# Metamaterials design w/ machine learning

## Where conventional design methods fail?

Structure-property relations of all existing (meta-)materials have primarily been explored in a forward fashion: given a microstructure, one extracts the effective properties by methods of homogenization. The inverse challenge – identifying a microstructural topology that meets the mechanical property requirements – has often been addressed by inefficient trial and error. Systematic approaches such as topology optimization and genetic algorithms are beneficial but also computationally expensive (relying on repeated sampling and computation of the effective properties), highly dependent on initial guesses, or may not be even compatible with complex design spaces (such as beam networks) or properties (e.g., curvature or buckling).

## What can Machine Learning (ML) do for inverse design?

Machine-learning-based surrogate models, which can bypass expensive simulations and experiments, have been developed for applications including metamaterials, composites, and nanomaterials. However, such approaches require solving an optimization problem based on the (forward-only) surrogate model, which prevents an on-demand inverse design framework. In addition, while the forward problem, i.e., mapping from topological parameters to property space, is well-defined, the inverse problem is ill-posed (multiple topologies can have the same or similar effective properties).

To this end, we introduced a novel data-driven approach based on the integration of two neural networks for the forward- and inverse-problems that renders the inverse design challenge well-posed. This approach, in contrast to surrogate optimization methods, provides a computationally inexpensive two-way relationship between design parameters and mechanical properties. We applied this framework to a diverse range of metamaterial classes for tunable mechanical response (anisotropic stiffness only being the tip of the iceberg) as shown below.

Schematics of the inverse design process (Kumar et al., 2020)

Spinodoid metamaterials

(Kumar et al., 2020)

Truss metamaterials

(Bastek et al., 2022)

Growth-based metamaterials

(Van 't Sant et al., 2023)

Shellular metamaterials

(Guo et al, 2023)

## When the design is non-parametric or uninterpretable?

Despite the advances, ML for metamaterials design is facing some critical bottlenecks namely, what if the design parameterization and property representation are

extremely high-dimensional,

uninterpretable (e.g., text- or graph-based) to a numerical optimization algorithm, and/or

discrete and discontinuous (e.g., ad-hoc truss- and plate-based lattices)?

For example, most truss lattices are based on well-known topologies such as the diamond, kagome, octahedron, or honeycomb lattices, which follow ad-hoc naming and hardly any rigorous design conventions. Specifically, each of those topologies can be described by a finite set of design parameters, which, however, do not generalize to other topologies. Consequently, most design optimization methods for truss lattices to date have focused on only a small pool of selected topologies, which leads to suboptimal solutions and fails to exploit the full range of available lattice structures. An alternative is the representation of a truss unit cell by voxels, which capture the presence (or lack) of material in a rasterized 3D space. Since the effective material properties are generally highly sensitive to the underlying structure, any such lattice parameterization based on voxelization requires high-resolution images and is hence computationally infeasible.

To address these challenges, we introduced a general ML framework for inverse design of metamaterials with high-dimensional and/or non-trivial parameterizations. The framework generally consists of two ML models – one to construct a low-dimensional, continuous latent representation unifying an enormous range of metamaterial designs; another one to map this unified latent space to properties of interest. Leveraging specially designed realizations of the general ML framework, we inverse designed metamaterials based on graph or textual representations. This is best represented by our recent work on designing graph-based truss lattices for metamaterials with customized mechanical properties in both the linear and nonlinear regimes, including designs exhibiting exceptionally stiff, auxetic, and tailored stress-strain behaviors.

Schematics of the ML framework to extract a latent design representation for graph-based truss lattices (Zheng et al., 2023)

## Publications

L. Zheng, K. Karapiperis, S. Kumar*, D.M. Kochmann*, Unifying the design space and optimizing linear and nonlinear truss metamaterials by generative modeling, Nature Communications, 14, 7563 (2023). [code] [data]

S. Van 't Sant*, P. Thakolkaran*, J. Martínez, S. Kumar, Inverse-designed growth-based cellular metamaterials, Mechanics of Materials, 182 (2023), 104668. [code] [data]

R.N. Glaesener, S. Kumar, C. Lestringant, T. Butruille, C.M. Portela, D.M. Kochmann, Predicting the influence of geometric imperfections on the mechanical response of 2D and 3D periodic trusses, Acta Materialia, 254 (2023), 118918.

J. H. Bastek, S. Kumar, B. Telgen, R. N. Glaesener, D. M. Kochmann, Inverting the structure–property map of truss metamaterials by deep learning, Proceedings of the National Academy of Sciences, 119 (1) e2111505119 (2022). [code] [data]

S. Kumar, D. M. Kochmann, What machine learning can do for computational solid mechanics, Current Trends and Open Problems in Computational Mechanics, Springer (2022). [Book chapter]

L. Zheng, S. Kumar, D. M. Kochmann, Data-driven topology optimization of spinodoid metamaterials with seamlessly tunable anisotropy, Computer Methods in Applied Mechanics and Engineering, 383 (2021), 113894. [code-topology] [code-ml] [data] (requires GIBBON: https://www.gibboncode.org/)

S. Kumar, S. Tan, L. Zheng, D. M. Kochmann, Inverse-designed spinodoid metamaterials, npj Computational Materials, 6 (2020), 73. [code-topology] [code-ml] [data] (requires GIBBON: https://www.gibboncode.org/)