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

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.

💡 Research Summary

The paper introduces a novel framework called Bayesian Physics‑Informed Neural Networks for Inverse Problems (BPINN‑IP), which integrates Bayesian inference into the Physics‑Informed Neural Network (PINN) paradigm to address linear inverse problems while providing principled uncertainty quantification. The authors start by formalizing the forward model as a linear operator (g = Hf + \varepsilon), where (g) denotes observed data, (f) the unknown field, (H) the forward operator (e.g., convolution, down‑sampling), and (\varepsilon) Gaussian noise. Gaussian priors are placed on both the unknown field (f) and the noise (\varepsilon), leading to a closed‑form Gaussian posterior for the linear case.

The core contribution lies in embedding these probabilistic components into the loss function of a neural network that maps observations (g) to an estimate (\hat f). The loss consists of three parts: (1) a data‑fit term penalizing the deviation between the network output and any available ground‑truth labels (when supervised data exist), (2) a physics‑consistency term enforcing that the forward model applied to the network output reproduces the measurements, and (3) a prior regularization term that reflects the chosen Gaussian prior on (f) and a sparsity‑inducing prior on the network weights. Both supervised and unsupervised training regimes are covered; in the unsupervised case the loss reduces to the physics‑consistency term plus an L1 weight penalty.

For Bayesian inference, the authors adopt two approximate strategies: variational inference (implemented via a mean‑field Gaussian approximation) and Monte‑Carlo dropout, which together enable sampling from an approximate posterior over network weights. By propagating these weight samples through the trained network, they obtain an empirical distribution of the reconstructed field, from which they extract the posterior mean (point estimate) and the diagonal of the covariance (pixel‑wise variance). This provides a practical way to visualize uncertainty maps alongside reconstructed images.

The methodology is demonstrated on two infrared imaging tasks: (i) deconvolution (image restoration) where the forward operator is a point‑spread function convolution, and (ii) super‑resolution where the forward operator combines down‑sampling and blurring. Synthetic datasets of 128 × 128 images are generated (1000 samples, 800 for training, 200 for testing). The network architecture is a modest convolutional neural network with three hidden layers and ReLU activations, trained with Adam optimizer. Results show that BPINN‑IP outperforms classical regularized inversion (e.g., Tikhonov) in terms of PSNR and SSIM, while simultaneously delivering variance maps that highlight regions of high uncertainty, especially where noise is dominant. Real‑world infrared images are also processed, confirming that the uncertainty estimates are meaningful in practice.

The paper’s contributions are threefold: (1) a unified Bayesian‑PINN formulation that naturally incorporates prior knowledge, measurement noise, and physical constraints; (2) an inference pipeline that yields both point estimates and calibrated uncertainty for high‑dimensional inverse problems; (3) empirical evidence that the approach reduces reliance on large labeled datasets and improves robustness to noise. Limitations include the current focus on linear forward models, the reliance on approximate inference (which may be sensitive to prior and architecture choices), and the computational overhead of sampling. Future work is outlined to extend the framework to nonlinear and dynamic inverse problems, to explore more efficient Bayesian inference techniques (e.g., Hamiltonian Monte Carlo, stochastic gradient MCMC), and to validate the method on larger, more complex real‑world datasets. Overall, BPINN‑IP represents a promising step toward uncertainty‑aware, physics‑consistent deep learning solutions for a broad class of inverse problems in science and engineering.