Beyond Forecast Accuracy: Evaluating Neural Operators for Differentiable PDE Control

Neural operators learn mappings between function spaces and are inexpensive to evaluate and differentiate, making them a natural choice of surrogate for controlling systems governed by partial differential equations (PDEs), where repeated numerical solves render online optimization expensive. Differentiable predictive control (DPC) exploits both properties: it trains a neural feedback policy offline by differentiating a control objective through surrogate rollouts, so deployment reduces to a single forward pass. The surrogate thus plays a dual role, predicting the consequences of actions and supplying the sensitivities that drive policy updates. Our framework trains a neural-operator surrogate on open-loop trajectories, freezes it, optimizes the policy through predicted closed-loop rollouts, and deploys it on the numerical solver. We compare Fourier, Recurrent, and Time-integrated DeepONet operators with single- and multi-step training on stabilization across linear, nonlinear, dispersive, and chaotic PDEs. We find that forecast accuracy alone is an insufficient proxy for controller quality: surrogates with comparable prediction errors can yield policies that differ by orders of magnitude in terminal tracking error. A policy-dependent transfer bound and a counterexample in which vanishing forward-model error coexists with persistent policy-gradient error formalize why this can occur. With multi-step training, DPC outperforms model-free reinforcement learning while avoiding the online cost of model predictive control.