Solving parametric Partial Differential Equations (PDEs) for a broad range of
parameters is a critical challenge in scientific computing. To this end, neural
operators, which learn mappings from parameters to solutions, have been
successfully used. However, the training of neural operators typically demands
large training datasets, the acquisition of which can be prohibitively
expensive. To address this challenge, physics-informed training can offer a
cost-effective strategy. However, current physics-informed neural operators
face limitations, either in handling irregular domain shapes or in
generalization to various discretizations of PDE parameters with variable mesh
sizes. In this research, we introduce a novel physics-informed model
architecture which can generalize to parameter discretizations of variable size
and irregular domain shapes. Particularly, inspired by deep operator neural
networks, our model involves a discretization-independent learning of parameter
embedding repeatedly, and this parameter embedding is integrated with the
response embeddings through multiple compositional layers, for more
expressivity. Numerical results demonstrate the accuracy and efficiency of the
proposed method.

参数化偏微分方程、神经算子、物理信息训练、不规则域形状和可变网格尺寸的研究

基于物理信息的无网格深度组合算子网络

Physics-informed Mesh-independent Deep Compositional Operator Network

Constructing first-principles models is usually a challenging and
time-consuming task due to the complexity of the real-life processes. On the
other hand, data-driven modeling, and in particular neural network models often
suffer from issues such as overfitting and lack of useful and highquality data.
At the same time, embedding trained machine learning models directly into the
optimization problems has become an effective and state-of-the-art approach for
surrogate optimization, whose performance can be improved by physics-informed
training. In this study, it is proposed to upgrade piece-wise linear neural
network models with physics informed knowledge for optimization problems with
neural network models embedded. In addition to using widely accepted and
naturally piece-wise linear rectified linear unit (ReLU) activation functions,
this study also suggests piece-wise linear approximations for the hyperbolic
tangent activation function to widen the domain. Optimization of three case
studies, a blending process, an industrial distillation column and a crude oil
column are investigated. For all cases, physics-informed trained neural network
based optimal results are closer to global optimality. Finally, associated CPU
times for the optimization problems are much shorter than the standard
optimization results.

提出了利用物理知识来升级神经网络模型以解决优化问题的方法，通过使用修正线性单元和分段线性逼近的双曲正切激活函数，针对三个不同的案例进行实验，结果表明这种升级模型比传统模型更接近于全局最优解，且更有效地优化了 CPU 时间。