  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.
