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Unifying Physical Backpropagation

Cyrill Bösch, Yigithan Gediz, Clara C. Wanjura, Hakan E. Türeci

arXiv:2608.11585v2cond-mat.dis-nncs.ETcs.LGphysics.optics

TL;DR

Digital-twin backpropagation is vulnerable to a model–reality gap, and a unifying account of when physical systems can compute exact gradients has been missing. The paper uses the adjoint method to identify sufficient same-hardware conditions and unify existing approaches. Linear and nonlinear systems require different experiments, while generalized intertwining conditions extend exact physical gradients beyond ordinary reciprocity.

  • Problem

    Digital-twin training is vulnerable to a model–reality gap, while formally exact on-device gradient methods lack a unifying account of when they apply.

    Source: §1 · p. 2

  • Method

    The paper unifies physical backpropagation through the adjoint method and identifies sufficient conditions for obtaining the adjoint field in a physical experiment on the same system.

    Source: §1 · p. 2

  • Results

    Linear reciprocal systems allow finite-amplitude adjoint experiments with damping or gain, whereas nonlinear trajectory systems require a time-reversal mirror, reciprocal linearized dynamics, and infinitesimal nudging.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Intertwining conditions extend exact physical gradients to some non-Hermitian, non-reciprocal linear systems and, with generalized time reversal, to nonlinear examples including PT-symmetric dynamics.

    Source: §10 · p. 28

    The framework requires model structure to determine which observable and overlap integral correspond to each parameter.

    Source: §4 · p. 9

Abstract

from arXiv · show

Physical computing systems exploit device dynamics for computation, but their gradient-based optimization is challenging: backpropagation through a digital twin suffers from a 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 (quantum) Equilibrium Propagation, Hamiltonian echo backpropagation, fully forward mode training and in situ gradient methods in integrated-photonic and free-space-optical systems. Finally, we show that reciprocity is a special case of more general intertwining conditions. For linear systems, these permit exact on-device gradients in a class of non-Hermitian, non-reciprocal systems. For nonlinear trajectories, they combine with generalized time-reversal mirrors to cover, e.g., PT-symmetric equations. The framework also includes time-dependent parameters and Onsager-reciprocal dynamics, providing a unified basis for formally exact physical learning.

1 Introduction

The paper unifies formally exact physical backpropagation methods and identifies conditions for generating the adjoint field on the same hardware. It distinguishes linear from nonlinear systems and extends reciprocity-based constructions to broader settings.

1 Introduction

  • The adjoint framework unifies formally exact physical backpropagation methods and identifies sufficient conditions for generating the adjoint field on the same hardware as the forward computation.Only initial conditions, external forces, and reversal of explicit time dependences may be changed between experiments.Source: §1 · p. 2
  • Linear reciprocal systems permit direct finite-amplitude adjoint experiments, with damping or gain allowed when reciprocity is preserved.The corresponding nonlinear trajectory construction is obstructed in linear reciprocal overdamped systems, where physical trajectory gradients can still be obtained.Source: §1 · p. 2
  • Nonlinear trajectory systems require infinitesimal nudging, a time-reversal mirror, and reciprocity of the linearized dynamics; ordinary damping or gain is generically obstructive.The time-reversal and linearized-reciprocity conditions are independent, and experimentally swapping damping and gain does not generally restore the construction.Source: §1 · pp. 2–3
  • The framework recovers physical gradient methods across equilibrium propagation, Hamiltonian systems, integrated photonics, free-space optics, mechanics, and resistor networks.For optical schemes using approximate gradients through nonlinear subsystems, it recovers exact backpropagation only through the linear layers.Source: §1 · p. 3
  • The paper does not cover heuristic or coupled learning rules, local-objective schemes, probabilistic machines, or systems that adapt themselves.It assumes an external teacher applies parameter updates from physically observed forward and adjoint fields.Source: §1 · p. 3

2 Introducing the adjoint method

The adjoint method computes gradients by solving for an adjoint field rather than differentiating the full trajectory separately for every parameter. The gradient follows from overlaps between that field, the forward trajectory, and parameter sensitivities.

2 Introducing the adjoint method

  • Loss derivatives set the adjoint forcing and initial conditions, while the adjoint dynamics use transposed operators and linearized internal forces along the time-reversed forward trajectory.Trajectory-loss derivatives provide forcing; final-state-loss derivatives set initial displacement and velocity.Source: §2 · p. 5
  • The adjoint method replaces differentiation of the full trajectory with respect to every parameter by a single adjoint solve.The adjoint dynamics generally differ from the forward dynamics because they involve transposes and derivatives of the internal forces.Source: §2 · pp. 5–6
  • Gradient components are obtained from overlap integrals of the forward and adjoint fields with parameter sensitivities; initial-condition parameters use endpoint overlaps.The adjoint field encodes sensitivity to forces applied during the forward evolution.Source: §2 · pp. 5–6
  • The adjoint field evolves backward in time, represented in forward time by changing variables from t to T − t.This time reversal is central to physically realizing the adjoint field.Source: §2 · p. 6
  • The standard algorithm solves the forward problem, solves the adjoint problem, evaluates overlap integrals, and then updates parameters.In digital optimization, the method requires two trajectory solves regardless of parameter count, unlike finite differences and forward sensitivity analysis, whose solve costs grow with parameter count.Source: §2 · p. 6

3 What is computed physically, under what conditions and how: an overview

The overview asks when and how a physical system can produce its own adjoint field. Its sufficient conditions and experiments differ between linear, nonlinear, and stationary systems.

3 What is computed physically, under what conditions and how: an overview

  • The common protocol records the forward trajectory, sets adjoint sources and initial conditions from loss derivatives, obtains the adjoint in a second experiment, and evaluates gradient overlaps.Quantum Equilibrium Propagation instead obtains gradients directly from changes in operator expectation values.Source: §3 · p. 7
  • Linear systems need reciprocity, not Hermiticity, and admit a direct finite-amplitude adjoint run with adjoint sources and initial conditions.Loss or gain is permitted, and explicit time dependence is played in reverse.Source: §3 · p. 7
  • Nonlinear trajectory systems require recoverability of the forward evolution by a time-reversal operation and reciprocity of the linearized dynamics around the reversed trajectory.The adjoint is the infinitesimal response of a nudged time-reversed trajectory, with explicit time dependence played in reverse.Source: §3 · p. 7
  • Stationary linear reciprocal fixed-point or Helmholtz-type problems yield the adjoint field through one direct finite-amplitude experiment.Source: §3 · p. 7
  • Stationary nonlinear systems yield the exact adjoint in the infinitesimal limit from the difference between free and nudged solutions, if they relax to a steady state with reciprocal linearized dynamics.This construction is Equilibrium Propagation [29].Source: §3 · p. 7

4 Local learning rules and “knowing the model” vs. “knowing the parameters”

When parameter sensitivities are local, gradient overlaps can reduce to the degrees of freedom coupled to each parameter. Linear parameter dependence also allows gradient evaluation without knowing parameter values, though the model structure remains necessary.

4 Local learning rules and “knowing the model” vs. “knowing the parameters”

  • A parameter's gradient depends only on the forward and adjoint dynamics of the degrees of freedom to which it couples when the parameter sensitivity is local.The overlap integral reduces to those degrees of freedom because the parameter sensitivity is supported only there.Source: §4 · p. 9
  • When parameters enter forward operators linearly, overlap integrals use measured trajectories without referencing parameter values, absorbing slow drift, aging, and unknown calibration offsets.Source: §4 · p. 9
  • The method still requires knowing the model structure to determine observables and which overlap integral belongs to each parameter.This distinguishes it from self-learning schemes that avoid explicit knowledge of the detailed energy function or Hamiltonian [5] [70].Source: §4 · p. 9

5 Second-order real-valued systems

The adjoint framework gives sufficient conditions for obtaining second-order real-valued systems’ adjoint fields on the same hardware. Linear and nonlinear systems require different experiments, with nonlinear trajectories imposing stricter conditions.

5 Second-order real-valued systems

  • Same-hardware adjoints require different sufficient conditions for linear and nonlinear second-order systems.Source: §5 · p. 9

5.2 Nonlinear

  • For nonlinear trajectories, a time-reversal mirror must recreate the reversed trajectory, and linearized reciprocity lets an infinitesimal nudge recover the adjoint.Source: §5 · p. 10, §5.2 · p. 12

5 Second-order real-valued systems

  • Time-varying internal-force schedules are allowed, but reciprocity must hold at every time and explicit time dependence must be replayed backward; mass and damping matrices must remain constant.Source: §5 · p. 10

5.1 Linear

  • Linear reciprocity lets the same damped hardware propagate the adjoint with adjusted source and initial conditions, without nudging.Source: §5.1 · p. 10, §5 · p. 10

5.2 Nonlinear

  • Ordinary damping obstructs the nonlinear same-hardware construction, while known damping can be compensated using forces applied at every damped degree of freedom.Compensation also requires accurate trajectory replay and calibrated external force control.Source: §5.2 · pp. 12–13
  • Opposite-sign nudges reduce extraction bias from O(ϵ) to O(ϵ^2), at the cost of one additional experiment.Source: §5.2 · p. 13

5.3 Summary: linear and nonlinear systems behave fundamentally differently

  • Linear systems admit simpler finite-amplitude adjoint constructions even with damping, whereas nonlinear trajectory systems require stricter conditions.Source: §5.3 · p. 13

6 First-order real-valued systems

First-order real-valued systems allow physical adjoints on the same hardware for linear reciprocal dynamics, but nonlinear trajectory gradients face a damping-versus-time-reversal obstruction. Steady-state nonlinear problems remain accessible under the conditions for Equilibrium Propagation.

6 First-order real-valued systems

  • First-order real-valued systems admit physical adjoints when reciprocity holds, with explicit time dependence replayed in reverse.The adjoint experiment uses the adjoint source and initial conditions.Source: §6 · p. 14

6.2 Nonlinear

  • Without prescribed compensation, nonlinear overdamped trajectories are obstructed, but a stable, time-independent fixed point remains accessible through Equilibrium Propagation.The fixed-point case requires convergence to a linearly exponentially stable steady state.Source: §6 · p. 14, §6.2 · p. 15

6.1 Linear

  • For linear reciprocal dynamics, the adjoint uses the same overdamped first-order form and hardware as the forward experiment.The experiment applies the negative adjoint drive, uses the loss-derived adjoint initial condition, and reverses explicit time dependence.Source: §6.1 · p. 14

6.2 Nonlinear

  • Nonlinear overdamped trajectory gradients are obstructed because time reversal requires flipping damping into gain, while the adjoint requires damping.Trajectory compensation can recover an infinitesimal adjoint response under stated conditions, but requires accurate velocity recording and external force control at relevant degrees of freedom.Source: §6.2 · pp. 14–15

6 First-order real-valued systems

  • The first-order ODE is the large-damping limit of the second-order ODE.The nonlinear obstruction follows from the same damping reversal conflict as in the second-order construction.Source: §6.2 · p. 14

7 Schr¨odinger-type complex systems

For complex Schrödinger-type systems, the paper gives conditions for obtaining the complex-conjugate adjoint on the same hardware. Linear systems require reciprocity, while nonlinear trajectories require both a time-reversal mirror and matching tangent dynamics.

7.1 Linear

  • For complex-linear systems, reciprocal symmetry—not Hermiticity—allows loss and gain while the same hardware propagates the adjoint.The experiment sets the drive to the negative adjoint source, imposes the adjoint initial condition, and replays explicit time dependence in reverse.Source: §7 · pp. 15–16, §7.1 · p. 16
  • Intensity-only overlap recovery requires time-reversal symmetry and linear superposition, and its site-wise overlap applies to on-site parameters but not coupling coefficients.Source and initial-condition gradients still require holographic measurements.Source: §7.1 · pp. 16–17
  • The framework recovers the in situ optical backpropagation rule for phase-only spatial light modulators, while cited nonlinear optical implementations approximate the required tangent.The reciprocal linear-adjoint construction is exact for ideal linear layers but approximate through nonlinear units with approximate saturable-absorber responses.Source: §7.1 · p. 17

7.2 Nonlinear

  • Nonlinear systems require a time-reversal mirror to regenerate the reversed background and tangent dynamics that match the adjoint.For ordinary conjugation time reversal, the linear term requires reciprocal K and the local Kerr coefficient must be real.Source: §7.2 · pp. 17–18
  • The nonlinear construction is exact in the infinitesimal-nudge limit; a one-sided difference has O(ϵ) bias, while opposite nudges reduce it to O(ϵ^2).The physical nudge is applied to a conjugation-reversed trajectory.Source: §7 · p. 16, §7.2 · pp. 17–18

8 Stationary problems and (quantum) Equilibrium Propagation

The stationary adjoint framework covers linear and nonlinear steady states, extending to Equilibrium Propagation and its quantum form. Linear systems use a finite-amplitude experiment, while nonlinear and quantum constructions use infinitesimal nudging.

8.1 Linear

  • Linear stationary systems obtain the adjoint by replacing the forward drive with the adjoint source when the operator is reciprocal.This is a single finite-amplitude experiment; damping or gain is allowed, and Hermiticity is not required.Source: §8.1 · p. 19

8.2 Nonlinear: Equilibrium Propagation

  • Nonlinear stationary systems recover the adjoint from the infinitesimal response of a nudged steady state when the tangent reciprocity conditions hold.The free and nudged experiments must settle on the same solution branch; the nudge must not cross a basin boundary or bifurcation.Source: §8 · p. 19, §8.2 · pp. 20–21
  • For real-valued fields, the nonlinear stationary construction reduces to Equilibrium Propagation [29].The Kerr example with Hermitian K and real g recovers the exact limit of Near-Equilibrium Propagation [30].Source: §8.2 · p. 21

8.3 Quantum Equilibrium Propagation

  • Quantum Equilibrium Propagation applies a Hamiltonian nudge to the normalized ground state, whose infinitesimal response supplies the adjoint on the tangent space orthogonal to that state.Hermiticity supplies the tangent reciprocity conditions, and a positive excitation gap makes the response unique.Source: §8.3 · p. 22
  • Quantum Equilibrium Propagation estimates parameter gradients from free- and nudged-state expectation values without reading out the adjoint field.For Aj = ∂H/∂pj, the expectation-value change pairs the free state with the state response to give the parameter-gradient overlap.Source: §8.3 · p. 22
  • Finite nudges introduce O(ϵ) bias for a one-sided estimate; opposite nudges reduce it to O(ϵ2), while repeated state preparations estimate expectation values.The nudge must remain in the linear-response regime and large enough to resolve changes above measurement noise.Source: §8.3 · p. 23

9 General intertwiners and twisted reciprocity

Known invertible transformations can relate forward and adjoint dynamics beyond ordinary reciprocity. For nonlinear trajectories, the construction additionally needs a time-reversal mirror and tangent matching; stationary Onsager dynamics provide a related case without trajectory reversal.

9 General intertwiners and twisted reciprocity

  • A known invertible intertwiner can map the adjoint operators onto forward operators, while nonlinear trajectories additionally require a time-reversal mirror to regenerate the reversed background.The tangent-intertwining condition is independent of the background-reversal condition.Source: §9 · p. 23

9.1 Linear Schrödinger systems: spatial symmetries and twisted reciprocity

  • For linear Schrödinger systems, a spatial symmetry can route the adjoint source to the input degrees of freedom, recovering fully forward mode training.The same approach permits exact gradients in certain non-Hermitian, non-reciprocal systems; the transformation must remain available across the trainable parameter family.Source: §9.1 · pp. 24–25
  • A mirror-symmetric Hatano–Nelson chain satisfies SKT S−1 = K, so the transformed adjoint propagates on the same chain with the same hopping bias.For this spatial-inversion construction, the on-site potential must be mirror-symmetric and the parameter ties must be preserved during training.Source: Figure 3 · p. 25, §9.1 · p. 26

9.2 Nonlinear Schrödinger systems: time-reversal mirrors and intertwined reciprocity

  • Nonlinear trajectory constructions combine a time-reversal mirror with an independent intertwiner that matches the tangent dynamics to the adjoint.The gradient is recovered in the infinitesimal-nudge limit; a PT mirror can regenerate the background while tangent matching remains a separate condition.Source: §9.2 · pp. 26–27
  • For the nonlinear Hatano–Nelson chain, a diagonal intertwiner handles arbitrary open-chain profiles but rescales amplitudes site by site.If hopping biases are trained, the intertwiner must be recalibrated after each update.Source: §9.2 · p. 27

9.3 Onsager reciprocity

  • Onsager reciprocity is a special case in which a constant physical mobility supplies the intertwiner for linear and nonlinear dynamics.For nonlinear trajectories, the time-reversal-mirror condition remains independent; stationary gradient flows instead need a stable operating point and continuation on the same branch.Source: §9.3 · p. 27, VFu · p. 28

10 Conclusions

The conclusions unify exact physical gradient methods through the adjoint framework and distinguish the conditions for linear, nonlinear-trajectory, and stationary systems. They also identify unresolved questions about how errors in model structure affect gradients.

10 Conclusions

  • The framework unifies formally exact physical gradient methods and identifies different sufficient conditions for obtaining adjoints on linear, nonlinear-trajectory, and stationary systems.Linear systems need reciprocity and a finite-amplitude experiment; nonlinear trajectories additionally need a time-reversal mirror, linearized reciprocity, and infinitesimal nudging.Source: §10 · p. 28
  • More general intertwiners extend exact physical gradients to some non-Hermitian, non-reciprocal linear systems and, with generalized time-reversal mirrors, to nonlinear trajectories including PT-symmetric systems.The framework also covers Onsager reciprocity, stationary gradient flows, and schedules replayed in reverse for trajectory adjoints.Source: §10 · pp. 28–29
  • How accurately the model structure must be known, and how gradients degrade when that structure is wrong, remain open questions.When parameters enter forward operators linearly, gradient overlaps use measured trajectories rather than parameter values, absorbing slow drift, aging, and calibration offsets.Source: §10 · p. 29

A Derivation of the adjoint state equation and the gradient formula

This section derives exact gradients for a parameterized second-order system using forward and adjoint solutions, and examines how time reversal relates the adjoint to forward-time propagation. It shows that on a gain-flipped device, damping causes a mismatch between the nudged response and the adjoint perturbation, which the source or initial conditions cannot remove.

A.1 State equation; A.2 Cost function

  • The forward problem is a parameterized second-order ODE with parametrized initial conditions, and the cost uses state outputs and possibly nonlinear functionals.The mass operator is assumed invertible for visited parameter values.Source: §A.1 · p. 38, §A.2 · p. 38

A.3 Differentiability assumptions

  • Differentiability assumptions make the state equation's solution unique and smoothly dependent on parameters.The force is twice differentiable in state and parameters and continuous in time, while the drive is twice differentiable in parameters.Source: §A.3 · p. 38

A.4–A.7 Constrained optimization, Lagrangian, and stationarity

  • Stationarity of the Lagrangian multipliers recovers the forward constraints, while choosing them to make Lu = 0 removes the need to compute du/dp.At that stationary point, the reduced-cost derivative equals the partial derivative Lp, which depends only on explicit parameter dependence.Source: A Derivation of the adjoint state… · p. 37, §A.6 · p. 39, §A.7 · p. 40

A.8 Stationarity with respect to the state: the adjoint problem

  • The adjoint is a unique terminal-value ODE solved backward from T, with the nonlinear force entering through its Jacobian along the forward trajectory.Its terminal conditions come from the terminal-cost derivatives; its values at t = 0 determine the initial-condition multipliers.Source: §A.8 · p. 41

A.9 The gradient

  • The gradient is a forward–adjoint overlap of explicit parameter derivatives, and costs one forward solve plus one backward solve regardless of parameter count.It incorporates parameter dependence in the mass, damping, force, drive, and initial conditions.Source: §A.9 · p. 42

A.10 Integrating the adjoints forward in time

  • Changing variables to forward time flips the damping sign, evaluates the linearization and source along the reversed trajectory, and converts terminal data into initial data.The adjoint field is a(t) = b(T − t).Source: §A.10 · p. 43

B. Physical adjoint for nonlinear second-order systems; B.1 Time reversal requires removing or flipping the damping

  • For nonlinear systems, reproducing the reversed trajectory requires zero damping or replacing D with −D; the gain-flipped device alone still does not deliver the adjoint.Time reversal changes D ˙u to −D ˙w, and the appendix proceeds to consider a nudged response on the gain-flipped device.Source: B Physical adjoint for nonlinear… · p. 43, §B.1 · p. 44

B.2 The nudged response on the gain-flipped device

  • On the gain-flipped device, the proposed nudged response adds an infinitesimal adjoint-source perturbation and initial-condition perturbations aligned with the adjoint data.The initial state and velocity are perturbed by ϵa(0) and ϵ ˙a(0), with |ϵ| ≪ 1.Source: §B.2 · p. 44
  • The velocity nudge includes the damping contribution DT a(0) required by the adjoint initial condition, which vanishes when D = 0.The passage specifies that the nudge reproduces the damping contribution demanded by initial condition (7).Source: §B.2 · p. 44
  • Subtracting the reversed state equation from the first-order force expansion yields the gain-flipped perturbation equation and its corresponding initial conditions.The initial conditions are obtained by subtracting the reversed-state initial conditions from those of the nudged response.Source: §B.2 · p. 44
  • The gain-flipped device’s response uses the time-reversed Jacobian appearing in the adjoint equation when its background follows the reversed trajectory.The passage identifies the linearization along w(t) = u(T − t) with the time-reversed Jacobian in the adjoint equation.Source: §B.2 · p. 44

B.3 Comparison with the adjoint equation: the damping obstruction

  • Matching the inertial and internal-force linearizations requires equal mass operators and reciprocal tangent dynamics along the reversed trajectory.The matching conditions are MT = M and FT u = Fu along the reversed trajectory.Source: §B.3 · p. 45
  • For reciprocal damping, the nudged response has −D˙ea while the adjoint has +DT ˙a, so their damping terms coincide only when D = 0.The two terms differ by 2D˙a when DT = D.Source: §B.3 · p. 45
  • Even with a gain flip, damping reverses the perturbation’s sign relative to the adjoint, and no source or initial-condition choice can repair this propagator mismatch.The same hardware cannot carry the reversed background and propagate the correct adjoint perturbation unless damping vanishes.Source: §B.3 · p. 45

B.4 The undamped reciprocal case

In the undamped reciprocal case, the infinitesimal response of a nudged time-reversed trajectory gives the adjoint and its gradient. The extensions treat mass-proportional damping through exponential weighting and Onsager reciprocity through a weighted adjoint.

B.4 The undamped reciprocal case

  • The adjoint field is the infinitesimal differential response of the nudged time-reversed trajectory, and the gradient follows from overlap integrals.For the undamped reciprocal construction, the reversed background and rescaled perturbation coincide with the adjoint dynamics.Source: §B.4 · p. 45

C Mass-proportional damping

  • Mass-proportional damping reconciles the damped and adjoint dynamics, but its construction still flips damping to gain, so it uses a different device from the forward pass.The reconciliation uses a scalar exponential weighting; it does not provide a same-device construction.Source: C Mass-proportional damping · p. 45

C.1 Exponential weighting interchanges damping and gain

  • A scalar exponential weighting maps the adjoint equation to anti-damped dynamics with the device's mass and stiffness operators, using an exponentially weighted adjoint source.For a terminal-only loss, the source vanishes and no drive weighting is needed.Source: §C.1 · p. 46

C.2 Initial conditions, recovery, and the physical algorithm

  • The physical algorithm measures the infinitesimal nudged response on the gain-flipped device, then removes the exponential weight before evaluating the gradient overlaps.The adjoint source is weighted by e^γt, and the recovered adjoint is a(t) = e^-γt ea(t).Source: §C.2 · pp. 46–47

D Onsager reciprocity

  • Onsager reciprocity is a special case of the general intertwining construction, with physical field d = Va and possible recovery of a = V^-1d depending on parameterization.Whether explicit recovery is needed depends on how trainable parameters enter the dynamics.Source: D Onsager reciprocity · p. 47

D.1.1 Assumptions

  • For second-order Onsager dynamics, the constant positive-definite symmetric mobility V must make the forward operators V-reciprocal; in the common symmetric case, this requires M and D to commute with V.The internal-force condition is evaluated along the time-reversed trajectory; for quadratic energy, the passage identifies the linear Onsager-reciprocal case as F = VHu.Source: §D.1.1 · p. 47

D.1.2 The weighted adjoint

  • Under V-reciprocity, the transformed field d = Va obeys the device's forward linearized operator, with adjoint sources and initial-condition kicks premultiplied by V.The method therefore uses the weighted adjoint source rather than directly realizing the untransformed adjoint.Source: §D.1.2 · p. 48

D.1.3 Physical realization and gradient recovery

This part develops physical gradient recovery for Onsager-weighted and first-order systems, including resistor–capacitor networks, and gives real and complex adjoint formulations. It specifies when hardware can realize a conjugated, time-reversed adjoint and how measured or computed fields yield parameter gradients through overlap terms, including sign and initial-condition factors.

D.1.3 Physical realization and gradient recovery

  • The Onsager-reciprocal extension uses V-weighted adjoint fields: linear systems admit finite-amplitude runs, while nonlinear gradient flows require infinitesimal nudging.For force parameters with fixed V, the weighting cancels in the gradient, so explicit recovery of the Euclidean adjoint is unnecessary; if a differentiated parameter changes V, its derivative must be included.Source: §D.1.3 · p. 48

D.2 First-order real-valued systems

  • In first-order systems, V-reciprocal linear dynamics permit a finite-amplitude adjoint run, but Onsager reciprocity alone does not remove the time-reversal obstruction for nonlinear trajectories.For nonlinear systems at linearly exponentially stable fixed points, a weighted nudge gives an equilibrium-propagation construction; both relaxations must settle on the same stable branch.Source: §D.2 · p. 49

D.2.1 Example: resistor–capacitor networks with heterogeneous capacitances

  • In heterogeneous-capacitance resistor–capacitor networks, the physical adjoint uses the circuit’s capacitance weighting, and conductance gradients require no digital multiplication by that weighting.The adjoint run applies current −q(t) and an initial capacitor-charge kick −qT; capacitance gradients require separate treatment.Source: §D.2.1 · pp. 49–50

E Adjoint state equation and gradient for first-order systems

E.7 Integrating the adjoint forward in time

  • Changing time to T − t converts the first-order backward adjoint into a forward-time equation along the reversed trajectory, with explicit time dependence replayed in reverse.This yields the adjoint source and initial condition used in the main text.Source: §E.7 · p. 53

F Wirtinger derivation of the Schrödinger adjoint

  • For complex Schrödinger-type dynamics, the Wirtinger derivation treats a field and its conjugate as independent, using a real-paired augmented Lagrangian to obtain the adjoint and gradient.This handles real-valued objectives that are generally non-holomorphic and supplies the forward-time conjugated field used for physical backpropagation.Source: F Wirtinger derivation of the… · p. 53, §F.1 · p. 54, §F.2 · pp. 54–55

F.2 Augmented Lagrangian and the Wirtinger adjoint

  • With b and λ chosen as in Definition F.2, Lu = 0, and identity (221) gives dχ/dp = Lp; explicit parameter dependence enters through constraint and initial-condition pairings.Source: §F.2 · p. 55
  • The external-drive gradient has the opposite sign to the internal-force gradient, while the initial-condition term carries the factor i from λ = −ib(0).These factors follow from the drive entering the residual as −f and substituting λ = −ib(0) into the gradient expression.Source: §F.2 · p. 55

F.3 Forward-time conjugated adjoint

  • The conjugated time-reversed field c(t) = b(T − t) obeys a forward-time equation that can be propagated on the same hardware, and b(t) = c(T − t) recovers the raw adjoint.Conjugating the adjoint equation flips the signs before the Wirtinger Jacobians; substituting the recovered adjoint gives gradient overlaps in terms of the measured field.Source: §F.3 · pp. 55–56

F.4 Proof of the nonlinear intertwined-adjoint construction

This section derives adjoint constructions and physical realizations across nonlinear intertwined systems, gyroscopic systems, free-space optics, stationary problems, and layered photonic networks. It describes how same-device physical adjoint solves in layered networks support layerwise backpropagation and local field-overlap measurements of parameter gradients.

F.4 Proof of the nonlinear intertwined-adjoint construction

  • The nonlinear intertwined construction recovers the adjoint by undoing the tangent intertwiner and time-reversal mirror in reverse order, with force covariance required across an operating domain.The force covariance must hold throughout the operating domain, not merely at isolated points on one trajectory.Source: §F.4 · pp. 56–57

G Second-order systems: gyroscopic PT mirrors

  • The adjoint is generated on the same hardware by reversing the explicit schedule and applying drive Πf(r), as a separate extension of the second-order adjoint derivation, not the nonlinear branch of Theorem 5.1.3.Source: G Second-order systems: gyroscopic PT… · pp. 57–58

H Free-space optics

  • In reciprocal scalar free-space optics, reciprocity means complex symmetry, allowing lossy or linear-gain pixels, while the Π-only construction excludes dissipative balanced gain and loss.The Π-only restriction may be relaxed for special cases by adding an intertwiner.Source: H Free-space optics · pp. 58–59

I Details for stationary problems

  • Stationary problems with an invertible tangent operator include static linear elasticity, resistor networks, and fixed-frequency scattering, with gradients given by a forward–adjoint overlap.The adjoint-variable sensitivity construction is standard for frequency-domain Maxwell and Helmholtz systems.Source: I Details for stationary problems · p. 60

I.1 The stationary adjoint

  • The stationary adjoint is chosen to cancel field-sensitivity terms, so the gradient depends on explicit parameter dependence without requiring one stationary solve per parameter.The adjoint condition is obtained by setting the Lagrangian derivative with respect to the field to zero.Source: §I.1 · p. 61

I.2 Linear systems

  • For reciprocal linear systems, the conjugated adjoint field is generated by solving the same forward problem with its source replaced by −Pψu.The reciprocal condition is K^T = K.Source: §I.2 · pp. 61–62

I.3 Real fields: the static adjoint at a fixed point

  • For real nonlinear dynamics, trajectory-based construction is obstructed, but at a stable fixed point reciprocity makes the static adjoint an infinitesimal difference between free and nudged equilibria.The resulting two-phase energy contrast gives the Equilibrium Propagation learning rule [29].Source: §I.3 · pp. 62–63

I.4 Layered photonic networks

  • In layered photonic networks, bilinear residual pairing yields transposed adjoint operators that a reciprocal device realizes through optical, pre-activation, and output-error variables.The optical adjoint field is accompanied by pre-activation error δℓ and output error γℓ.Source: §I.4 · p. 63
  • Physical backpropagation starts from the output error, pulls it through the activation, solves the adjoint field equation, and reads off the preceding layer’s error.The procedure is applied from layer L down to 1.Source: §I.4 · p. 63
  • The physical adjoint solve implements the transpose of the forward optical map in the same device, excited from the output side.The adjoint source is P^T_ℓδℓ.Source: §I.4 · pp. 63–64
  • When trainable parameters enter through the optical operators, their gradients reduce to local overlaps of forward and adjoint fields.For a local permittivity perturbation in a phase-shifter region, the overlap is measured locally because the operator perturbation is supported only there.Source: §I.4 · p. 64

I.5 Equivalence with scattering-matrix learning rules

The section identifies scattering-matrix learning rules as adjoint experiments and derives quantum equilibrium propagation as a tangent-space stationary-adjoint method. It also gives finite-nudge measurement considerations and a Taylor-test diagnostic for checking gradients.

I.5 Equivalence with scattering-matrix learning rules

  • The scattering rule of Wanjura and Marquardt is the adjoint identity evaluated at individual ports, using measured forward and adjoint fields from the same reciprocal device.The drive and readout each address one port, and the backward pass is a second scattering experiment on the same hardware.Source: §I.5 · pp. 64–65

I.6 Tangent-space adjoint for quantum equilibrium states

  • Quantum Equilibrium Propagation identifies ground-state observable responses with the constrained stationary adjoint and relates tangent reciprocity to the Onsager symmetry used in Ref. [31].Source: §I.6 · p. 65

I.6.1–I.6.3 State equation, readout, and projected adjoint

  • The normalized ground-state residual is nonlinear through its Rayleigh quotient; fixed-observable loss signals source a tangent-projected adjoint with reciprocal projected Jacobians.Source: §I.6.1 · p. 65, §I.6.2 · p. 66, §I.6.3 · p. 67

I.6.4 Spectral-gap invertibility

  • A positive spectral gap makes the restricted tangent operator invertible and the adjoint unique, but does not establish exponential stability of physical ground-state preparation.Source: §I.6.4 · p. 67

I.6.5 The Hamiltonian nudge generates the adjoint

  • A Hamiltonian nudge generates the reduced adjoint equation, provided the nondegenerate ground state follows a smooth branch for sufficiently small nudges.Source: §I.6.5 · p. 68

I.6.6–I.6.7 Observable response and Onsager reciprocity

  • The QEP gradient is read from observable expectation changes, and the static susceptibility symmetry follows from tangent-space Onsager reciprocity.The response measurement does not require phase-resolved measurements of the state or adjoint.Source: §I.6.6 · p. 68, §I.6.7 · p. 69

I.6.8 Finite nudges and measurement noise

  • Smaller nudges reduce finite-difference bias but increase estimator variance, which scales as σ^2/(ϵ^2M); centered nudging requires two nudged settings plus the free preparation.Source: §I.6.8 · p. 70

J How to test a gradient: the Taylor test

  • In the nonlinear Schrödinger test, correct-gradient R1 falls with slope 2 over eight decades before round-off saturation, while breaking condition (A1) leaves slope 1.The displayed Taylor test contrasts the correct and condition-violating gradients against slope-1 and slope-2 references.Source: J How to test a gradient: the Taylor… · p. 70, Figure 4 · p. 71

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