Model Based and Physics Informed Deep Learning Neural Network Structures

TL;DR

The work addresses how to choose neural network architectures for inverse problems in signal and image processing, leveraging forward models and priors. It proposes a taxonomy of five structure families—explicit analytical solutions, transform-domain processing, operator decomposition, optimization algorithm unfolding, and physics-informed NN (PINN)—and grounds the discussion with infrared and acoustical imaging. It derives concrete NN templates for each category, including linear analytic mappings, FFT/Wavelet-based nets, encoder–decoder architectures, and unfolded ISTA networks, all linked to the Bayesian objective $J(f)=||g-Hf||^2+λ||f||^2$ and the forward model $g=Hf+ε$. The framework provides principled guidance for designing physics-informed DL solutions in inverse problems, with potential improvements in reconstruction quality, interpretability, and integration of prior knowledge.

Abstract

Neural Networks (NN) has been used in many areas with great success. When a NN's structure (Model) is given, during the training steps, the parameters of the model are determined using an appropriate criterion and an optimization algorithm (Training). Then, the trained model can be used for the prediction or inference step (Testing). As there are also many hyperparameters, related to the optimization criteria and optimization algorithms, a validation step is necessary before its final use. One of the great difficulties is the choice of the NN's structure. Even if there are many "on the shelf" networks, selecting or proposing a new appropriate network for a given data, signal or image processing, is still an open problem. In this work, we consider this problem using model based signal and image processing and inverse problems methods. We classify the methods in five classes, based on: i) Explicit analytical solutions, ii) Transform domain decomposition, iii) Operator Decomposition, iv) Optimization algorithms unfolding, and v) Physics Informed NN methods (PINN). Few examples in each category are explained.

Figures (13)

• Figure 1: Infrared simplified Forward model: Real temperature distribution as input $f(x,y)$, nonlinear emissivity and environment perturbation function $\phi(f)$ and the convolution operation to simulate the diffusion process, and finally the measured infrared camera output $g(x,y)$.
• Figure 2: Infrared imaging: Forward model and Inverse problem resolved either by mathematical inversion or by a Neural Network
• Figure 3: Forward model in acoustical imaging: Each microphone receives the sum of the delayed sources sounds.
• Figure 4: Acoustical imaging Inverse problem via Beamforming (BF) and Deconvolution (Dec):: In a first step, the received data are used to obtain an image ${\color{green}b}(x,y)$ by BF, then an inversion by Dec can result to a good estimate of the sources.
• Figure 5: Basic Bayesian approach illustration for the case of Gaussian priors.