Bayesian Physics-Informed Neural Networks for Inverse Problems (BPINN-IP): Application in Infrared Image Processing

TL;DR

This work introduces BPINN-IP, a BayesianPhysics-Informed Neural Network tailored for inverse problems, by formulating a probabilistic data-generation process that integrates the forward operator and measurement uncertainties. It shows that PINNs are recovered as the MAP solution within this framework and extends them to full posterior inference for both the unknown fields and the neural network parameters, enabling uncertainty quantification. The method is demonstrated on infrared image restoration and super-resolution, using synthetic and real data, with MC dropout providing practical UQ and improved robustness over traditional PINNs. The approach offers a principled bridge between traditional Bayesian inference and physics-informed learning, with potential applicability to a broad class of linear inverse problems governed by forward models.

Abstract

Inverse problems arise across scientific and engineering domains, where the goal is to infer hidden parameters or physical fields from indirect and noisy observations. Classical approaches, such as variational regularization and Bayesian inference, provide well established theoretical foundations for handling ill posedness. However, these methods often become computationally restrictive in high dimensional settings or when the forward model is governed by complex physics. Physics Informed Neural Networks (PINNs) have recently emerged as a promising framework for solving inverse problems by embedding physical laws directly into the training process of neural networks. In this paper, we introduce a new perspective on the Bayesian Physics Informed Neural Network (BPINN) framework, extending classical PINNs by explicitly incorporating training data generation, modeling and measurement uncertainties through Bayesian prior modeling and doing inference with the posterior laws. Also, as we focus on the inverse problems, we call this method BPINN-IP, and we show that the standard PINN formulation naturally appears as its special case corresponding to the Maximum A Posteriori (MAP) estimate. This unified formulation allows simultaneous exploitation of physical constraints, prior knowledge, and data-driven inference, while enabling uncertainty quantification through posterior distributions. To demonstrate the effectiveness of the proposed framework, we consider inverse problems arising in infrared image processing, including deconvolution and super-resolution, and present results on both simulated and real industrial data.

Figures (11)

• Figure 1: Infrared simplified forward model: from left to right. True temperature distribution $f(x,y)$; emissivity and environmental mapping $\phi(f)$; diffusion–convolution via the PSF $h$; observed IR image $g$.
• Figure 2: Examples of synthetic IR training data. Top: true temperature images ${{\bm f}}_{Ti}$. Bottom: blurred and noisy observations ${{\bm g}}_{Ti}$.
• Figure 3: Examples of validation images used during training. Top: true images; Bottom: blurred and noisy observations.
• Figure 4: Examples of test images. Top: ground truth; Bottom: degraded inputs.
• Figure 5: Training and validation losses, PSNR, and SSIM for the simple NN model.