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.