用神经网络从数据中学习偏微分方程里的未知函数
Learning functional components of PDEs from data using neural networks
- 将神经网络嵌入偏微分方程，通过数据训练逼近未知函数
- 在稳态数据下成功恢复了相互作用核与外势函数
- 适合需要建模复杂非线性过程的科研人员使用
偏微分方程常包含难以直接测量的未知函数，阻碍模型预测能力。已有方法可从数据中恢复标量参数，本文展示如何将类似流程扩展至恢复函数。具体地，将神经网络嵌入PDE，训练过程中可任意逼近未知函数。以非局部聚集-扩散方程为例，从稳态数据中恢复了相互作用核与外部势函数。研究考察了可用解的数量、性质、采样密度及测量噪声等因素对函数恢复效果的影响。该方法优势在于可沿用标准参数拟合流程，且训练后的PDE可作为常规方程用于系统预测。
原文摘要 · Abstract (English)
Partial differential equations often contain unknown functions that are difficult or impossible to measure directly, hampering our ability to derive predictions from the model. Workflows for recovering scalar PDE parameters from data are well studied: here we show how similar workflows can be used to recover functions from data. Specifically, we embed neural networks into the PDE and show how, as they are trained on data, they can approximate unknown functions with arbitrary accuracy. Using nonlocal aggregation-diffusion equations as a case study, we recover interaction kernels and external potentials from steady state data. Specifically, we investigate how a wide range of factors, such as the number of available solutions, their properties, sampling density, and measurement noise, affect our ability to successfully recover functions. Our approach is advantageous because it can utilise standard parameter-fitting workflows, and in that the trained PDE can be treated as a normal PDE for purposes such as generating system predictions.
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