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

Technical Summary: Thermal Analog Computing via Inverse-Designed Metastructures

Problem Statement
Modern classical computing relies on binary logic and digital switching, which incurs fundamental energy limits due to resistive losses and repeated memory access. While analog computing paradigms (e.g., memristors, photonics) offer energy-efficient alternatives for matrix–vector multiplication (MVM)—a core operation in machine learning and signal processing—heat remains an underexplored information carrier. Existing thermal metamaterials primarily focus on guiding heat flow (e.g., cloaking) rather than performing computation. Furthermore, traditional thermal logic gates operate in discrete "hot/cold" states, mirroring digital limitations. This work addresses the challenge of performing continuous MVMs using heat conduction as the signal carrier, specifically tackling the inverse problem: determining the geometric structure of a metastructure that yields a desired conductance matrix mapping input temperatures to output power.

Methodology
The authors propose a framework termed thermal analog computing, where input vectors are encoded as temperature fields applied to thermalized metal ports, and output vectors are represented by the collected thermal power at designated terminals. The system operates under the linear regime of Fourier's law, establishing a relationship $P_{out} = M T_{in}$, where $M$ is the conductance matrix dependent solely on geometry and material distribution.

Key methodological components include:

• Inverse Design via Topology Optimization: The authors employ density-based topology optimization to synthesize 2D metastructures. The material density $\rho(x,y)$ is treated as a continuous variable transitioning between solid (conductive) and void.
• Handling Negative Matrix Elements: Since thermal conductance cannot be negative, the framework decomposes any target matrix $M$ into a weighted sum of $K$ physically realizable positive components: $M = \sum z_i A(\rho_i)$. Positive and negative coefficients ($z_i$) are realized by separate physical structures, with the final result obtained via digital post-subtraction of their respective power outputs.
• Differentiable Solver Pipeline: The optimization utilizes an in-house, JAX-based differentiable finite-volume thermal solver. This enables an end-to-end differentiable pipeline where gradients of the cost function with respect to design parameters (densities and thicknesses) are computed via automatic differentiation.
• Optimization Constraints: The cost function minimizes the weighted $L_2$ norm between the target and reconstructed matrices. Additional constraints include a subpixel-smoothed projection (SSP) to ensure manufacturable binary structures and regularization to prevent unrealistic heat flux concentrations.

Key Results
The framework was validated across a pool of matrices with dimensions $2\times2$ and $3\times3$, achieving accuracy exceeding 99% in most cases.

• Single-Structure Designs: Simple matrices (Identity, Directional, Two-Branch) were realized with a single structure ($K=1$), demonstrating errors $<1\%$. However, matrices with uniform element magnitudes or dominant off-diagonal terms proved challenging for single structures.
• Multi-Structure Designs: Complex matrices, including those with mixed signs and large magnitude contrasts, were successfully realized using multiple structures ($K=2$ to $K=4$).
  • Hadamard Matrix: Realized with $K=2$ structures (error $\approx 1\%$).
  • Rotational Matrices: A $2\times2$ rotation ($\theta=\pi/3$) and a $3\times3$ Rotational Toeplitz matrix were achieved with $K=3$ structures (errors $\approx 3\%$).
  • Discrete Fourier Transform (DFT): The real and imaginary parts of a $3\times3$ DFT matrix were decomposed and realized with $K=4$ and $K=2$ structures respectively, yielding errors of 3.55% and 1.13%.
• Performance Metrics:
  • Bandwidth: Limited by thermal diffusion, estimated at $\sim 1.9$ kHz for a $100\,\mu\text{m}$ lateral dimension.
  • Energy Efficiency: Estimated at $\sim 9.3$ nJ/MAC for a $3\times3$ matrix. While higher than electronic or photonic accelerators, the system is fundamentally energy-passive, requiring no active power for the matrix operation itself.
  • Robustness: The system maintains linearity (within 1–5% error) for temperature differences up to 100 K. Signal-to-noise ratios (SNR) due to thermal fluctuations are estimated at $\sim 10^6$ at room temperature.

Significance and Claims
The paper claims to establish a proof-of-concept for thermal analog computing, demonstrating that heat flow can be engineered to perform high-fidelity linear transformations. The significance of this work lies in:

1. Paradigm Shift: Moving beyond using heat as a byproduct or a binary switch to utilizing it as a continuous analog signal carrier.
2. In-situ Computation: Enabling computation in thermally active environments (e.g., microelectronics) where heat is already present, potentially allowing for temperature-gradient sensing and thermal control without additional energy overhead for the computation itself.
3. Design Methodology: Providing a robust, differentiable inverse-design pipeline capable of synthesizing complex thermal metastructures for arbitrary matrix operations.

The authors modestly note that while these structures will not replace high-throughput electronic or photonic accelerators due to speed and energy density limitations, they open a complementary regime where computation and dissipation coincide. Future work is directed toward extending the methodology to nanoscales (solving the phonon Boltzmann transport equation) and scaling to higher-dimensional matrices.