Data-driven discovery of thermal illusions through latent-space geometry

TL;DR

The paper addresses the non-uniqueness problem in thermal configurations that yield identical external responses, which complicates inverse design based on explicit transformations. It introduces a data-driven pipeline using a beta-VAE to compress the temperature field $T$ into a latent coordinate $z$, revealing a single dominant degree of freedom and non-unique mappings from $(kappa_r, kappa_theta, kappa_C)$ to $z$. A latent-space cloaking/illusion metric is defined as $S(kappa_r, kappa_theta) = \mathrm{mean}_{kappa_C} | Z(kappa_r, kappa_theta, kappa_C) - z_b |$ with $z_b = Z(kappa_b, kappa_b, kappa_b)$, enabling identification of robust shell designs; a representative cloak at $(kappa_r, kappa_theta) = (0.2, 4.8)$ reproduces the background across variations in $kappa_C$, and illusion is achieved by targeting $z_{target}$ for a chosen $kappa_{target}$. This framework provides a unifying, interpretable geometric approach to inverse design in thermal metamaterials and is extendable to other classical wave systems.

Abstract

Illusion effects-where one object appears as another-arise from the non-uniqueness of physical systems, in which different material configurations yield identical external responses. Conventional approaches, such as coordinate transformation, map equivalent configurations but provide only specific solutions, while analytical or numerical optimization methods extend these designs by minimizing scattering yet remain constrained by model assumptions and computational cost. Here, we exploit this non-uniqueness through a data-driven framework that uses a variational autoencoder to compress high-dimensional thermal-field data into a compact latent space capturing geometrical relations between configurations and observations. In this latent space, thermal illusion corresponds to finding configurations that minimize geometric distance to a target configuration, with thermal cloaking as a special case where the target is free space. Specifically, we demonstrate the concept in a cylindrical shell with anisotropic thermal conductivities enclosing a core of arbitrary conductivity, achieving robust thermal illusion and cloaking using only positive conductivities. Such a latent-space distance approach provides a refreshed perspective for achieving illusion and can be applied to inverse-design problems in other classical wave systems.

Figures (4)

• Figure 1: (a) Schematic of the thermal illusion framework. Heat flows from the left boundary at high temperature $T_{\mathrm{high}}$ to the right boundary at $T_{\mathrm{low}}$ through a cylindrical shell with anisotropic thermal conductivities $(\kappa_r, \kappa_\theta)$ enclosing a core of conductivity $\kappa_C$. The task is to find the material profile of the shell such that the overall system behaves like a core of target conductivity $\kappa_{\mathrm{target}}$ embedded in the background of conductivity $\kappa_b$, with thermal cloaking as the special case $\kappa_{\mathrm{target}}=\kappa_b$. Temperature fields are measured in the external region (gray area) outside the shell. (b) Data-driven framework for analyzing equivalence in thermal responses. A variational autoencoder (VAE) encodes the measured temperature data into a compact latent space and decodes it to reconstruct the temperature fields. The latent representation provides a geometric measure of similarity between configurations, enabling the identification of cloaking and illusion solutions.
• Figure 2: (a) Statistical evaluation of latent variables. The standard deviation (std) of $\mu_i$ and the mean of $\sigma_i$ across the training dataset indicate whether a latent variable is meaningful. A meaningful variable corresponds to a well-defined generative factor, exhibiting low mean of $\sigma_i$ and a large std of $\mu_i$, while a meaningless one shows high $\sigma_i$ and small $\mu_i$ variations. In this case, only the second latent variable satisfies these criteria, suggesting that the temperature data are governed by a single effective degree of freedom. (b) Relationships between the meaningful latent variable $z$ and the material parameters $\kappa_r$, $\kappa_\theta$, and $\kappa_C$. Different parameter combinations yield the same $z$, revealing non-unique configurations that produce identical external temperature fields. The gray dashed line at $z = 2.14$ corresponds to the homogeneous background response, indicating potential cloaking configurations associated with low $\kappa_r$ values.
• Figure 3: (a) The cloaking metric $S$ as a function of $\kappa_r$ and $\kappa_\theta$, where smaller values (bright regions) indicate configurations that best reproduce the background response. Green dashed line indicates the condition $\kappa_r \kappa_\theta = 1$. (b) Temperature contours of the selected configuration, marked by the blue triangle in (a) with $(\kappa_r, \kappa_\theta) = (0.2, 4.8)$, under three different core conductivities $\kappa_C = 0.1$, $5$, and $10$. The nearly vertical isothermal lines outside the shell demonstrate that the heat flux remains undisturbed, achieving the desired cloaking effect that makes the core appear as part of the background regardless of its thermal conductivity.
• Figure 4: (a) The illusion metric $S(\kappa_r,\kappa_\theta;\kappa_{\mathrm{target}}{=}9.88)$ shown as a colormap, where bright regions represent configurations that closely reproduce the response of a bare core with $\kappa_{\mathrm{target}}=9.88$ embedded in the background. (b) The temperature field of the bare core with $\kappa_{\mathrm{target}}=9.88$ in the background. (c) The selected configuration (marked by the blue triangle in (a)) with $(\kappa_r,\kappa_\theta)=(0.397,5.644)$ produces nearly identical temperature patterns to (b), even when the actual core conductivity varies across $\kappa_C=0.1$, $5$, and $10$, misleading external observers to perceive a bare core of $\kappa_{\mathrm{target}}=9.88$.