Microstructure-based Variational Neural Networks for Robust Uncertainty Quantification in Materials Digital Twins

TL;DR

This work tackles the challenge of irreducible microstructure-induced variability (aleatoric uncertainty) in materials digital twins. It introduces the Variational Deep Material Network (VDMN), a physics-informed surrogate that embeds hyper-variational distributions within a Deep Material Network framework to propagate uncertainty analytically and via sampling. The approach is validated on synthetic microstructures and demonstrated in two digital-twin tasks: forward uncertainty quantification for nonlinear mechanical responses of 3D-printed polymers and inverse uncertainty quantification to disentangle multiple uncertainty sources, including constitutive properties. The results establish VDMN as a scalable, uncertainty-robust foundation for real-time materials digital twins with robust interpretability and broader multimodal-extension potential.

Abstract

Aleatoric uncertainties - irremovable variability in microstructure morphology, constituent behavior, and processing conditions - pose a major challenge to developing uncertainty-robust digital twins. We introduce the Variational Deep Material Network (VDMN), a physics-informed surrogate model that enables efficient and probabilistic forward and inverse predictions of material behavior. The VDMN captures microstructure-induced variability by embedding variational distributions within its hierarchical, mechanistic architecture. Using an analytic propagation scheme based on Taylor-series expansion and automatic differentiation, the VDMN efficiently propagates uncertainty through the network during training and prediction. We demonstrate its capabilities in two digital-twin-driven applications: (1) as an uncertainty-aware materials digital twin, it predicts and experimentally validates the nonlinear mechanical variability in additively manufactured polymer composites; and (2) as an inverse calibration engine, it disentangles and quantitatively identifies overlapping sources of uncertainty in constituent properties. Together, these results establish the VDMN as a foundation for uncertainty-robust materials digital twins.

Figures (9)

• Figure 1: Modeling Aleatoric Uncertainty in Materials Behavior with Variational Deep Material Networks (VDMN).Top Row: Motivation. (a) Microstructural variability in two-phase (red/blue) composites arises naturally in advanced manufacturing and represents a key source of aleatoric uncertainty. (b) This variability induces distributions in homogenized elastic stiffness coefficients ($C_{11}^h, C_{12}^h$ in Voigt notation). (c) Downstream, such variability propagates to variability in macroscopic linear and nonlinear responses under prescribed loading. Bottom Row: VDMN framework. (d) The VDMN augments the Deep Material Network by replacing deterministic network parameters (interface normals $\theta_{i,j}$ and volume fraction fluctuations $\delta f_{N,j}$) with variational Gaussian distributions parameterized by means $\mu_{i,j}$ and variances $S_{i,j}$. Here $i$ denotes the layer index and $j$ the node index in an $N$-depth VDMN (a depth-2 example is shown). (e) A probabilistic homogenization algorithm recursively propagates these uncertainties through the laminate tree to yield distributions of homogenized responses, supporting both analytic (closed-form propagation of distributions) and sampling (deterministic DMNs are drawn from the hyper-variational distributions) modes. (f) Network parameters $\phi$ are trained by minimizing the negative log-likelihood of elastic homogenization data. Despite training on elastic responses only, the resulting model generalizes to quantify uncertainty in nonlinear regimes..
• Figure 2: Overview of Validation of Offline (Linear) Performance. (a)–(d) Parity plots comparing VDMN mean and covariance predictions for four homogenized stiffness components against FFT-based ground truth; parity lines lie within three standard deviations, with marginal distributions shown for clarity. (e)–(h) Simulation-based calibration tests (see Supplementary Note 2 in Supplementary Information) evaluating uncertainty prediction quality; black lines show normalized rank statistics relative to the VDMN Empirical Cumulative Distribution Function (ECDF), red lines indicate perfect calibration, and purple envelopes denote the 95% confidence interval. (i) Probabilistic relationship between input stiffness ratio $C^{1}_{11}/C^{2}_{11}$ and homogenized output $C^h_{11}$ for the training set (black) and three unseen test inputs, illustrating microstructure-induced variability. The three selected values were not contained in the training dataset. The full set of $30$ homogenized outputs -- computed on each microstructure in the ensemble -- is included for each of the three selected test inputs. (j)–(l) Comparison of the ground-truth output distributions (black histograms) from FFT-based homogenization over the microstructure ensemble with analytic VDMN predictions (solid lines) computed using Algorithm \ref{['alg:stochastic_homogenization']} and sampling-based VDMN results (colored histograms) obtained from 25 hyper-variational samples.
• Figure 3: Overview of Validation of Online (i.e., Non-Linear) Performance. (a) Comparison of sample-based nonlinear VDMN predictions (i.e., predictions made by repeatedly sampling DMNs from the VDMN and running standard nonlinear predictions) against direct numerical solutions for the mechanical response of the microstructure ensemble in two different strain rate controlled boundary conditions (pure $\dot{\varepsilon}_{xx}$ and $\dot{\varepsilon}_{xy}$). (b), (c) marginal distributions comparing the sample-based predicted stress against the direct numerical solver at an applied strain of $\varepsilon_{ij}=0.02$.
• Figure 4: Comparison of VDMN Predictions with Experimental Tensile Measurements on 3D Printed Two-Phase Composites. Four representative tensile bars (from twelve) fabricated via polymer additive manufacturing are shown, each derived from a distinct spinodal unit cell to emulate microstructural variability. For each specimen, the base unit cell and a segment of the printed bar are displayed. Insets show VDMN ensemble predictions (turquoise) of tensile stress-strain responses, compared with experimental measurements (black). The experimentally measured stress-strain responses of the matrix phase (blue) and particle phase (red) are also depicted. These measurements were performed prior to the VDMN predictions and used to calibrate the Norton elastoviscoplastic constitutive models inputted to the VDMN, see Supplementary Information. Marginal stress distributions at two strain levels highlight agreement between predicted and measured variability.
• Figure 5: Uncertainty-Aware Calibration of Constitutive Properties. (a) Three primary sources of uncertainty (experimental, mesoscale microstructural, and microscale constitutive uncertainty) arise during mechanical testing (indentation shown as an example). The underlying constitutive property distributions are often difficult to measure directly. (b) The VDMN disentangles these uncertainty sources and estimates unknown constitutive distributions by identifying parameters that maximize the likelihood of observed homogenized measurements under a VDMN-based total probability model (see Supplementary Note 6 in Supplementary Information). (c) Comparison between predicted constitutive distributions (solid lines) and ground truth (histograms) for two isotropic phases where only the Young's moduli $E$ vary (constant Poisson's ratio, $\nu=0.3$). (d) Effect of increasing experimental noise (variance) on the predicted mean Young's modulus $E$ for each phase. (e–g) Negative log-likelihood landscapes of homogenized measurements with respect to unknown constitutive parameters, computed using the VDMN total probability model. Phase 1 corresponds to the matrix phase (blue). These landscapes quantify parameter confidence and guide experimental design (see Supplementary Note 6 in Supplementary Information).