Abstract
Convolutional neural operator is a CNN-based architecture recently proposed to enforce structure-preserving continuous-discrete equivalence and enable the genuine, alias-free learning of solution operators of PDEs. This neural operator was demonstrated to outperform for certain cases some baseline models such as DeepONet, Fourier neural operator, and Galerkin transformer in terms of surrogate accuracy. The convolutional neural operator, however, seems not to be validated for few-shot learning. We extend the model to few-shot learning scenarios by first pre-training a convolutional neural operator using a source dataset and then adjusting the parameters of the trained neural operator using only a small target dataset. We investigate three strategies for adjusting the parameters of a trained neural operator, including fine-tuning, low-rank adaption, and neuron linear transformation, and find that the neuron linear transformation strategy enjoys the highest surrogate accuracy in solving PDEs such as Kuramoto-Sivashinsky equation, Brusselator diffusion-reaction system, and Navier-Stokes equations.
Abstract (translated)
卷积神经算子是一种最近提出的基于CNN（卷积神经网络）的架构，旨在强制执行保持结构连续-离散等价性，并使PDE（偏微分方程）解算器的学习真正无混叠。这种神经算子在某些情况下展示出优于一些基准模型（如DeepONet、傅立叶神经算子和Galerkin变换器）的替代准确性。然而，卷积神经算子似乎尚未在少量样本学习中得到验证。我们通过首先使用源数据集对卷积神经算子进行预训练，然后仅利用小规模目标数据集调整已训练神经算子的参数的方式，将该模型扩展到了少量样本学习场景下。 为了调整已训练神经算子的参数，我们研究了三种策略：微调、低秩适应和神经元线性变换。实验结果表明，在求解柯特沃什尼-西瓦什金斯基方程（Kuramoto-Sivashinsky equation）、布鲁塞尔扩散反应系统（Brusselator diffusion-reaction system）以及纳维叶-斯托克斯方程（Navier-Stokes equations）等PDE时，神经元线性变换策略在替代准确性方面表现出最高的性能。
URL
https://arxiv.org/abs/2512.17969
