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Label Propagation Training Schemes for Physics-Informed Neural Networks and Gaussian Processes

Ming Zhong, Dehao Liu, Raymundo Arroyave, Ulisses Braga-Neto

TL;DR

The paper tackles the difficulty of propagating information from limited labeled data in physics-informed learning by introducing semi-supervised label propagation for PINNs and PIGPs. It develops self-training and co-training algorithms to train these models in isolation and in hybrid PINN–PIGP configurations, enabling uncertainty-aware predictions via co-trained PIGPs. Key contributions include the first PINN–PIGP hybrid, systematic evaluation across parabolic and elliptic PDEs, and evidence that co-training improves forward-time information transfer while providing uncertainty quantification. This approach offers data-efficient, uncertainty-aware physics-informed modeling with potential for broader applicability to stiff PDEs and complex dynamical systems.

Abstract

This paper proposes a semi-supervised methodology for training physics-informed machine learning methods. This includes self-training of physics-informed neural networks and physics-informed Gaussian processes in isolation, and the integration of the two via co-training. We demonstrate via extensive numerical experiments how these methods can ameliorate the issue of propagating information forward in time, which is a common failure mode of physics-informed machine learning.

Label Propagation Training Schemes for Physics-Informed Neural Networks and Gaussian Processes

TL;DR

The paper tackles the difficulty of propagating information from limited labeled data in physics-informed learning by introducing semi-supervised label propagation for PINNs and PIGPs. It develops self-training and co-training algorithms to train these models in isolation and in hybrid PINN–PIGP configurations, enabling uncertainty-aware predictions via co-trained PIGPs. Key contributions include the first PINN–PIGP hybrid, systematic evaluation across parabolic and elliptic PDEs, and evidence that co-training improves forward-time information transfer while providing uncertainty quantification. This approach offers data-efficient, uncertainty-aware physics-informed modeling with potential for broader applicability to stiff PDEs and complex dynamical systems.

Abstract

This paper proposes a semi-supervised methodology for training physics-informed machine learning methods. This includes self-training of physics-informed neural networks and physics-informed Gaussian processes in isolation, and the integration of the two via co-training. We demonstrate via extensive numerical experiments how these methods can ameliorate the issue of propagating information forward in time, which is a common failure mode of physics-informed machine learning.
Paper Structure (11 sections, 11 equations, 16 figures, 5 tables, 2 algorithms)

This paper contains 11 sections, 11 equations, 16 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Contributions of this Work
  4. Methodology
  5. Propagation of Labels
  6. Numerical Experiments
  7. PINN Self Trains
  8. PIGP Self Trains
  9. Co-train: PINN trains with GP
  10. Co-train: PINN Trains PIGP
  11. Co-Train: PIGP Trains PINN

Figures (16)

  • Figure 1: Points Setup for vBurgers for PINN.
  • Figure 2: No self training, $L_2$-Err = $3.8 \cdot 10^{-3}$ and $L_{\infty}$-Err = $1.9 \cdot 10^{-2}$.
  • Figure 3: $5$ Self training, $L_2$-Err = $1.6 \cdot 10^{-3}$ and $L_{\infty}$-Err = $9.5 \cdot 10^{-3}$.
  • Figure 4: Propagation of Labeled Points in self train PINN on solving vBurgers.
  • Figure 5: Points Setup for vBurgers for PIGP.
  • ...and 11 more figures