Pietro Zanotta

Pietro Zanotta, Dibakar Roy Sarkar, Honghui Zheng et al.

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.

Pietro Zanotta, Panos Stinis, Ján Drgoňa

Certifying the Region of Attraction (ROA) for high-dimensional nonlinear dynamical systems remains a severe computational bottleneck. Traditional deterministic verification methods, such as Sum-of-Squares (SOS) programming and Satisfiability Modulo Theories (SMT), provide hard guarantees but suffer from the curse of dimensionality, typically failing to scale beyond 20 dimensions. To overcome these limitations, we propose SCORE, a statistical certification framework that shifts from seeking deterministic guarantees to bounding the worst-case safety violation with high statistical confidence. By integrating Projected Stochastic Gradient Langevin Dynamics (PSGLD) with Extreme Value Theory (EVT), we frame ROA certification as a constrained extreme-value estimation problem on the sublevel set boundary. We theoretically demonstrate that modeling the optimization process as a stochastic diffusion on a compact manifold places the local maxima of the Lyapunov derivative into the Weibull maximum domain of attraction. Since the Weibull domain features a finite right endpoint, we can compute a rigorous statistical upper bound on the global maximum of the Lyapunov derivative. Numerical experiments validate that our EVT-based approach achieves certification tightness competitive to exact SOS programming on a 2D Van der Pol benchmark. Furthermore, we demonstrate unprecedented scalability by successfully certifying a dense, unstructured 500-dimensional ODE system up to a confidence level of 99.99\%, effectively bypassing the severe combinatorial constraints that limit existing formal verification pipelines.