PINN for Dynamical Partial Differential Equations is Not Training Deeper Networks Rather Learning Advection and Time Variance

TL;DR

This paper investigates the limitations of physics-informed neural networks (PINNs) in solving dynamical partial differential equations, highlighting that shallow networks often outperform deeper ones and proposing reservoir computing as a promising alternative.

Contribution

The study identifies key limitations of PINNs in handling advection and long-term dynamics, and suggests reservoir computing architectures for improved robustness.

Findings

Shallow networks often outperform deeper PINNs in dynamical problems.

PINNs struggle with strong advection and long time durations.

Reservoir computing is recommended for better dynamical PDE solutions.

Abstract

The concepts and techniques of physics-informed neural networks (PINNs) is studied and limitations are identified to make it efficient to approximate dynamical equations. Potential working research domains are explored for increasing the robustness of this technique for the solvability of partial differential equations. It is identified that PINNs potentially fails to stronger advection and longer time duration. Also, optimization function and constraint posing needs to be smarter. Even a shallow network is good for a lot of problems while powerful deeper network fails. Reservoir computing based recurrent neural network architecture is recommended to solve dynamical problems.