Thermal Analog Computing: Application to Matrix-vector Multiplication with Inverse-designed Metastructures

TL;DR

This work demonstrates MVM using inverse-designed metastructures that exploit heat conduction as the signal carrier, and applies this methodology--termed thermal analog computing--to realize matrices relevant to practical tasks, including the discrete Fourier transform and convolutional filters.

Abstract

The rising computational demand of modern workloads has renewed interest in energy-efficient paradigms such as neuromorphic and analog computing. A fundamental operation in these systems is matrix-vector multiplication (MVM), ubiquitous in signal processing and machine learning. Here, we demonstrate MVM using inverse-designed metastructures that exploit heat conduction as the signal carrier. The proposed approach is based on a generalization of effective thermal conductivity to systems with multiple input and output ports: The input signal is encoded as a set of applied temperatures, while the output is represented by the power collected at designated terminals. The metastructures are obtained via density-based topology optimization, enabled by a differentiable thermal transport solver and automatic differentiation, achieving an accuracy $> 99\%$ in most cases across pool of matrices with dimensions $2 \times 2$ and $3 \times 3$. We apply this methodology--termed thermal analog computing--to realize matrices relevant to practical tasks, including the discrete Fourier transform and convolutional filters. These results suggest new opportunities for analog information processing in environments where temperature gradients naturally arise, such as device hotspots and thermal controllers

Figures (9)

• Figure 1: (a) Input temperatures $\mathbf{T}_{\mathrm{in}}$ are applied through thermalized ports, and output powers $\mathbf{P}_{\mathrm{out}}$ are collected from the opposite side. For linear, temperature-independent conductivities, the steady-state relation $\mathbf{P}_{\mathrm{out}} = \mathbf{M}\mathbf{T}_{\mathrm{in}}$ defines a conductance matrix $\mathbf{M}$ that depends solely on the geometry and material distribution. (b) In simulations, each metastructure is represented as a binary 2D map $\rho(x,y)$, where $\rho=1$ denotes conductive material and $\rho=0$ denotes vacuum. The simulated 2D power response $\mathbf{P}_{\mathrm{2D}} = \mathbf{A}(\rho)\mathbf{T}_{\mathrm{in}}$ defines a geometry-dependent operator $\mathbf{A}(\rho)$, which is later scaled by the wafer width $z$ to obtain the 3D conductance matrix. (c) Because thermal conduction cannot produce negative flux, we represent arbitrary real-valued matrices as the weighted sum of physically realizable positive components. The proposed infrastructure in a realistic setting is that structures positive $z_i$ (blue) and negative $z_i$ (red) are operated separately; their respective outputs, $\mathbf{P}_+$ and $\mathbf{P}_-$, are then digitally subtracted to yield the effective analog operation $\mathbf{M}\mathbf{T}_{\mathrm{in}} = \mathbf{P}_+ - \mathbf{P}_-$.
• Figure 2: (a) Overview of the optimization workflow for a 3$\times$3 thermal metastructure performing matrix--vector multiplication (MVM). The device geometry is parameterized by the material density field $\rho$, where $\rho = 1$ and $\rho = 0$ correspond to solid and void regions, respectively. At each iteration, three steady-state heat conduction simulations are performed using orthogonal temperature inputs $\mathbf{T}_0$, $\mathbf{T}_1$, and $\mathbf{T}_2$ (blue arrows), representing the basis vectors of the input space. The corresponding output powers $\mathbf{P}_0$, $\mathbf{P}_1$, and $\mathbf{P}_2$ (red arrows) form the columns of the 2D conductance matrix $\mathbf{A}$, which is used to calculate $\mathbf{M}_{out}$. From the reconstructed matrix, the cost function is calculated. The optimizer updates the density field $\boldsymbol{\rho}$ using the computed objective function $g(\rho)$ and its gradient $\nabla_{\rho} g$, generating the next iteration $\boldsymbol{\rho}_{i+1}$. (b) Evolution of the density field during optimization, showing the emergence of the final topology. Intermediate structures gradually converge toward a well-defined heat-guiding geometry, where conductive paths (white) connect the thermal ports (orange).
• Figure 3: Optimization trajectories and final inverse-designed structures for four target matrices: a)Identity, b) Directional, c)Two-branch, and d) StraightPath. Each subplot shows the evolution of the cost function over optimization iterations. Vertical dashed lines indicate updates of the regularization parameter $\beta$. The right panels display the final material distributions $\rho(\mathbf{r})$ after convergence. For each design, the effective propagation length $z_0$ and final response matrix $\mathbf{A}(\rho_0)$ are reported, representing the physical length and two-dimensional transfer matrix derived from the optimized structure.
• Figure 4: Each panel shows the current normalized to maximum 1 in log scale. For each structure, we drive the two inports separatly. Red bars mark driven (hot) ports; gray bars mark thermalized ports. a)Identity—heat splits symmetrically and exits through collinear ports, approximating an identity map. b)Directional—anisotropy in $\mathbf{A}(\mathbf{x})$ suppresses one branch and funnels heat along a single corridor (near-selector). c)Two Branches—two comparable channels divide the flux between outputs (operator with sizable off–diagonal weights). d) Straight Path—a reinforced backbone routes heat along a near-straight channel (strongly diagonal map). Across all cases, the spatially varying conductivity tensor $\mathbf{A}(\mathbf{x})$ sculpts low-resistance corridors that realize the desired boundary map $\mathbf{M}_{\text{out}}$ while suppressing leakage.
• Figure 5: Each row shows the multi-structure representation of a given targed matrix. Beneath each structure, the correposnding effective thickness $Z_i$ is shown. Blue and orange structures are associated with positive and negative $z_i$, respectively. (a) Positive-Off Diagonal ($2\times2$), (b) Signed Off-Diagonal ($2\times2$), (c) Large Magnitude Contrast ($2\times2$), and (d) General Coupled ($3\times3$).