Solving second-order nonlinear evolution partial differential equations using deep learning**The project is supported by the National Natural Science Foundation of China (No. 11675054), Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things (Grant No. ZF1213) and Science and Technology Commission of Shanghai Municipality (No. 18dz2271000).

Abstract

Solving nonlinear evolution partial differential equations has been a longstanding computational challenge. In this paper, we present a universal paradigm of learning the system and extracting patterns from data generated from experiments. Specifically, this framework approximates the latent solution with a deep neural network, which is trained with the constraint of underlying physical laws usually expressed by some equations. In particular, we test the effectiveness of the approach for the Burgers’ equation used as an example of second-order nonlinear evolution equations under different initial and boundary conditions. The results also indicate that for soliton solutions, the model training costs significantly less time than other initial conditions.