Scientific machine learning is creating new opportunities for controlling systems governed by partial differential equations (PDEs). In this talk, we present a differentiable programming framework that combines Differentiable Predictive Control (DPC) with neural operators for end-to-end learning of feedback policies for PDE-constrained control. DPC reformulates parametric model predictive control as an offline gradient-based policy optimization problem, enabling policies to be optimized by backpropagating task objectives and constraint penalties through differentiable PDE solvers and surrogates [1]. To overcome the limitations of fixed-dimensional policy representations, we further cast PDE control as an operator learning problem that maps state fields to continuous control functions, enabling policies that naturally adapt to varying sensor, actuator, and multi-agent configurations [2]. Remarkably, policies trained on small agent populations exhibit cardinality invariance, enabling zero-shot transfer to significantly larger populations and robustness to partial agent failure. We empirically validate the framework on tracking, stabilization, and density transport across linear, nonlinear, chaotic, and turbulent PDE systems. References [1] Dibakar Roy Sarkar, Jan Drgoňa, Somdatta Goswami, Learning to Control PDEs with Differentiable Predictive Control and Time-Integrated Neural Operators, arXiv:2511.08992 2025 [2] Pietro Zanotta, Dibakar Roy Sarkar, Honghui Zheng, Somdatta Goswami, Jan Drgona, Cardinality-Invariant Neural Operator Policies for Scalable PDE Control, ICML, 2026