Unifying Physical Backpropagation

This paper develops a unifying theoretical framework for formally exact physical backpropagation methods based on the adjoint method from PDE-constrained optimization. The authors state: "The main contribution of this work is twofold. First, we show that seemingly disparate physical backpropagation algorithms are instances of a single adjoint construction: platform-specific methods become one family in a common language that lets those algorithms be compared and extended. Second, we identify sufficient conditions under which the required adjoint field—and hence the cost function gradient—can be obtained from a physical experiment that runs in the same system."

The only allowed modifications between forward and adjoint experiments are changes of initial conditions, external forces, and reversal of explicit time dependences.

The paper considers N-dimensional discretized systems governed by ODEs, with physical state u(t) ∈ R N and trainable parameters p ∈ R M. The cost function is:

χ[u] = ∫0T θ[Pu(t)] dt + ψ[Pu(T), Pu̇(T)]

where P projects onto measured output degrees of freedom. The forward dynamics for a second-order real-valued system is:

M(p M)ü(t) + D(p D)u̇(t) + F[u(t), p F, t] − f(t, p f) = 0

with initial conditions u(0) = u0(p u0), u̇(0) = v0(p v0).

The adjoint field a(t) satisfies (in forward time):

MTä(t) + DTȧ(t) + FT u[u(T−t), p F, T−t]a(t) + f adj(t) = 0

with adjoint source f adj(t) = Pθ u[Pu(T−t)] and adjoint initial conditions MTa(0) = −Pψ u̇[Pu(T), Pu̇(T)] and MTȧ(0) = −DTa(0) − Pψ u[Pu(T), Pu̇(T)].

The gradient is obtained from overlap integrals, e.g., dχ/dp F = ∫0T ⟨a(T−t), F p F[u(t), p F, t]⟩ dt.

The authors note: "The advantage of the adjoint method in digital optimization is that the cost of this gradient is essentially independent of the number of parameters... The adjoint method, by contrast, requires only two trajectory solves regardless of how many parameters are trained."

Reciprocity is sufficient. For linear systems, the paper states: Reciprocal linear systems admit direct, finite-amplitude adjoint experiments, and damping or gain is compatible with exact physical gradients as long as reciprocity is preserved.

For second-order systems, the conditions are: MT = M, DT = D, and K(p K, t)T = K(p K, t). The adjoint experiment has exactly the same damped second-order form as the forward experiment.

For first-order systems (u̇ + Ku − f = 0), reciprocity KT = K suffices. The authors note: To the best of the authors' knowledge this is the first theoretical demonstration that in linear, reciprocal but overdamped dynamics not only fixed points but also full trajectory gradients can be obtained physically.

For Schrödinger-type systems (iu̇ + Ku − f = 0), the condition is K(p K, t)T = K(p K, t) — symmetry in the reciprocal sense, not Hermiticity: loss and gain are compatible with the physical adjoint as long as reciprocity is preserved.

Nonlinear systems are more restrictive: they require infinitesimal nudging, multiple experiments, a particular form of time reversal symmetry and reciprocity of the linearized dynamics—the latter two are independent conditions.

For second-order systems, the conditions are: D = 0, MT = M, and FT u[u(t), p F, t] = F u[u(t), p F, t] for all t ∈ [0,T]. The adjoint is obtained as the infinitesimal response of a nudged time-reversed trajectory.

The physical algorithm: "One first runs the forward dynamics and records u(t); the time-reversed trajectory w(t) = u(T−t) is then available directly from this recording and need not be regenerated physically. The adjoint requires only one further experiment, in which the TRM and the nudge are applied together."

The adjoint field is recovered as a(t) = lim ϵ→0 [w ϵ(t) − w(t)]/ϵ, where w ϵ is the nudged trajectory. Symmetric nudging with two experiments gives O(ϵ2) bias instead of O(ϵ).

The damping obstruction: Ordinary damping or gain is generically obstructive in the nonlinear case. The paper shows that even granting a gain-flipped run (D → −D), the perturbation dynamics differs from the adjoint equation by 2Dȧ, so the same hardware cannot simultaneously reproduce the reversed nonlinear background trajectory and propagate the correct adjoint perturbation around it, except in the undamped case.

For nonlinear first-order (overdamped) systems: trajectories are obstructed, the steady state is not. Time reversal requires F → −F while the adjoint requires damping.

For nonlinear Schrödinger systems, the conditions are PT-symmetry along the forward trajectory and reciprocity of the parity-pulled-back Wirtinger Jacobians: ΠF u[w, w̄, p F, t]Π = F u[u, ū, p F, t]T and ΠF ū[w, w̄, p F, t]Π = F u[u, ū, p F, t]T.

For linear stationary problems (Ku* = f), reciprocity KT = K suffices: no nudging is required, the adjoint source is applied at finite amplitude in a single experiment, and damping or gain are permissible: reciprocity, not Hermiticity, is what is asked of the hardware.

For nonlinear stationary problems, the condition is reciprocity of tangent operators at the steady state: F u[u*, p F]T = F u[u*, p F] and F ū[u*, p F]T = F u[u*, p F]. This recovers Equilibrium Propagation: a = lim ϵ→0 (u* ϵ − u*)/ϵ, where u* ϵ is the steady state under an adjoint-source nudge.

The paper generalizes reciprocity using an Onsager mobility V = VT ≻ 0: We call A V-reciprocal if it is self-adjoint in this Onsager metric, VATV−1 = A. With V = I, this reduces to Euclidean reciprocity.

For example, in resistor–capacitor networks with heterogeneous capacitances, K = C−1G is not Euclidean-symmetric but is V-reciprocal with V = C−1. The hardware propagates the V-weighted adjoint field d = Va, and the Euclidean adjoint is recovered as a = V−1d.

The paper shows reciprocity is only the simplest instance of a more general intertwining condition. The sufficient condition is: there exists a constant invertible matrix S such that S K(p K, t)T S−1 = K(p K, t) for all times and parameter values.

This recovers Fully Forward Mode (FFM) training as a special case: Reciprocity does the existence work and the symmetry performs the relocation of the injection ports to the experimentally convenient side of the device. The paper extends FFM from static optics to time-dependent trajectory learning.

For non-reciprocal systems satisfying this condition (called twisted reciprocal), the paper demonstrates exact physical gradients in a Hatano–Nelson chain with asymmetric hopping: "For h ≠ 0 rightward hops are amplified and leftward hops attenuated. Hence, KT ≠ K... The spatial inversion resolves this: transposition reverses the arrow of every hop and the inversion reverses it again, so condition (81) holds, provided the on-site potential is mirror-symmetric."

The framework recovers: "Equilibrium Propagation [30], Hamiltonian echo backpropagation and its recurrent variants [26, 31], the Fully Forward Mode (FFM) algorithm [32], early reciprocity-based recurrent learning [20], and physical gradient algorithms from integrated photonics [23, 33], free-space and diffractive optics [21, 22, 34–36], linear wave scattering [29, 37], static and topological mechanics [24, 38], and the projector-based analytical gradient for passive resistor networks [25]."

When parameters enter linearly, "the gradient overlaps are assembled entirely from measured forward and adjoint trajectories and never reference the parameter values themselves. In an experiment the gradient is thus evaluated at whatever parameter values the device happens to sit at, so slow drift, aging, and unknown calibration offsets are absorbed automatically instead of biasing the gradient estimate."

However, knowing which observable to record, and which overlap integral belongs to which parameter, is a statement about the structure of F and therefore still requires a model.

The authors conclude: "Because on-device gradients are far easier to obtain in linear than in nonlinear hardware, encoding the inputs in the parameters of a linear system, a scheme known as structural nonlinearity [53–55], is a particularly attractive route to physical computing: the device computes nonlinearly in its inputs, yet its adjoint field, and with it the gradient, remains accessible at finite amplitude and with damping present [37]."

The paper also notes: The non-autonomous generalization makes the temporal modulation of the internal forces itself a trainable resource and may thereby enable novel ways of computing with physical systems.

Improvement 1: Physics-Aware Gradient Computation for Neural ODEs and Differentiable Simulators

The improved AI system can compute exact gradients for neural ODEs and differentiable physics simulators by automatically detecting whether the system satisfies reciprocity (e.g., symmetric mass, damping, and stiffness matrices) or twisted reciprocity (e.g., via a constant intertwiner S). If satisfied, the system uses a single physical adjoint experiment with finite-amplitude sources, eliminating the need for backpropagation through time or checkpointing. This reduces memory and compute costs by orders of magnitude for long-horizon trajectory optimization, enabling training of recurrent physical models on sequences of length 105+ without gradient explosion or memory blowup.

Improvement 2: Self-Calibrating On-Device Learning for Analog and Neuromorphic Hardware

The improved AI system can train neural networks deployed on analog or neuromorphic chips (e.g., resistive RAM, photonic processors) using local, on-device gradient rules derived from the paper’s overlap integrals. Because these gradients are evaluated at the device’s current parameters, the system automatically compensates for drift, aging, and manufacturing tolerances—no calibration needed. It can also handle non-reciprocal hardware by applying the twisted reciprocity condition (e.g., using spatial inversion to make a Hatano–Nelson chain trainable), enabling exact gradient descent on devices previously considered untrainable.

Improvement 3: Damping-Tolerant Training for Nonlinear Physical Neural Networks

The improved AI system can train nonlinear physical networks (e.g., mechanical, electrical, or optical) even in the presence of damping or gain, by switching between two modes: (a) for linear systems, it uses finite-amplitude adjoint experiments with damping present; (b) for nonlinear systems, it uses the paper’s infinitesimal nudging method with time-reversed trajectories, but only when damping is zero. To handle damped nonlinear systems, the system automatically reformulates the task as a stationary problem (e.g., via structural nonlinearity) or uses a gain-flipped run with a correction term, thereby extending exact physical training to realistic, lossy hardware.

Improvement 4: Unified Training Framework for Hybrid Digital-Physical AI

The improved AI system can train models that combine digital neural networks with physical computational substrates (e.g., optics, mechanics, electronics) by using the paper’s adjoint framework as a common language. It automatically chooses the optimal gradient strategy per subsystem: digital parts use backpropagation, linear physical parts use finite-amplitude adjoints, and nonlinear physical parts use nudged time-reversal (if undamped) or stationary equilibrium propagation. This enables end-to-end training of hybrid systems with mixed parameter types, such as a digital encoder feeding a photonic linear processor and a mechanical nonlinear layer, with exact gradients and minimal hardware experiments.

Improvement 5: Time-Modulated and Non-Autonomous Physical Learning

The improved AI system can exploit time-varying internal forces (e.g., modulated springs, optical potentials, or electrical impedances) as trainable parameters, using the paper’s non-autonomous adjoint generalization. This allows the system to learn dynamic control policies directly on physical hardware—e.g., training a robotic arm’s joint stiffness profiles or a metamaterial’s time-dependent elastic moduli—by running just two experiments (forward and adjoint) per gradient step, regardless of the number of time-varying parameters. The system can also discover novel computational primitives by treating temporal modulation as a resource, potentially enabling new forms of physical reservoir computing or time-multiplexed neural networks.

Improvement 6: Robust Training Under Model Mismatch and Partial Observability

The improved AI system can train physical models even when the exact dynamics are unknown, as long as the reciprocity structure is known (or inferred). It uses the paper’s insight that gradient overlaps depend only on measured trajectories, not on parameter values, so the system can learn from noisy or partial observations of the physical state (via the projection operator P). This enables training of black-box physical systems (e.g., biological networks, fluidic devices) where only a few output channels are measurable, by constructing the adjoint source from those outputs and recovering gradients without a full model—only a structural model of which parameters affect which forces is needed.