CINOC: Cardinality-Invariant Neural Operator Policies for Scalable PDE Control

Controlling partial differential equations (PDEs) with learning-based policies remains fundamentally limited by fixed-dimensional representations: policies trained for a specific sensor, actuator, or agent configuration typically fail when the configuration changes. This limitation is particularly severe in multi-agent PDE control, where policies do not scale across population sizes without retraining. We address this challenge by introducing Cardinality Invariant Neural Operator Control (CINOC), reformulating PDE control as an operator learning problem that maps state fields to continuous control functions and trains them end-to-end through differentiable PDE solvers, yielding policies that naturally adapt to varying sensor and actuator configurations. Remarkably, CINOC policies trained on small swarms exhibit cardinality invariance, allowing for zero-shot transfer to significantly larger populations as well as robustness to partial agent failure. This scalability arises from agents sharing a common policy and coordinating through their physical environment, which produces an emergent self-normalization effect. To explain this phenomenon, we provide a theorem grounded in mean-field theory demonstrating that policy gradients computed from finite-agent systems converge to those of a continuous control limit. Empirically, we validate CINOC on tracking, stabilization, and density transport across linear, nonlinear, chaotic, and turbulent PDEs.

Controlling complex physical systems using artificial intelligence is traditionally limited by rigid designs. Standard AI models are built for a specific, fixed number of acting agents. This rigidity becomes a critical flaw in real-world applications: if a controller is programmed for a network of fifty actuators, the system typically breaks down if a partial failure occurs and several units unexpectedly go offline or if the number of actuators is increased. Our approach, CINOC, resolves this by equipping all agents with a shared, continuous policy rather than relying on a centralized, fixed-size command structure. Because agents coordinate naturally by reacting to their shared physical environment, the system automatically adapts to any population size. To guarantee this scaling is reliable, we ground our framework in mean-field theory. This mathematical approach models the overall distribution of the group rather than tracking individual agents. It mathematically proves that the learning process for a small, discrete group perfectly aligns with the macroscopic behavior of a massive, continuous population. Consequently, a model trained on a small swarm can handle phased deployments or unexpected hardware losses without requiring any retraining.