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Bayesian PINNs for uncertainty-aware inverse problems (BPINN-IP)

Ali Mohammad-Djafari

TL;DR

BPINN-IP tackles linear inverse problems by merging physics-informed neural networks with hierarchical Bayesian inference to obtain point estimates and posterior uncertainty for reconstructions. The framework handles both supervised and unsupervised data and uses Bayesian updates for the latent variables, with uncertainty propagated through the forward model; NN weights are also treated in a Bayesian manner to yield predictive uncertainty. Applied to infrared image processing, BPINN-IP demonstrates deconvolution and super-resolution capabilities with accompanying uncertainty maps, using both synthetic and real data and showing robustness with limited labels. This approach offers a practical, uncertainty-aware tool for ill-posed inverse problems across imaging and related domains.

Abstract

The main contribution of this paper is to develop a hierarchical Bayesian formulation of PINNs for linear inverse problems, which is called BPINN-IP. The proposed methodology extends PINN to account for prior knowledge on the nature of the expected NN output, as well as its weights. Also, as we can have access to the posterior probability distributions, naturally uncertainties can be quantified. Also, variational inference and Monte Carlo dropout are employed to provide predictive means and variances for reconstructed images. Un example of applications to deconvolution and super-resolution is considered, details of the different steps of implementations are given, and some preliminary results are presented.

Bayesian PINNs for uncertainty-aware inverse problems (BPINN-IP)

TL;DR

BPINN-IP tackles linear inverse problems by merging physics-informed neural networks with hierarchical Bayesian inference to obtain point estimates and posterior uncertainty for reconstructions. The framework handles both supervised and unsupervised data and uses Bayesian updates for the latent variables, with uncertainty propagated through the forward model; NN weights are also treated in a Bayesian manner to yield predictive uncertainty. Applied to infrared image processing, BPINN-IP demonstrates deconvolution and super-resolution capabilities with accompanying uncertainty maps, using both synthetic and real data and showing robustness with limited labels. This approach offers a practical, uncertainty-aware tool for ill-posed inverse problems across imaging and related domains.

Abstract

The main contribution of this paper is to develop a hierarchical Bayesian formulation of PINNs for linear inverse problems, which is called BPINN-IP. The proposed methodology extends PINN to account for prior knowledge on the nature of the expected NN output, as well as its weights. Also, as we can have access to the posterior probability distributions, naturally uncertainties can be quantified. Also, variational inference and Monte Carlo dropout are employed to provide predictive means and variances for reconstructed images. Un example of applications to deconvolution and super-resolution is considered, details of the different steps of implementations are given, and some preliminary results are presented.
Paper Structure (7 sections, 26 equations, 4 figures)

This paper contains 7 sections, 26 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Proposed Hierarchical Bayesian PINN framework
  3. Supervized training data
  4. Unsupervised training data
  5. Inference and uncertainty quantification
  6. Application in infrared image processing
  7. Conclusion

Figures (4)

  • Figure 1: An example of synthetic IR image restoration: A synthetic example used for testing the trained NN. From left to right: Original, Blurred, Estimated.
  • Figure 2: An example of synthetic IR image super-resolution: A synthetic example used for testing the trained NN. From left to right: Original high resolution (HR), Low resolution obtained by a downsampling of factor 2, Estimated.
  • Figure 3: An example of IR image restoration: from left to right: Original, Blurred, Estimated by BPINN, mean and variances.
  • Figure 4: An example of IR image super-resolution: from left to right: original High Resolution (HR), Low resolution image at the input of the trained model, output of the model.