Generalized Transferable Neural Networks for Steady-State Partial Differential Equations

TL;DR

This paper introduces GTransNet, an advanced neural network architecture that enhances the original TransNet for solving steady-state PDEs, improving accuracy and stability in complex solution scenarios.

Contribution

The paper proposes GTransNet, a generalized neural network with additional layers and symmetry constraints, extending TransNet's capabilities for more accurate PDE solutions.

Findings

GTransNet improves accuracy over TransNet in oscillatory PDE solutions.

The architecture maintains interpretability while enhancing stability.

Experimental results demonstrate better performance in complex PDE problems.

Abstract

Deep learning has emerged as a compelling framework for scientific and engineering computing, motivating growing interest in neural network-based solvers for partial differential equations (PDEs). Within this landscape, network architectures with deterministic feature construction have become an appealing approach, offering both high accuracy and computational efficiency in practice. Among them, the transferable neural network (TransNet) is a special class of shallow neural networks (i.e., single-hidden-layer architectures), whose hidden-layer parameters are predetermined according to the principle of uniformly distributed partition hyperplanes. Although TransNet has demonstrated strong performance in solving PDEs with relatively smooth solutions, its accuracy and stability may deteriorate in the presence of highly oscillatory solution structures, where activation saturation and system conditioning issues become limiting factors. In this paper, we propose a generalized transferable neural network (GTransNet) for solving steady-state PDEs, which augments the original TransNet design with additional hidden layers while preserving its interpretable feature-generation mechanism. In particular, the first hidden layer of GTransNet retains TransNet's parameter sampling strategy but incorporates an additional symmetry constraint on the neuron biases, while the subsequent hidden layers omit bias terms and employ a variance-controlled sampling strategy for selecting neuron weights.