Network scaling and scale-driven loss balancing for intelligent poroelastography

TL;DR

The physics-based dynamic scaling approach for adaptive loss balancing is presented, first presented in a generic form for multi-physics and multi-scale PDE systems, and then applied through a set of numerical experiments to poroelastography.

Abstract

A deep learning framework is developed for multiscale characterization of poroelastic media from full waveform data which is known as poroelastography. Special attention is paid to heterogeneous environments whose multiphase properties may drastically change across several scales. Described in space-frequency, the data takes the form of focal solid displacement and pore pressure fields in various neighborhoods furnished either by reconstruction from remote data or direct measurements depending on the application. The objective is to simultaneously recover the six hydromechanical properties germane to Biot equations and their spatial distribution in a robust and efficient manner. Two major challenges impede direct application of existing state-of-the-art techniques for this purpose: (i) the sought-for properties belong to vastly different and potentially uncertain scales, and~(ii) the loss function is multi-objective and multi-scale (both in terms of its individual components and the total loss). To help bridge the gap, we propose the idea of \emph{network scaling} where the neural property maps are constructed by unit shape functions composed into a scaling layer. In this model, the unknown network parameters (weights and biases) remain of O(1) during training. This forms the basis for explicit scaling of the loss components and their derivatives with respect to the network parameters. Thereby, we propose the physics-based \emph{dynamic scaling} approach for adaptive loss balancing. The idea is first presented in a generic form for multi-physics and multi-scale PDE systems, and then applied through a set of numerical experiments to poroelastography. The results are presented along with reconstructions by way of gradient normalization (GradNorm) and Softmax adaptive weights (SoftAdapt) for loss balancing. A comparative analysis of the methods and corresponding results is provided.

Figures (14)

• Figure 1: Synthetic experiments simulating wave motion in the poroelastic domain $\mathcal{B}$: (a) the model is harmonically excited at frequency $\omega$ by a fluid body force $\delta(\boldsymbol{x}-\boldsymbol{x}_\circ)$ and the response is computed in a neighborhood of the source point $\boldsymbol{x}_\circ$ i.e., in the square $\mathcal{B}_\circ$ of side $5 \ell_r$, (b) the $x$ component of effective body force $\boldsymbol{f}^{u}$ in the first of Biot equations in \ref{['eq:Biot']}, and (c) the fluid source term $f^p$ in the generalized Darcy's law i.e., the second of \ref{['eq:Biot']}. Here, $\mathfrak{R}$ and $\mathfrak{I}$ respectively indicate the real and imaginary parts of a complex-valued quantity.
• Figure 2: Simulated poroelastic response to the excitation shown in Fig. \ref{['problem_statement1']} in focal area $\boldsymbol{\xi}_1$: real (top raw) and imaginary (bottom row) of (a) solid displacement $u_x$, (b) solid displacement $u_y$, and (c) interstitial pore pressure $p$.
• Figure 3: Application of the proposed approach for intelligent poroelastography: (a) mapping of the unknown hydromechanical properties in each focal neighborhood $\boldsymbol{\xi}_i$, $i = 1,2$, by a multiscale MLP such that the network parameters (i.e., weights and biases) remain of O(1) while the outputs are properly scaled by the last layer, (b) identifying the network by minimizing the multi-objective loss function $\mathcal{L}$ comprised of weighted PDE residuals associated with the Biot equations, and (c) the training dataset entailing the harmonic source function $\delta_i$ of frequency $\omega$ in each focal region and the associated displacement and pressure fields $[\boldsymbol{u}_i^j, p_i^j]$ at $j = 1,\ldots, N_{p_i}$ points in the vicinity of each source along with their spatial derivatives $D_{kl}$.
• Figure 4: Network-predicted poroelastic properties vs. number of epochs $N_{e}$ when the reconstruction is simultaneously conducted in the focal regions $\{ \boldsymbol{\xi}_i \}$, $i =1,2$. The network is a scaled MLP and the loss is balanced using the proposed Dynamic Scaling (DS) approach: (a, g) drained shear modulus $\mu^{\text{\tiny DS}}_i$, (b, h) Biot modulus $M^{\text{\tiny DS}}_i$, (c, i) porosity $\phi^{\text{\tiny DS}}_i$, (d, j) drained first Lamé parameter $\lambda^{\text{\tiny DS}}_i$, (e, k) Biot effective stress coefficient $\alpha^{\text{\tiny DS}}_i$, (f, l) permeability coefficient $\kappa^{\text{\tiny DS}}_i$.
• Figure 6: Network-predicted poroelastic properties vs. the number of epochs $N_{e}$ when the reconstruction is conducted in the high-permeability neighborhood $\boldsymbol{\xi}_1$. The network is a scaled MLP and the loss is balanced using the proposed Dynamic Scaling (DS) approach: (a) drained shear modulus $\mu^{\text{\tiny DS}}_1\!$, (b) drained first Lamé parameter $\lambda^{\text{\tiny DS}}_1\!$, (c) Biot modulus $M^{\text{\tiny DS}}_1\!$, (d) Biot effective stress coefficient $\alpha^{\text{\tiny DS}}_1\!$, (e) porosity $\phi^{\text{\tiny DS}}_1\!$, (f) permeability coefficient $\kappa^{\text{\tiny DS}}_1\!$, (g) DS weights $\text{w}^{\text{\tiny DS}}_i$, $i = 1, \ldots, 6$, versus the minimizer step $N_{e}$, (h) weighted total loss $\text{log}(\mathscr{L}_1^{\text{\tiny DS}})$ trajectory against the number of epochs.

Theorems & Definitions (1)

• Remark 1