通过后处理重训练提升物理神经网络求解精度
Improving the accuracy of physics-informed neural networks via last-layer retraining
- 用函数空间最优逼近作为后处理，优化PINN输出
- 误差降低四到五个数量级，跨架构和维度有效
- 可复用基函数实现迁移学习，适合复杂方程求解
物理信息神经网络（PINNs）是科学机器学习中求解偏微分方程（PDEs）的有力工具。然而，其训练策略尚不明确，通常只能得到中等精度解。本文提出一种方法：将PINNs与后处理步骤结合，寻找与网络相关函数空间中的最优近似。结果表明，该方法在多种架构和维度下，使误差比原始PINNs降低四到五个数量级。此外，可复用线性空间的基函数于时变及非线性问题中，支持迁移学习。该方法还提供基于残差的度量，用于最优选择基函数数量。
原文摘要 · Abstract (English)
Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not obvious, with the result that they typically yield moderately accurate solutions. In this article, we propose a method for improving the accuracy of PINNs by coupling them with a post-processing step that seeks the best approximation in a function space associated with the network. We find that our method yields errors four to five orders of magnitude lower than those of the parent PINNs across architectures and dimensions. Moreover, we can reuse the basis functions for the linear space in more complex settings, such as time-dependent and nonlinear problems, allowing for transfer learning. Our approach also provides a residual-based metric that allows us to optimally choose the number of basis functions employed.
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