Plane-Wave Decomposition and Randomised Training; a Novel Path to Generalised PINNs for SHM

TL;DR

This work addresses the limitation of standard PINNs that typically learn a single BC-specific solution by introducing plane-wave activations and a training regime with randomly sampled BCs to learn a general solution form. The approach decouples the solution from boundary conditions, enabling rapid evaluation for arbitrary BCs once the network is trained, demonstrated on a toy system of two coupled oscillators. The results compare conventional PINNs, unique-solution plane-wave PINNs, and generalised-solution plane-wave PINNs, showing substantial reductions in training time and strong generalisation across unseen BCs, including extrapolation beyond the training range. This framework offers a pathway to scalable, generalised PINN solvers for SHM and other physical systems where many BCs must be evaluated efficiently, leveraging Fourier-like decompositions to capture the solution structure.

Abstract

In this paper, we introduce a formulation of Physics-Informed Neural Networks (PINNs), based on learning the form of the Fourier decomposition, and a training methodology based on a spread of randomly chosen boundary conditions. By training in this way we produce a PINN that generalises; after training it can be used to correctly predict the solution for an arbitrary set of boundary conditions and interpolate this solution between the samples that spanned the training domain. We demonstrate for a toy system of two coupled oscillators that this gives the PINN formulation genuine predictive capability owing to an effective reduction of the training to evaluation times ratio due to this decoupling of the solution from specific boundary conditions.

Figures (16)

• Figure 1: A depiction of an archetypal fully connected artificial neural network (ANN), where the coloured circles represent individual neurons. Layers are represented by vertical stacks of neurons. The training data is supplied to the network at the left- and right-most nodes from the $X$ and $Y$ datasets respectively, where $X$ is the set of independent data and $Y(X)$ is the set of dependent data. It is the task of the ANN to "learn" the mapping $Y(X)$ between the training datasets.
• Figure 2: A system of two coupled oscillators, with rigid boundaries at $x=\{0,L\}$. $a$ and $b$ represent the equilibrium positions of the masses. $x_1$ and $x_2$ represent the displacements of the oscillators $m_1$ and $m_2$ from their equilibrium positions respectively. $k_1$, $k_2$ and $k_3$ represent the spring constants associated with each spring in the system.
• Figure 3: Network architecture for a conventional PINN. Apart from the input and output layers, the architecture depicted here is representative only.
• Figure 4: Network architecture for a plane-wave PINN. Apart from the input and output layers, the architecture depicted here is representative only.
• Figure 5: Result A. Solution to a system of two coupled oscillators using a conventional PINN, trained using fixed boundary conditions. $x_{\{1,2\},P}$ and $x_{\{1,2\},N}$ denote the PINN and numerical solutions for both coordinates $x_1$ and $x_2$ respectively.