MSPINN: Multiple scale method integrated physics-informed neural networks for reconstructing transient natural convection

TL;DR

This paper introduces a multi-scale physics-informed neural network approach to improve the reconstruction of transient natural convection flow fields from temperature data, addressing challenges like vanishing gradients and phase-dependent errors.

Contribution

The study proposes a novel multi-scale PINN framework that enhances reconstruction accuracy in transient convection problems by mitigating vanishing gradient issues and adapting to multiple spatial scales.

Findings

Reconstruction errors are higher during the incipient phase and smaller during the quasi-steady phase.

The multi-scale approach improves maximum errors by 72.2% and mean errors by 6.4%.

Errors tend to accumulate in regions with very small spatial gradients.

Abstract

This study employs physics-informed neural networks (PINNs) to reconstruct multiple flow fields in a transient natural convection system solely based on instantaneous temperature data at an arbitrary moment. Transient convection problems present reconstruction challenges due to the temporal variability of fields across different flow phases. In general, large reconstruction errors are observed during the incipient phase, while the quasi-steady phase exhibits relatively smaller errors, reduced by a factor of 2 to 4. We hypothesize that reconstruction errors vary across different flow phases due to the changing solution space of a PINN, inferred from the temporal gradients of the fields. Furthermore, we find that reconstruction errors tend to accumulate in regions where the spatial gradients are smaller than the order of $10^{-6}$, likely due to the vanishing gradient phenomenon. In convection phenomena, field variations often manifest across multiple scales in space. However, PINN-based reconstruction tends to preserve larger-scale variations, while smaller-scale variations become less pronounced due to the vanishing gradient problem. To mitigate the errors associated with vanishing gradients, we introduce a multi-scale approach that determines scaling constants for the PINN inputs and reformulates inputs across multiple scales. This approach improves the maximum and mean errors by 72.2% and 6.4%, respectively. Our research provides insights into the behavior of PINNs when applied to transient convection problems with large solution space and field variations across multiple scales.