Optimal control problems (OCPs) involve finding a control function for a dynamical system such that a cost functional is optimized. It is central to physical systems in both academia and industry. In this paper, we propose a novel instance-solution control operator perspective, which solves OCPs in a one-shot manner without direct dependence on the explicit expression of dynamics or iterative optimization processes. The control operator is implemented by a new neural operator architecture named Neural Adaptive Spectral Method (NASM), a generalization of classical spectral methods. We theoretically validate the perspective and architecture by presenting the approximation error bounds of NASM for the control operator. Experiments on synthetic environments and a real-world dataset verify the effectiveness and efficiency of our approach, including substantial speedup in running time, and high-quality in- and out-of-distribution generalization.

Despite extensive research, recovering PDE expressions from experimental observations often involves symbolic regression. This method generally lacks the incorporation of meaningful physical insights, resulting in outcomes lacking clear physical interpretations. Recognizing that the primary interest of Machine Learning for Science (ML4Sci) often lies in understanding the underlying physical mechanisms or even discovering new physical laws rather than simply obtaining mathematical expressions, this paper introduces a novel ML4Sci task paradigm. This paradigm focuses on interpreting experimental data within the framework of prior physical hypotheses and theories, thereby guiding and constraining the discovery of PDE expressions. We have formulated this approach as a nonlinear mixed-integer programming (MIP) problem, addressed through an efficient search scheme developed for this purpose. Our experiments on newly designed Fluid Mechanics and Laser Fusion datasets demonstrate the interpretability and feasibility of this method.

We present a novel framework, StochastIc Network Graph Evolving operatoR (SINGER), for learning the evolution operator of high-dimensional partial differential equations (PDEs). The framework uses a sub-network to approximate the solution at the initial time step and stochastically evolves the sub-network parameters over time by a graph neural network to approximate the solution at later time steps. The framework is designed to inherit the desirable properties of the parametric solution operator, including graph topology, semigroup, and stability, with a theoretical guarantee. Numerical experiments on 8 evolution PDEs of 5,10,15,20-dimensions show that our method outperforms existing baselines in almost all cases (31 out of 32), and that our method generalizes well to unseen initial conditions, equation dimensions, sub-network width, and time steps.

Neural operators have emerged as a promising approach for solving high-dimensional partial differential equations (PDEs). However, existing neural operators often have difficulty in dealing with constrained PDEs, where the solution must satisfy additional equality or inequality constraints beyond the governing equations. To close this gap, we propose a novel neural operator, Hyper Extended Adaptive PDHG (HEAP) for constrained high-dim PDEs, where the learned operator evolves in the parameter space of PDEs. We first show that the evolution operator learning can be formulated as a quadratic programming (QP) problem, then unroll the adaptive primal-dual hybrid gradient (APDHG) algorithm as the QP-solver into the neural operator architecture. This allows us to improve efficiency while retaining theoretical guarantees of the constrained optimization. Empirical results on a variety of high-dim PDEs show that HEAP outperforms the state-of-the-art neural operator model.