This study investigates the potential accuracy boundaries of physics-informed
neural networks, contrasting their approach with previous similar works and
traditional numerical methods. We find that selecting improved optimization
algorithms significantly enhances the accuracy of the results. Simple
modifications to the loss function may also improve precision, offering an
additional avenue for enhancement. Despite optimization algorithms having a
greater impact on convergence than adjustments to the loss function, practical
considerations often favor tweaking the latter due to ease of implementation.
On a global scale, the integration of an enhanced optimizer and a marginally
adjusted loss function enables a reduction in the loss function by several
orders of magnitude across diverse physical problems. Consequently, our results
obtained using compact networks (typically comprising 2 or 3 layers of 20-30
neurons) achieve accuracies comparable to finite difference schemes employing
thousands of grid points. This study encourages the continued advancement of
PINNs and associated optimization techniques for broader applications across
various fields.

研究通过改进优化算法和调整损失函数，得出物理感知神经网络在多个领域具有与有限差分方案相当的准确性，鼓励进一步推动 PINNs 和相关优化技术在各个领域的应用。

揭示物理任务导向神经网络的优化过程：PINN 能有多准确和有竞争力？

Unveiling the optimization process of Physics Informed Neural Networks:  How accurate and competitive can PINNs be?

In recent years, Physics-informed neural networks (PINNs) have been widely
used to solve partial differential equations alongside numerical methods
because PINNs can be trained without observations and deal with continuous-time
problems directly. In contrast, optimizing the parameters of such models is
difficult, and individual training sessions must be performed to predict the
evolutions of each different initial condition. To alleviate the first problem,
observed data can be injected directly into the loss function part. To solve
the second problem, a network architecture can be built as a framework to learn
a finite difference method. In view of the two motivations, we propose
Five-point stencil CNNs (FCNNs) containing a five-point stencil kernel and a
trainable approximation function for reaction-diffusion type equations
including the heat, Fisher's, Allen-Cahn, and other reaction-diffusion
equations with trigonometric function terms. We show that FCNNs can learn
finite difference schemes using few data and achieve the low relative errors of
diverse reaction-diffusion evolutions with unseen initial conditions.
Furthermore, we demonstrate that FCNNs can still be trained well even with
using noisy data.

本文介绍了一种利用五点卷积神经网络解决物理对偶方程的方法，并展示了该方法可以在少量数据下学习有限差分方案，实现不同反应扩散演化的低相对误差，并且即使使用有噪声的数据也能很好地训练。