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

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.

References in corpus (13)

• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• DGM: A deep learning algorithm for solving partial differential equations
• On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks
• Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks
• A Physics Informed Neural Network for Time-Dependent Nonlinear and Higher Order Partial Differential Equations
• A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks
• Thermodynamic Stability of Driven Open Systems and Control of Phase Separation by Electro-autocatalysis
• Reliable extrapolation of deep neural operators informed by physics or sparse observations
• How Neural Networks Extrapolate: From Feedforward to Graph Neural Networks
• A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions
• A practical PINN framework for multi-scale problems with multi-magnitude loss terms
• Pre-training strategy for solving evolution equations based on physics-informed neural networks
• A deep learning method for multi-material diffusion problems based on physics-informed neural networks