Plasma feedback control drops full phase-space sensing: macroscopic currents and field gradients alone hold instabilities down under substantial delay

Feedback control of a kinetic plasma can now run on macroscopic observables alone, without observing or reconstructing the full phase-space distribution at deployment. An earlier operator-based feedback framework took the full phase-space perturbation as the controller input, while field-cancellation schemes from linear analysis fixed the control fields from the initial data and did not update them from the evolving state. In numerical experiments on two-stream and Weibel-type current-filamentation instabilities, the learned cancellation–dissipation controller substantially suppresses growth of the kinetic perturbation and electromagnetic energy under substantial feedback delay and remains effective well beyond the optimization horizon.

The controller combines an instantaneous cancellation–dissipation feedback law derived from the nonlinear perturbation balance with a learned residual term that compensates delay errors. Pure field cancellation works with instantaneous observations, but evaluating the cancellation from delayed observations leaves a residual that noticeably weakens suppression. Maxwell's equations identify the delay-induced cancellation errors as governed by currents and spatial gradients of the transverse fields; these quantities feed a low-rank linear neural operator whose kernels are parameterized by neural networks and trained with an adjoint-state method on kinetic simulations.

The choice of feedback inputs, not only the network architecture, shapes long-time stabilization. A control variant using electromagnetic field components as inputs also suppresses the perturbation at short times but degrades significantly beyond the optimization horizon. In both the two-stream and Weibel-type current-filamentation cases, keeping network architecture and training settings unchanged and replacing only the last two feedback variables, the gradient-based inputs maintained substantially stronger long-time suppression across the two instability mechanisms.

The demonstrated scope is a one-dimensional-in-space, two-velocity Vlasov–Maxwell reduced model with periodic boundaries, where the control objective is to stabilize the distribution toward a prescribed spatially homogeneous equilibrium; the two test cases are the two-stream instability and a Weibel-type current-filamentation instability, and the conclusions come from numerical experiments rather than experimental measurement.

Under delay mismatch, the learned dissipation underestimates the error and compensates insufficiently when the actual delay is smaller than the training reference, and can overcompensate and degrade stability at larger delays; rescaling the dissipation network output by the delay ratio mitigates this without retraining. Linear and nonlinear operator architectures perform comparably at the reference delay, and the difference becomes more pronounced under delay mismatch, with the precise source not yet clear and possibly reflecting both architectural and optimization effects. Spatial differentiation can amplify measurement noise, and noisy observations with regularized derivative estimation are not yet part of training.