Data-efficient inverse design of spinodoid metamaterials

TL;DR

This work addresses the data-inefficient inverse design of spinodoid metamaterials by introducing a permutation-equivariant neural network surrogate that maps a four-parameter spinodoid description to the effective elasticity tensor. The surrogate enforces physical symmetries and positive semidefiniteness, and extends to rotations to account for orientation, enabling differentiable, gradient-based inverse design. Across three inverse-design tasks, the authors demonstrate that a remarkably small training set (as few as 75 data points) yields accurate structure–property mappings, with three progressively complex examples validating the approach. The framework leverages FFT-based homogenization for data generation and shows promise for extending to nonlinear and inelastic behavior, potentially enabling data-efficient experimental surrogate calibration. Overall, the method achieves reliable inverse design with substantially reduced data requirements and improved computational efficiency, broadening the practical applicability of metamaterial design.

Abstract

We create an data-efficient and accurate surrogate model for structure-property linkages of spinodoid metamaterials with only 75 data points -- far fewer than the several thousands used in prior works -- and demonstrate its use in multi-objective inverse design. The inverse problem of finding a material microstructure that leads to given bulk properties is of great interest in mechanics and materials science. These inverse design tasks often require a large dataset, which can become unaffordable when considering material behavior that requires more expensive simulations or experiments. We generate a data-efficient surrogate for the mapping between the characteristics of the local material structure and the effective elasticity tensor and use it to inversely design structures with multiple objectives simultaneously. The presented neural network-based surrogate model achieves its data efficiency by inherently satisfying certain requirements, such as equivariance with respect to permutations of structure parameters, which avoids having to learn them from data. The resulting surrogate of the forward model is differentiable, allowing its direct use in gradient-based optimization for the inverse design problem. We demonstrate in three inverse design tasks of varying complexity that this approach yields reliable results while requiring significantly less training data than previous approaches based on neural-network surrogates. This paves the way for inverse design involving nonlinear mechanical behavior, where data efficiency is currently the limiting factor.

Figures (11)

• Figure 1: Overview of the workflow used here: (a) A dataset $\mathcal{D}$ consisting of structure parameters $\mathcal{S}^\alpha$ and corresponding effective elasticity tensors $\bar{\mathbbm{C}}^\alpha$ is generated. Points in the four-dimensional parameter space $\mathbb{S}$ are sampled appropriately, the corresponding geometries $G^\alpha$ are created, and the effective elasticity tensor is determined computationally. Geometry generation and simulation are abbreviated as $f : \mathcal{S}\mapsto\bar{\mathbbm{C}}$. (b) The generated dataset is used to calibrate the proposed surrogate model $\tilde{f}$, which replaces $f$. (c) Inverse design is formulated as the minimization of an objective function $\ell$ with respect to the structure parameters $\mathcal{S}$ and additionally rotations $\mathbf{Q}$, which can be solved efficiently using the surrogate model.
• Figure 2: Generation of spinodoid structures using a 2d example: (a) The allowed sampling space of wave vectors is constrained by the half-angles $\theta_i$. (b) A maximum number of cosine waves is sampled within the allowed region. (c) These cosine waves are superimposed to form a Gaussian random field. (d) Another parameter $\rho$ determines how much of the resulting structure is occupied by Material 1. Based on this parameter, the Gaussian random field is thresholded at a specific function value, and (e) everything below this threshold is interpreted as Material 1 (blue), while everything above is interpreted as Material 2 (red).
• Figure 3: (a) A permutation equivariant layer, that maps three first order matrices $\mathbf{x}^i$, i.e., $r_{\mathbf{x}}=1$, to three matrices of (b) rank one, i.e., $r_{\mathbf{y}}=1$ or (c) rank two, i.e., $r_{\mathbf{y}}=2$. The weights and biases of the layer are summarized in $\mathcal{W}$. Each connection $(i,j)$ is independent of the others, but is restricted internally to enforce the equivariance condition. More precisely, the weight matrix $\mathbf{w}^{22}$ is constructed such that every orbit of input-output index pairs $\mathcal{I}\times\mathcal{J}$ receives an independent weight. In case of rank one output matrices (b), this results in only two independent weights, one for horizontal connections and one for the diagonal ones. For a rank two output matrix (c), there are five weights. The same idea applies to any ranks $r_{\mathbf{x}}$ and $r_{\mathbf{y}}$.
• Figure 4: Architecture of the surrogate model $\tilde{f}$: (a) The four structure parameters $\mathcal{S}$ are the input of $\tilde{f}$. (b) A neural network, that is permutation equivariant w.r.t. $\theta_i$ is employed. This network additionally enforces the othorhombic symmetry of the output $\mathbbm{t}^{\text{NN}}$, as well as minor and major symmetry. (c) To ensure, that the output is isotropic if $\rho=1$ or any $\theta_i=90°$, the ansisotropic part is filtered out if needed and $\mathbbm{t}$ is obtained. (d) Squaring $\mathbbm{t}$ ensures positive semidefiniteness and (e) the output is interpreted as elasticty tensor $\bar{\mathbbm{C}}$.
• Figure 5: Sampling strategy of the $\theta_i$ on the example of only $\theta_1$ and $\theta_2$: Purely random sampling in a quadratic and a triangular domain, Latin Hypercube (LH) sampling in the triangular domain and finally with a bias towards smaller values, as used for the training data sets.