An extrapolation-driven network architecture for physics-informed deep learning

TL;DR

This paper introduces an extrapolation-driven neural network architecture for physics-informed deep learning that improves the stability, continuity, and efficiency of solving time-dependent PDEs over large domains by leveraging extrapolation properties.

Contribution

It proposes a novel network architecture that couples parameters with time, enabling sequential solving of PDEs with guaranteed continuity and smoothness at interval nodes.

Findings

The method maintains local solutions from previous intervals.

It ensures strict continuity and smoothness at interval nodes.

Numerical experiments confirm improved performance over traditional PINNs.

Abstract

Current PINN implementations with sequential learning strategies often experience some weaknesses, such as the failure to reproduce the previous training results when using a single network, the difficulty to strictly ensure continuity and smoothness at the time interval nodes when using multiple networks, and the increase in complexity and computational overhead. To overcome these shortcomings, we first investigate the extrapolation capability of the PINN method for time-dependent PDEs. Taking advantage of this extrapolation property, we generalize the training result obtained in a specific time subinterval to larger intervals by adding a correction term to the network parameters of the subinterval. The correction term is determined by further training with the sample points in the added subinterval. Secondly, by designing an extrapolation control function with special characteristics and combining it with a correction term, we construct a new neural network architecture whose network parameters are coupled with the time variable, which we call the extrapolation-driven network architecture. Based on this architecture, using a single neural network, we can obtain the overall PINN solution of the whole domain with the following two characteristics: (1) it completely inherits the local solution of the interval obtained from the previous training, (2) at the interval node, it strictly maintains the continuity and smoothness that the true solution has. The extrapolation-driven network architecture allows us to divide a large time domain into multiple subintervals and solve the time-dependent PDEs one by one in a chronological order. This training scheme respects the causality principle and effectively overcomes the difficulties of the conventional PINN method in solving the evolution equation on a large time domain. Numerical experiments verify the performance of our method.