non-Hermitian physics.Physical Review X, 9(4):041015, 2019

    2019

  90. [98]

    Non-reciprocal robotic metamaterials.Nature Communications, 10(1):4608, 2019

    Martin Brandenbourger, Xander Locsin, Edan Lerner, and Corentin Coulais. Non-reciprocal robotic metamaterials.Nature Communications, 10(1):4608, 2019

    2019

  91. [99]

    Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial

    Ananya Ghatak, Martin Brandenbourger, Jasper van Wezel, and Corentin Coulais. Observation of non-Hermitian topology and its bulk–edge correspondence in an active mechanical metamaterial. Proceedings of the National Academy of Sciences, 117(47):29561–29568, 2020

    2020

  92. [100]

    Adjoint computation of Berry phase gradients.Journal of Sound and Vibration, 619:119357, 2025

    Cyrill B¨ osch, Marc Serra-Garcia, Christian B¨ ohm, and Andreas Fichtner. Adjoint computation of Berry phase gradients.Journal of Sound and Vibration, 619:119357, 2025

    2025

  93. [101]

    Quantum-optical analogies using photonic structures.Laser & Photonics Reviews, 3(3):243–261, 2009

    Stefano Longhi. Quantum-optical analogies using photonic structures.Laser & Photonics Reviews, 3(3):243–261, 2009

    2009

  94. [102]

    M. D. Feit and J. A. Fleck, Jr. Light propagation in graded-index optical fibers.Applied Optics, 17(24):3990–3998, 1978

    1978

  95. [103]

    Automatic adjoint-based inversion schemes for geodynamics: reconstructing the evolution of Earth’s mantle in space and time.Geoscientific Model Development, 17(13):5057–5086, 2024

    Sia Ghelichkhan, Angus Gibson, D Rhodri Davies, Stephan C Kramer, and David A Ham. Automatic adjoint-based inversion schemes for geodynamics: reconstructing the evolution of Earth’s mantle in space and time.Geoscientific Model Development, 17(13):5057–5086, 2024. 32

    2024

  96. [104]

    Automated derivation of the adjoint of high-level transient finite element programs.SIAM Journal on Scientific Computing, 35(4):C369–C393, 2013

    Patrick E Farrell, David A Ham, Simon W Funke, and Marie E Rognes. Automated derivation of the adjoint of high-level transient finite element programs.SIAM Journal on Scientific Computing, 35(4):C369–C393, 2013. Contents of the appendices A Derivation of the adjoint state equa...

    2013

  97. [105]

    Run the forward dynamics and recordu(t), then formw(t) =u(T−t)

  98. [106]

    Flip the damping to gain,D=γM7→−γM, so the device reproduces the reversed backgroundw

  99. [107]

    On the same gain-flipped device, apply the TRM and nudge together: initialize withw ϵ(0) =u(T)− ϵM−1Pψ ˙u and ˙wϵ(0) =− ˙u(T)−ϵM −1Pψu, drive withf(T−t)−ϵe γtPθu[Pu(T−t)], and reverse the explicit time dependence ofF

  100. [108]

    Measure ea(t) = limϵ→0[wϵ(t)−w(t)]/ϵand form the adjointa(t) =e −γtea(t)

  101. [109]

    hockey-stick

    Evaluate the gradient from (8)–(13), equivalently (150). Two remarks. First, the construction hinges onγbeing a knownscalar: only then ise −γt a scalar weight that commutes withMandF u and leaves the stiffness untouched in (146). A dampingDthat is not a scalar multiple ofMwoul...

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.