Near-Equilibrium Propagation training in nonlinear wave systems

I. INTRODUCTION

• The method addresses difficulties in physical-network training: backpropagation assumes an explicit computational graph and accurate adjoint dynamics, while nonlinear complex-valued dynamics couple amplitude and phase to gain and loss.These assumptions can be fragile in analog hardware with hidden couplings, latency, and dissipation.Source: §I · p. 1
• Equilibrium Propagation estimates gradients locally by contrasting free and nudged steady states, and has been explored across several network types and physical platforms.Its experimental demonstrations include mechanical, resistor, and memristor networks.Source: §I · p. 1
• NEP extends Equilibrium Propagation to weakly pumped and dissipative oscillatory or wave systems, trains local parameters instead of inter-node weights, and supports discrete and continuous complex-valued fields.This broadens its application to physical systems without well-defined nodes.Source: §I · p. 1
• Numerical studies demonstrate stable NEP training on XOR and MNIST using a driven–dissipative Gross–Pitaevskii model of exciton-polariton condensates.The model includes pumping geometry and output nudging.Source: §I · p. 2

II. NEAR-EQUILIBRIUM PROPAGATION IN COMPLEX-VALUED SYSTEMS.

• NEP assumes the free dynamics reach an oscillating steady state whose frequency is locked to the resonant drive, independent of parameters and inputs.In the rotating frame, this becomes a stationary fixed point.Source: §II · p. 2
• Training uses a free phase followed by a target-nudged phase, with the nudged perturbation confined to the output region.The cost measures output–target discrepancy, and the perturbation is proportional to its gradient.Source: §II · p. 2
• A small imaginary-factor perturbation based on the cost’s Wirtinger derivative produces the nudged steady state used for in-situ gradient descent.The paper focuses on amplitude outputs, although complex-field outputs can depend on both amplitude and phase.Source: §II · p. 2
• Parameter updates follow the steepest cost gradient and can be applied per sample, batch, or epoch.The cited derivations cover continuous and discrete cases.Source: §II · p. 2
• NEP generalizes to complex wave equations without requiring an energy functional, provided the dynamics’ derivatives satisfy near-equilibrium conditions.Equilibrium here means a stationary fixed point; the stated derivative condition may follow from a Hamiltonian functional, and training can still converge when it is not exactly met.Source: §II · p. 2

III. IMPLEMENTATION IN EXCITON-POLARITON SYSTEMS.

• The setup uses resonant beams for input pumping and output nudging, while an SLM controls the spatial potential landscape.Trainable parameters include software input weights and the physical potential.Source: §III · p. 3
• The generalized continuous Gross–Pitaevskii model includes dissipation, complex potential, resonant pumping, and density- or saturation-dependent nonlinear response.The complex potential and pumping weights are the trainable parameters; NEP remains accurate when dissipation and the potential’s imaginary part are small relative to other terms.Source: §III · p. 3
• Potential and pump-weight updates use local steady-state fields from the free and driven phases, with nudging tuned to balance derivative accuracy and measurable wavefunction change.The nudging strength must be small enough to approximate the derivative at zero but large enough to produce a detectable change.Source: §III · p. 3

• For XOR inputs (00, 01, 10, 11), the trained output intensities are |Ψ4|2 = (0.00, 0.92, 1.06, 0.01).Both the potential V and pump weights w are trained.Source: §IV · p. 3, Figure 3 · p. 4
• XOR training converges after about 10 epochs, and small fixed random potential fluctuations do not affect convergence.The reported setup trains both V and w, while Appendix B also reports effective optimization with only V trained and w fixed.Source: §IV · p. 4
• The MNIST network reaches ≈90.3% test accuracy after about 20 epochs on 1,000 samples per digit.The 15 × 150 grid replicates each rescaled image across ten 15 × 15 cells, with ten central output nodes representing the digits.Source: §IV · p. 4, Figure 4 · p. 4
• Varying the nonlinearity g identifies a moderate g = 0.04 as optimal, corresponding to the mean nonlinearity-to-loss ratio ⟨g|Ψ|2⟩; larger g values tend to destabilize learning.Source: §IV · p. 4
• At ℏγ ≈0.3, validation accuracy is ≈91%; the backpropagation-trained linear benchmark reaches ≈89.9% on the same MNIST test subset.The reported accuracy also varies with the mean nonlinearity-to-loss ratio.Source: §IV · p. 4, Figure 5 · p. 5

• Tphys ≈ 0.1 ms to 10 ms is the estimated full training time for typical γ−1 ≈1 ps to 100 ps.The estimate is for the MNIST task with 1,000 samples per digit.Source: §V · p. 5
• The estimate uses two steady-state measurements per sample, each requiring ≈5000 RK4 steps with dt = 0.1.Source: §V · p. 5
• The estimated training time is nearly six orders of magnitude faster than GPU-based numerical integration on an NVIDIA GeForce RTX 2080 Ti.Source: §V · p. 5

• The continuous-variable derivation applies the chain rule for functional derivatives to relate cost changes to a trainable parameter θ(⃗r).The derivation concerns an arbitrary complex wave system.Source: Appendix A · p. 6
• The derivation tracks how variations in β and θ propagate through steady-state fields via the system’s linearized response near equilibrium.Source: Appendix A · pp. 6–7
• The stated update-rule condition for the polariton GPE holds when the imaginary potential and dissipation γ are negligible in the near-equilibrium regime.The passage attributes this condition to spatial symmetry of the Liouvillian and diagonal Jacobians.Source: Appendix A · p. 7
• For arbitrary nonlinearity f(Ψ), the appendix states that condition (A7) remains valid under an additional condition, but the supplied passage does not specify it.Source: Appendix A · p. 7

Appendix B: Comparison with the vector-field Equilibrium Propagation

• Directly applying Ref. [40] to the real and imaginary parts of a wave function does not produce a useful update rule for complex wave systems.The vector-field derivation requires a symmetric Jacobian.Source: Appendix B · pp. 7–8
• For real V, the GPE has opposite-sign mixing terms between real and imaginary components, unlike the symmetric coupling matrix assumed in Ref. [40].This sign difference prevents the required symmetric-Jacobian condition.Source: Appendix B · pp. 7–8
• NEP nudges each component according to the error in its conjugate variable, a prescription that differs from Ref. [40] and yields a symmetric Jacobian for conservative wave equations.The real component responds to the imaginary-output error, and the imaginary component responds oppositely to the real-output error.Source: Appendix B · p. 8

XOR task

• With near-optimal fixed pump weights, training only V minimizes loss and reproduces the XOR outputs, though convergence saturates later than when both V and w are optimized.The fixed pump weights are w2 = 1 and w6 = −0.75.Source: Appendix C · p. 8, Figure 7 · p. 8
• Jointly training V and w still performs the XOR mapping with inputs at sites 1 and 3 and output at site 5 in a reduced 7-node network.The asymmetric placement shows that spatial symmetry of inputs is unnecessary for this task.Source: Appendix C · p. 8, Figure 7 · p. 8

MNIST task

• ≈80% test accuracy with both V and w trained versus ≈53% with V-only training on MNIST demonstrates purely optical optimization.The task uses MNIST after PCA on a 5 × 50 polariton grid.Source: Appendix C · p. 8
• Figure 6 reports ≈80% validation accuracy with both V and w trainable and ≈53% when w is fixed to unities.The plot tracks validation accuracy and CCE loss over 50 training epochs.Source: Figure 6 · p. 8