We present a data-driven control framework for partial differential equations (PDEs). Our approach integrates Time-Integrated Deep Operator Networks (TI-DeepONets) as differentiable PDE surrogate models within the Differentiable Predictive Control (DPC)-a self-supervised learning framework for constrained neural control policies. The TI-DeepONet architecture learns temporal derivatives and couples them with numerical integrators, while the DPC algorithm uses automatic differentiation to compute policy gradients by backpropagating the expectations of the optimal control loss through the learned TI-DeepONet. This approach enables efficient offline optimization of neural policies without the need for online optimization or supervisory controllers. We empirically demonstrate the proposed method across diverse PDE systems, including the heat, the nonlinear Burgers', and the reaction-diffusion equations. The learned policies achieve target tracking, constraint satisfaction, and curvature minimization objectives, while generalizing across distributions of initial conditions and parameters. Moreover, we demonstrate four orders of magnitude acceleration at inference time compared to nonlinear model predictive control benchmarks. These results highlight the promise of operator learning for scalable model-based control of PDEs.

翻译：我们提出了一种面向偏微分方程（PDE）的数据驱动控制框架。该方法将时域积分深度算子网络（TI-DeepONets）作为可微PDE替代模型，集成至可微预测控制（DPC）——一种面向约束神经控制策略的自监督学习框架中。TI-DeepONet架构通过学习时间导数并将其与数值积分器耦合，而DPC算法利用自动微分技术，通过将最优控制损失的期望值反向传播至学习得到的TI-DeepONet，从而计算策略梯度。该方法无需在线优化或监督控制器，即可实现神经策略的高效离线优化。我们在多种PDE系统上进行了实证验证，包括热传导方程、非线性Burgers方程及反应扩散方程。学习得到的控制策略可实现目标跟踪、约束满足与曲率最小化等目标，同时能够泛化至不同初始条件与参数分布。此外，与非线性模型预测控制基准相比，我们在推理阶段实现了四个数量级的加速。这些结果凸显了算子学习在实现可扩展的基于模型的PDE控制中的潜力。