SONets: Sub-operator learning enhanced neural networks for solving parametric partial differential equations

Abstract

Complex nonlinear partial differential equations (PDEs) can be decomposed into subsystems comprising certain linear and nonlinear sub-operators. We usually have fundamental solutions to the linear subsystems and have a good knowledge of their uniqueness and regularity. The nonlinear subsystems have more straightforward formulas than the original complex nonlinear PDEs, even with reduced dimensions. Solutions of the complex nonlinear PDEs based on simpler subsystems are investigated in this study. We derive a proper subsystem decomposition of the nonlinear PDEs into linear and nonlinear subsystems to get unique solutions. We propose the sub-operator learning enhanced neural networks (SONets) based on this proper subsystem decomposition to solve nonlinear PDEs. The solutions of nonlinear subsystems with generated force terms are modeled via operator learning. In contrast, only the cumbersome integration and series parts of fundamental solutions of linear subsystems are modeled. The complex nonlinear PDEs are then solved by finding the force terms to minimize the physics-informed loss functions. It is also convenient to generalize SONets for a new PDE parameter, e.g., the diffusion coefficient. Next, we extend the SONets to a meta version (meta-SONets) for parametric PDEs by transfer learning. In meta-SONets, the initial conditions are explicitly encoded via operator learning neural networks (ENNs) to output force terms, followed by the whole SONets to solve PDEs. In the pre-training stage of meta-SONets, parametric PDEs given randomly generated initial conditions are solved by training ENNs to minimize the physics-informed loss functions. At the fine-tuning stage, the trainable parameters of ENNs are fine-tuned to solve the PDE given a new initial condition. Numerical experiments are conducted to solve the Burgers equation, the nonlinear diffusion-reaction system, and the Allen-Cahn equation. The numerical results indicate that SONets with neural network-parameterized force terms obtain solutions with good accuracy. With transfer learning, the computational cost of meta-SONets for solving the nonlinear PDEs with a new initial condition is dramatically reduced.