Predicting implicit operators among solutions for certain differential equations systems with DeepONets

Abstract

Solving multivariate ordinary differential equations (ODE) systems and partial differential equations (PDE) systems is the key to many complex physics and chemistry problems, such as the combustion in process of reacting flow. However, the traditional numerical methods in solving multivariate ODE and PDE systems are limited by computational cost, and sometimes its impossible to obtain the solution due to the high stiffness of ODE or PDE. Coincident with the development of machine learning has been a growing appreciation of applying neural networks in solving physics models. DeepM&M net was proposed to address complicated problems in fluid mechanics based on another neural network: DeepONet, which is used to predict functional nonlinear operators. Inspired by these two nets, a machine learning way of solving certain ODE and PDE systems is proposed with a similar framework to the DeepM&M net, which takes inputs of the initial conditions and outputs the corresponding solutions. The main ideas of this framework are first to explore the relations among solutions of the system by DeepONets and then to train a deep neural network with the assistance of trained DeepONets. The implicit operators between variables in certain ODE systems are verified to have existed and are well predicted by the DeepONet. The feasibility of the proposed framework is implied by the success in building blocks.