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Understanding the training of PINNs for unsteady flow past a plunging foil through the lens of input subdomain level loss function gradients

Rahul Sundar, Didier Lucor, Sunetra Sarkar

TL;DR

The paper addresses how spatial subdomains influence the training dynamics of moving-boundary MB-PINNs for unsteady flow past a plunging foil. It introduces a zonal gradient-analysis framework that partitions the domain into moving body, wake, and outer zones, and defines metrics to quantify each zone's contribution to the Bulk and Physics loss gradients. Across three training scenarios, the study shows that physics loss relaxation alone biases learning toward the wake, while combining relaxation with vorticity-based undersampling shifts dominance to the moving body and alleviates vanishing gradients. The proposed zonal-analysis approach enables design of spatially aware loss weighting and can be extended to other input subdomains and physics-informed learning problems.

Abstract

Recently immersed boundary method-inspired physics-informed neural networks (PINNs) including the moving boundary-enabled PINNs (MB-PINNs) have shown the ability to accurately reconstruct velocity and recover pressure as a hidden variable for unsteady flow past moving bodies. Considering flow past a plunging foil, MB-PINNs were trained with global physics loss relaxation and also in conjunction with a physics-based undersampling method, obtaining good accuracy. The purpose of this study was to investigate which input spatial subdomain contributes to the training under the effect of physics loss relaxation and physics-based undersampling. In the context of MB-PINNs training, three spatial zones: the moving body, wake, and outer zones were defined. To quantify which spatial zone drives the training, two novel metrics are computed from the zonal loss component gradient statistics and the proportion of sample points in each zone. Results confirm that the learning indeed depends on the combined effect of the zonal loss component gradients and the proportion of points in each zone. Moreover, the dominant input zones are also the ones that have the strongest solution gradients in some sense.

Understanding the training of PINNs for unsteady flow past a plunging foil through the lens of input subdomain level loss function gradients

TL;DR

The paper addresses how spatial subdomains influence the training dynamics of moving-boundary MB-PINNs for unsteady flow past a plunging foil. It introduces a zonal gradient-analysis framework that partitions the domain into moving body, wake, and outer zones, and defines metrics to quantify each zone's contribution to the Bulk and Physics loss gradients. Across three training scenarios, the study shows that physics loss relaxation alone biases learning toward the wake, while combining relaxation with vorticity-based undersampling shifts dominance to the moving body and alleviates vanishing gradients. The proposed zonal-analysis approach enables design of spatially aware loss weighting and can be extended to other input subdomains and physics-informed learning problems.

Abstract

Recently immersed boundary method-inspired physics-informed neural networks (PINNs) including the moving boundary-enabled PINNs (MB-PINNs) have shown the ability to accurately reconstruct velocity and recover pressure as a hidden variable for unsteady flow past moving bodies. Considering flow past a plunging foil, MB-PINNs were trained with global physics loss relaxation and also in conjunction with a physics-based undersampling method, obtaining good accuracy. The purpose of this study was to investigate which input spatial subdomain contributes to the training under the effect of physics loss relaxation and physics-based undersampling. In the context of MB-PINNs training, three spatial zones: the moving body, wake, and outer zones were defined. To quantify which spatial zone drives the training, two novel metrics are computed from the zonal loss component gradient statistics and the proportion of sample points in each zone. Results confirm that the learning indeed depends on the combined effect of the zonal loss component gradients and the proportion of points in each zone. Moreover, the dominant input zones are also the ones that have the strongest solution gradients in some sense.
Paper Structure (11 sections, 9 equations, 7 figures, 4 tables)

This paper contains 11 sections, 9 equations, 7 figures, 4 tables.

Table of Contents

  1. INTRODUCTION
  2. METHODOLOGY
  3. The plunging foil system
  4. Moving boundary enabled PINNs (MB-PINNs)
  5. Zonal splitting of layer-wise gradients
  6. Metrics to diagnose zonal imbalances
  7. RESULTS AND DISCUSSION
  8. Case 1: Baseline ($\lambda_{Phy} = 1)$
  9. Case 2: With physics loss relaxation ($\lambda_{Phy} = 0.001$)
  10. Case 3: With global physics loss relaxation and vorticity cutoff based under sampling ($\lambda_{Phy} = 0.01, S_{\omega_z} = 5\%$)
  11. CONCLUSIONS

Figures (7)

  • Figure 1: Problem setup (Image adapted from Sundar et al.)sundar2023physicsinformed.
  • Figure 2: MB-PINN schematic (Image adapted from Sundar et al.sundar2023physicsinformed.)
  • Figure 3: Spatial grids of (a) CI and (b) CI-S5 data sets and the corresponding (c-d) zonal splitting of the grids into $\text{Z1}$- moving body, $\text{Z2}$ - wake and $\text{Z3}$ - outer zones, respectively
  • Figure 4: Comparison of (a-b) true and (c-e) MB-PINN predicted velocity $x$-component, corresponding point-wise maximum value normalised absolute errors and vorticity contours at $t/T = 1.0.$
  • Figure 5: First layer (top row) and last layer (bottom row) $\nabla_{\theta}\mathcal{L}_{Bulk}$ and $\nabla_{\theta}\mathcal{L}_{Phy}$ distributions obtained after $1e06$ training iterations for (a,d) Case 1: $\lambda_{Phy}=1$, (b,e) Case 2: $\lambda_{Phy} = 0.001$ and $S_{\omega_z} = 100\%$ and (c,f) Case 3: $\lambda_{Phy} = 0.01$ and $S_{\omega_z} = 5\%.$
  • ...and 2 more figures