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Physical Adjoint Learning in Wave Systems with Asymmetric Couplings

Lucas J. Fernández-Alcázar, John Guillamon, Krishna Joshi, Tsampikos Kottos

arXiv:2610.34815v1physics.opticscond-mat.dis-nncs.SYeess.SPeess.SYnlin.AO

TL;DR

In-situ adjoint methods can avoid reliance on accurate numerical models, but reciprocity-based implementations generally do not extend to nonreciprocal systems. This paper reconstructs adjoint fields for asymmetric wave systems through similarity transformations and forward experiments on the same device. The authors report broadband transmission with suppressed reflection, directional transport, and spatially selective sensing in optimized networks.

  • Problem

    Reciprocity-based in-situ adjoint methods generally do not apply to nonreciprocal systems, where the adjoint field cannot be directly accessed through the original device.

    Source: §I · p. 2

  • Method

    The framework uses similarity transformations between forward and transposed operators to reconstruct adjoint fields through forward experiments on the original device.

    Source: §I · p. 2

  • Results

    Optimized networks show broadband high transmission with suppressed reflection, a directional transmission window, and a sensitivity peak at the target site.

    Source: A. Broadband transmission engineering · p. 5, C. Broadband nonreciprocal transport… · p. 7, E. Spatially selective sensing in a… · p. 9

  • Takeaways & Limitations

    The protocol provides a framework for in-situ adjoint optimization of nonreciprocal wave-scattering systems, including systems with inaccessible or unknown parts.

    Source: §V · p. 11

    Applicability depends on the required transformations being compatible with available sources and measurements.

    Source: §V · p. 11

Abstract

from arXiv · show

Developing self-optimizing physical systems that learn directly from their response to external stimuli offers a promising route toward adaptive and intelligent machines. A key ingredient is the ability to evaluate optimization gradients within the physical hardware, bypassing the need for accurate numerical models. In wave systems, \textit{in-situ} adjoint methods provide this capability by exploiting reciprocity to physically realize the adjoint dynamics. However, this approach generally breaks down in nonreciprocal systems, where the adjoint field cannot be directly accessed through the original device. Here, we develop an \textit{in-situ} adjoint optimization framework that extends physical gradient computation to wave systems with asymmetric couplings through computationally low-cost similarity transformations between the forward and transposed operators. We utilize available degrees of freedom like tailored injected excitations, targeted measurements, and tunable parameters, to enable adjoint field reconstruction through forward experiments on the same physical platform. We demonstrate the versatility of the protocol through broadband transmission and isolation, coherent perfect absorption, programmable wavefront shaping, and spatially selective sensing. Our results provide a route to self-optimization in complex nonreciprocal wave systems through efficient physical gradient-based learning. The number of experiments required by the protocol does not increase with the number of optimization parameters, and the method remains applicable even when parts of the system are completely inaccessible.

I. INTRODUCTION

Complex wave scattering makes physical self-optimization a high-dimensional problem, while model-based gradients depend on accurate digital twins. The paper extends in-situ adjoint optimization to asymmetrically coupled systems by reconstructing adjoint fields through forward experiments on the same device.

I. INTRODUCTION

  • Adjoint sensitivity analysis obtains the full objective gradient from one forward and one adjoint solution, independent of parameter count, whereas finite differences scale with tunable parameters.Numerical forward and adjoint solutions require an accurate digital twin, which can be difficult to develop and can yield inaccurate gradients when the model differs from the device.Source: §I · p. 1
  • Reciprocity-based in-situ methods do not generally apply to nonreciprocal systems because their adjoint dynamics reverse asymmetric couplings and are not directly accessible on the original device.The protocol addresses this challenge without modifying the device structure during optimization.Source: §I · p. 2
  • The protocol reconstructs adjoint fields for asymmetric systems through forward experiments using similarity transformations, designed sources, and targeted measurements.It supports optimization without complete system modeling, including when parts of the system are inaccessible or unknown.Source: §I · p. 2, Figure 1 · p. 2
  • The framework applies to objectives spanning multiple frequencies, inputs, perturbations, or system configurations, and the paper demonstrates broadband transmission, isolation, coherent perfect absorption, wavefront shaping, and spatially selective sensing.It also demonstrates applicability to coupled optical ring resonators with asymmetric effective scattering equations.Source: §I · p. 2

II. PHYSICAL ADJOINT OPTIMIZATION IN ASYMMETRIC WAVE SYSTEMS

  • The formalism describes finite-dimensional scattering with an internal operator and external-channel coupling, and allows objectives over multiple system configurations.A configuration may represent a frequency, perturbation, input state, or boundary condition; the forward field is represented using the Green’s function.Source: §II · pp. 2–3

A. Adjoint gradients for multi-configuration objectives.

The section derives gradients for objectives spanning multiple configurations, then identifies when the adjoint fields can be physically reconstructed. Similarity transformations make that reconstruction possible through forward experiments when the transformed source is physically realizable.

A. Adjoint gradients for multi-configuration objectives.

  • The full gradient requires one forward and one adjoint solution per configuration, irrespective of the number of optimization parameters.Thus, computational cost scales with the number of configurations in the objective rather than parameter-space dimensionality.Source: A. Adjoint gradients for… · p. 3
  • In generic nonreciprocal systems, the adjoint field follows the transposed Green’s function, which need not match the forward propagator.The physical question is whether the adjoint evolution can be reconstructed through realizable operations.Source: A. Adjoint gradients for… · p. 3

B. Physical reconstruction of adjoint fields.

  • Transpose symmetry can make the adjoint propagator equal the forward propagator; under magnetic bias, reversing the bias can also realize adjoint propagation.These correspondences also apply to reciprocal gain or loss invariant under transposition; generic asymmetric systems need not have a direct correspondence.Source: B. Physical reconstruction of adjoint… · p. 3
  • When the internal operator and channel coupling meet the stated similarity-transformation conditions, the adjoint field can be obtained from forward propagation using a transformed source and a known transformation.The transformed source must be physically synthesizable for this protocol to be realizable.Source: B. Physical reconstruction of adjoint… · pp. 3–4

C. Diagonal similarity transformations: construction and applicability

Diagonal similarity transformations are useful for in-situ optimization because they act locally and can remain fixed while onsite parameters are tuned. Their construction and applicability depend on network topology and coupling conditions.

C. Diagonal similarity transformations: construction and applicability

  • A diagonal mapping rescales each site independently, allowing local probes to implement it through excitation and readout rescaling at accessible sites.It can remain unchanged under independent variation of every onsite term when internal couplings are fixed.Source: C. Diagonal similarity… · p. 4
  • For chains and trees, a nonzero root weight determines all other diagonal weights along unique paths, provided each bond has nonzero coupling in both directions.The weights follow from products of edge ratios.Source: C. Diagonal similarity… · p. 4
  • Networks with loops require consistent directional-coupling ratios around every independent cycle; these constraints restrict admissible couplings but leave onsite terms unrestricted.When the conditions hold, weights can be constructed along a spanning tree; increasing cycle count makes the constraints more restrictive.Source: C. Diagonal similarity… · p. 4
  • Fixed directional-coupling ratios allow the mapping to be constructed once for onsite optimization; if ratios are optimized, weights can be updated from their current values using path products, subject to loop consistency.This construction avoids solving a general matrix-similarity problem anew.Source: C. Diagonal similarity… · p. 4

III. SELF-OPTIMIZATION OF ASYMMETRIC WAVE NETWORKS

The section applies the in-situ adjoint formulation to asymmetric wave networks, including broadband transmission engineering. In the lattice example, adding spectral derivative constraints produces broader, flatter transmission alongside suppressed reflection.

A. Broadband transmission engineering

  • The lattice objective compares target-frequency transmission with objectives that incorporate neighboring frequencies and penalties on first- and second-order spectral derivatives.The system is an asymmetric tight-binding chain with optimized onsite potentials and weak random onsite losses.Source: A. Broadband transmission engineering · pp. 4–5
  • Adding first- and second-order spectral derivative constraints produces a broad, flat high-transmission plateau with strongly suppressed reflectance across the same frequency range.The target-frequency-only objective gives a narrow resonant peak; adding first- and then second-order constraints progressively broadens and flattens the response.Source: A. Broadband transmission engineering · p. 5, Figure 2 · p. 5
  • Near the target frequency, two eigenvalues of the auxiliary Hamiltonian approach one another, indicating a near second-order reflectionless degeneracy.The authors identify this proximity as a mechanism for suppressing reflection and enhancing transmission over a finite spectral interval despite disorder and asymmetry.Source: A. Broadband transmission engineering · p. 5

B. Coherent perfect absorption

The asymmetric-lattice optimization approaches coherent perfect absorption in an unknown absorbing subsystem. In coupled asymmetric rings, the protocol produces broadband directional transport and preserves strong isolation even when control is restricted to one ring.

B. Coherent perfect absorption

  • The asymmetric lattice approaches near-perfect absorption by jointly optimizing its onsite potentials and second coherent source while the first source remains fixed.The optimized field builds up near the unknown absorbing site Nabs = 15, directing energy toward the target region.Source: B. Coherent perfect absorption · p. 6
  • The CPA state combines coherent interference, asymmetric transport, and adaptive lattice optimization to funnel and trap wave energy in the prescribed absorbing region.The optimization minimizes g = R11 + TN1, thereby maximizing absorption α = 1 − R11 − TN1.Source: B. Coherent perfect absorption · p. 6

C. Broadband nonreciprocal transport in coupled rings

  • T ←≈4.9 and T →≈0.06 near ω = 0, with low reflection over the optimized bandwidth, show broadband isolation close to the preferred-direction target Ttar = 5.The coupled-ring network satisfies cycle-consistency constraints for diagonal similarity.Source: C. Broadband nonreciprocal transport… · p. 7, Figure 4 · p. 7
  • In the coupled-ring network, the optimized onsite potentials develop a nonuniform profile, suggesting that the broadband response relies on collective interference rather than changes localized to one region.The objective converges while cycle-consistency constraints are maintained.Source: C. Broadband nonreciprocal transport… · pp. 6–7
  • Control of only six of eighteen onsite potentials still yields a broadband transmission window and strong isolation, although the maximum transmittance falls to T ←≈0.27.The reported counterpropagating transmission is T ←≫T →≈0.003.Source: C. Broadband nonreciprocal transport… · p. 7, Figure 4 · p. 7

D. Coherent wavefront shaping in asymmetric trees

In asymmetric tree networks, the protocol shapes output intensity and phase, including multi-focus wavefronts. In a Hatano–Nelson chain, it enhances sensitivity at a target site while suppressing response over most of the penalized region, though boundary sensitivities persist.

D. Coherent wavefront shaping in asymmetric trees

  • Tree topology removes diagonal-similarity cycle-consistency constraints, allowing asymmetric couplings to vary independently when both directional couplings remain nonzero.The similarity transformation is updated from the instantaneous hopping amplitudes as coupling ratios change; control can also be restricted to the last two layers.Source: D. Coherent wavefront shaping in… · pp. 7–8
  • The optimized tree-network wavefront closely matches both a triangular target intensity and a linear phase profile, and also reproduces prescribed phases at three focal positions.The three foci are at n = (5, 15, 27), with target phases θm = (0, π/2, π).Source: D. Coherent wavefront shaping in… · pp. 8–9, Figure 5 · p. 8

E. Spatially selective sensing in a Hatano-Nelson chain

  • Sensitivity is probed by applying two distinct local perturbations at each site and measuring transmission from the left input to the right output.The objective maximizes sensitivity at target m while penalizing off-target response outside a five-site window centered on m.Source: E. Spatially selective sensing in a… · p. 9
  • The optimized Hatano–Nelson chain has a pronounced sensitivity peak at the target site and comparatively low response over most of the penalized region.Sizeable sensitivities nevertheless persist near the lead-coupled boundaries, consistent with the chain’s strong boundary sensitivity.Source: E. Spatially selective sensing in a… · p. 9, Figure 6 · p. 9

IV. EXTENDING THE ADJOINT OPTIMIZATION TO ASYMMETRIC SCATTERING OPERATORS

The section extends in-situ adjoint optimization to asymmetric scattering operators by reconstructing adjoint responses from physically accessible forward responses. In a coupled-ring example, approximate gradients improve transmission across network sizes despite discrepancies from the full-matrix reference.

IV. EXTENDING THE ADJOINT OPTIMIZATION TO ASYMMETRIC SCATTERING OPERATORS

  • For N = 3, 5, and 7, the proposed method increases transmission and provides effective optimization directions despite gradient discrepancies from the full-matrix reference.The figure compares optimization trajectories, transmission evolution, and gradient discrepancies; the N = 5 and N = 7 landscapes show two-dimensional cross sections.Source: Figure 7 · p. 10, §IV · p. 11
  • Eliminating complementary ring fields makes the exact reduced operator asymmetric and upper-Hessenberg, with longer-range elements generated by successive scattering events.These higher-order terms are not direct non-neighbor couplings in the physical network.Source: §IV · p. 10
  • The ring-network objective is output transmission, optimized by tuning inter-resonator coupling coefficients, while coupling coefficients are tunable and cavity lengths and losses are fixed.The Lumerical implementation uses N = 3, 5, and 7 rings; weakly coupled probes monitor intracavity amplitudes.Source: §IV · pp. 10–11
  • In the weak-coupling regime, a tridiagonal approximation admits a diagonal similarity transformation, reconstructing the adjoint response from forward measurements by site-dependent rescaling.The transformation is approximate for the full reduced operator, but its resulting gradients still provide effective optimization directions.Source: §IV · p. 11
  • Successful ring-network optimization supports applying the methodology to systems described by asymmetric reduced operators.Source: §IV · p. 11

V. CONCLUSIONS

The conclusions present a same-platform protocol for reconstructing adjoint fields in asymmetrically coupled wave networks. They summarize its implementation conditions, parameter-dependence considerations, and demonstrated optimization tasks.

V. CONCLUSIONS

  • The protocol reconstructs adjoint fields through designed forward experiments on the same platform, using a low-cost diagonal similarity transformation derived from directional-coupling ratios.The conclusions distinguish parameters that leave the transformation unchanged from directional couplings that require updating it at every iteration.Source: §V · p. 11
  • Its stated viability conditions cover chains, trees, networks with loops, and systems with unknown portions, subject to compatible sources and measurements.Source: §V · p. 11
  • Demonstrated tasks include broadband transmission and isolation, coherent perfect absorption with an unknown or inaccessible absorber, wavefront shaping, and spatially selective hypersensitive sensing.Source: §V · p. 11

A. Adjoint sources for spectrally constrained objectives

For objectives combining measurements across frequencies, each sampled frequency contributes an adjoint source. The section specifies source construction for first- and second-order spectral objectives using forward measurements.

A. Adjoint sources for spectrally constrained objectives

1. Objective involving a first-order spectral derivative.

  • For a first-order spectral derivative objective, adjoint sources are constructed from complex transmitted amplitudes measured in forward experiments at ω0 and ω0 + δω.The sources are recalculated after each pair of forward measurements.Source: §1 · p. 14

2. Objective involving second-order spectral derivatives.

  • For a second-order spectral derivative objective, three forward scattering measurements at ω0 − δω, ω0, and ω0 + δω supply the quantities for constructing adjoint sources.The first-derivative penalty alters the two side-frequency sources with opposite signs, while the second-derivative penalty modifies all three; no direct spectral-derivative measurement is required.Source: §2 · pp. 14–15, Figure S1 · p. 15

B. Scattering formulation for a chain of coupled ring resonators

The scattering formulation reduces a coupled-ring chain to equations for one retained field amplitude per resonator. Eliminating complementary fields yields an exact asymmetric operator whose longer-range terms arise from multiple scattering.

B. Scattering formulation for a chain of coupled ring resonators

  • Eliminating complementary amplitudes ηn from the coupler and boundary relations yields a reduced scattering problem for the retained resonator fields.The formulation generalizes from three rings to an arbitrary chain of N resonators.Source: B. Scattering formulation for a chain… · pp. 15–16
  • The exact reduced operator is upper-Hessenberg, with longer-range elements generated by successive cross-coupling events rather than direct non-neighbor couplings.For example, the three-resonator element A13 is proportional to κ1κ2.Source: B. Scattering formulation for a chain… · p. 16
  • Neglecting products of two or more cross-coupling amplitudes while retaining through-coupling coefficients reduces the problem to a tridiagonal matrix with nearest-neighbor interactions.Source: B. Scattering formulation for a chain… · p. 16

C. General similarity transformations and experimental accessibility

The section establishes when general similarity transformations can reconstruct adjoint fields through forward experiments on the original device. Complete channel access guarantees a dense transformation, while practical implementations require accessible sources and measurements and benefit from transformations fixed under tuning.

C. General similarity transformations and experimental accessibility

  • When onsite resonant frequencies are training parameters, fixed transformations are diagonal, reducing adjoint-field reconstruction to local rescaling.This is a practical special case of the broader transformation conditions.Source: C. General similarity transformations… · p. 16
  • An in-situ mapping requires that the transformed adjoint source be generable and the gradient-relevant adjoint field components measurable; invariance of D^T D lets the internal-operator relation extend to the full scattering operator.The mapping must also remain fixed as control parameters are tuned for efficient optimization.Source: C. General similarity transformations… · p. 16

1. Complete channel access

  • With complete channel access, a dense Ξ is guaranteed, including for networks that violate cycle consistency, and coherent multichannel excitation implements the auxiliary field.With one channel per site, any adjoint source can be constructed and the full auxiliary field measured.Source: §1 · p. 17

2. Structural symmetry with partial access

With partial channel access, a similarity transformation must preserve accessible and inaccessible sites so the required adjoint source does not demand excitation at inaccessible locations. Such transformations can be structural permutations, but their validity restricts which parameters may be tuned independently; arbitrary independent onsite tuning instead requires diagonal mappings.

2. Structural symmetry with partial access

  • With partial access, a dense mapping can require excitation at inaccessible sites; a sufficient remedy is a mapping that keeps accessible and inaccessible sites separate.The objective must depend only on fields at accessible sites for the transformed source to remain supported there.Source: §2 · p. 17
  • A site permutation can reconstruct the adjoint field using channel measurements alone when it preserves each access set and maps operator entries to their transposed counterparts.The resulting permutation matrix satisfies Ξ−1AΞ = A^T.Source: §2 · p. 17
  • Permutation-based mappings restrict optimization parameters: paired sites must share resonance frequencies and intrinsic decay rates, and the permuted directed couplings must match their reverse-direction counterparts.These quantities must therefore be tuned together rather than independently.Source: §2 · p. 17

3. Fixed mappings under independent onsite tuning

  • If every onsite term can be varied independently, a fixed similarity transformation must be diagonal; independent tuning at only some sites forces the corresponding rows and columns to have no off-diagonal entries.Diagonal mappings satisfying the base-operator relation remain valid throughout arbitrary onsite tuning.Source: §3 · p. 18

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